Showing papers in "Mathematical Notes in 2018"

The Jordan property for Lie groups and automorphism groups of complex spaces

Abstract: We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifolds are Jordan.

Contrast Structures in Problems for a Stationary Equation of Reaction-Diffusion-Advection Type with Discontinuous Nonlinearity

Abstract: A singularly perturbed boundary-value problem for a nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems with discontinuous advective and reactive terms is considered. The existence of contrast structures in problems of this type is proved, and an asymptotic approximation of the solution with an internal transition layer of arbitrary order of accuracy is obtained.

Upper Bounds for the Chromatic Numbers of Euclidean Spaces with Forbidden Ramsey Sets

Abstract: The chromatic number of a Euclidean space ℝ n with a forbidden finite set C of points is the least number of colors required to color the points of this space so that no monochromatic set is congruent to C. New upper bounds for this quantity are found.

On a Series Representation for Integral Kernels of Transmutation Operators for Perturbed Bessel Equations

Abstract: A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.

Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

Abstract: Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.

Lagrangian manifolds related to the asymptotics of Hermite polynomials

Abstract: We discuss two approaches that can be used to obtain the asymptotics of Hermite polynomials. The first, well-known approach is based on the representation of Hermite polynomials as solutions of a spectral problem for the harmonic oscillator Schrodinger equation. The second approach is based on a reduction of the finite-difference equation for the Hermite polynomials to a pseudodifferential equation. Associated with each of the approaches are Lagrangian manifolds that give the asymptotics of Hermite polynomials via the Maslov canonical operator.

Liouville-Type Theorem for a Nonlinear Degenerate Parabolic System of Inequalities

Abstract: In this paper, we establish some new Liouville-type results for solutions of nonlinear degenerate parabolic system of inequalities. Nonexistence of nontrivial global solutions to initial-value problems is studied by using scaling transformations and test functions.

On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients

Abstract: Asymptotic formulas as x→∞ are obtained for a fundamental system of solutions to equations of the form $$l\left( y \right): = {\left( { - 1} \right)^n}{\left( {p\left( x \right){y^{\left( n \right)}}} \right)^{\left( n \right)}} + q\left( x \right)y = \lambda y,x \in [1,\infty )$$ , where p is a locally integrable function representable as $$p\left( x \right) = {\left( {1 + r\left( x \right)} \right)^{ - 1}},r \in {L^1}\left( {1,\infty } \right)$$ , and q is a distribution such that q = σ(k) for a fixed integer k, 0 ≤ k ≤ n, and a function σ satisfying the conditions $$\sigma \in {L^1}\left( {1,\infty } \right)ifk < n,$$ $$\left| \sigma \right|\left( {1 + \left| r \right|} \right)\left( {1 + \left| \sigma \right|} \right) \in {L^1}\left( {1,\infty } \right)ifk = n$$ . Similar results are obtained for functions representable as $$p\left( x \right) = {x^{2n + v}}{\left( {1 + r\left( x \right)} \right)^{ - 1}},q = {\sigma ^{\left( k \right)}},\sigma \left( x \right) = {x^{k + v}}\left( {\beta + s\left( x \right)} \right)$$ , for fixed k, 0 ≤ k ≤ n, where the functions r and s satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression l(y) (for real functions p and q) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case n = 1.

Finding Solution Subspaces of the Laplace and Heat Equations Isometric to Spaces of Real Functions, and Some of Their Applications

Abstract: We single out subspaces of harmonic functions in the upper half-plane coinciding with spaces of convolutions with the Abel–Poisson kernel and subspaces of solutions of the heat equation coinciding with spaces of convolutions with the Gauss–Weierstrass kernel that are isometric to the corresponding spaces of real functions defined on the set of real numbers. It is shown that, due to isometry, the main approximation characteristics of functions and function classes in these subspaces are equal to the corresponding approximation characteristics of functions and function classes of one variable.

On the Number of Independent Sets in Simple Hypergraphs

Abstract: Extremal problems on the number of j-independent sets in uniform simple hypergraphs are studied. Nearly optimal results on the maximum number of independent sets for the class of simple regular hypergraphs and on the minimum number of independent sets for the class of simple hypergraphs with given average degree of vertices are obtained.

On the Absolute Matrix Summability Factors of Fourier Series

Abstract: In this paper, two known theorems on |N, p n | k summability methods of Fourier series have been generalized for |A, p n | k summability factors of Fourier series by using different matrix transformations. New results have been obtained dealing with some other summability methods.

Existence of Infinitely Many Solutions for Δγ-Laplace Problems

Abstract: In this article, we study the existence of infinitelymany solutions for the boundary–value problem $$ - {\Delta _\gamma }u + a\left( x \right)u = f\left( {x,u} \right)in\Omega ,u = 0on\partial \Omega $$ , where Ω is a bounded domain with smooth boundary in ℝ N (N ≥ 2) and Δγ is a subelliptic operator of the form $${\Delta _\gamma }: = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}} \right)} ,{\partial _{{x_j}}}: = \frac{\partial }{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \cdots ,\gamma N} \right)$$ . Our main tools are the local linking and the symmetric mountain pass theorem in critical point theory.

On the Recovery of an Integer Vector from Linear Measurements

Abstract: Let 1 ≤ 2l ≤ m < d. A vector x ∈ ℤd is said to be l-sparse if it has at most l nonzero coordinates. Let an m × d matrix A be given. The problem of the recovery of an l-sparse vector x ∈ Zd from the vector y = Ax ∈ Rm is considered. In the case m = 2l, we obtain necessary conditions and sufficient conditions on the numbers m, d, and k ensuring the existence of an integer matrix A all of whose elements do not exceed k in absolute value which makes it possible to reconstruct l-sparse vectors in ℤd. For a fixed m, these conditions on d differ only by a logarithmic factor depending on k.

Distance-Regular Shilla Graphs with b2 = c2

Abstract: A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ1 equal to a3. For a Shilla graph, let us put a = a3 and b = k/a. It is proved in this paper that a Shilla graph with b2 = c2 and noninteger eigenvalues has the following intersection array: $$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$ If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array $$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$ and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array $$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$ The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.

Diameter of the Berger Sphere

Abstract: We consider the Lie group $SU_2$ endowed with a left-invariant axisymmetric Riemannian metric. This means that a metric has eigenvalues $I_1 = I_2, I_3 > 0$. We give an explicit formula for the diameter of such metric. Other words, we compute the diameter of Berger's sphere.

Bilinear Hardy–Steklov Operators

Abstract: The weighted inequality in Lebesgue norms with bilinear Hardy–Steklov operators is characterized.

On an Example of the Nikishin System

Abstract: An example of a Markov function f = const + $$\hat \sigma $$ such that the three functions f, f2, and f3 constitute a Nikishin systemis given. It is conjectured that there exists aMarkov function f such that, for each n ∈ N, the system of f, f2,..., fn is a Nikishin system.

On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections

Abstract: It is well known that Givental’s toric Landau–Ginzburg models for Fano complete intersections admit Calabi–Yau compactifications. We give an alternative proof of this fact. As a consequence of this proof, we obtain a description of the fibers over infinity of the compactified toric Landau–Ginzburg models.

Stability Analysis of Distributed-Order Hilfer–Prabhakar Systems Based on Inertia Theory

Abstract: The notion of a distributed-order Hilfer–Prabhakar derivative is introduced, which reduces in special cases to the existing notions of fractional or distributed-order derivatives. The stability of two classes of distributed-order Hilfer–Prabhakar differential equations, which are generalizations of all distributed or fractional differential equations considered previously, is analyzed. Sufficient conditions for the asymptotic stability of these systems are obtained by using properties of generalized Mittag-Leffler functions, the final-value theorem, and the Laplace transform. Stability conditions for such systems are introduced by using a new definition of the inertia of a matrix with respect to the distributed-order Hilfer–Prabhakar derivative.

On Singular Perturbations of Quantum Dynamical Semigroups

Abstract: We consider two examples of quantum dynamical semigroups obtained by singular perturbations of a standard generator which are special case of unbounded completely positive perturbations studied in detail in [11]. In Sec. 2, we propose a generalization of an example in [15] aimed to give a positive answer to a conjecture of Arveson. In Sec. 3 we consider in greater detail an improved and simplified construction of a nonstandard dynamical semigroup outlined in our short communication [12].

Exponential Stability of a Certain Semigroup and Applications

Abstract: The uniform exponential stability of a C0-semigroup with generator of a special form is proved. Such semigroups arise in the study of various problems of the theory of viscoelasticity. The proved statement is applied to the study of the asymptotic behavior of solutions in the problem of small motions of a viscoelastic body subject to driving forces of a special form.

On Regularity of Positive Quadratic Doubly Stochastic Operators

Abstract: Author Information Reprint Address: Saburov, M (reprint author) Int Islamic Univ Malaysia, Kuantan, Malaysia. Organization-Enhanced Name(s) International Islamic University Malaysia Reprint Address: Saburov, M (reprint author) Amer Univ Middle East, Coll Engn & Technol, Egaila, Kuwait. Addresses: [ 1 ] Int Islamic Univ Malaysia, Kuantan, Malaysia Organization-Enhanced Name(s) International Islamic University Malaysia [ 2 ] Amer Univ Middle East, Coll Engn & Technol, Egaila, Kuwait E-mail Addresses: msaburov@gmail.com

On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

Abstract: It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ⊂ Rn is complete in L2(D), where D is a bounded domain in Rn, n ≥ 3, such that $$\bar D$$ ∩ L = ∅. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R3.

On a Kantorovich Problem with a Density Constraint

Abstract: The Kantorovich optimal transport problemwith a density constraint onmeasures on an infinite-dimensional space is considered. In this setting, the admissible transport plan is nonnegative and majorized by a given constraint function. The existence and the uniqueness of a solution of this problem are proved.

New Criteria for the Existence of a Continuous ε -Selection

Abstract: We study sets admitting a continuous selection of near-best approximations and characterize those sets in Banach spaces for which there exists a continuous e-selection for each e > 0. The characterization is given in terms of P-cell-likeness and similar properties. In particular, we show that a closed uniqueness set in a uniformly convex space admits a continuous e-selection for each e > 0 if and only if it is B-approximately trivial. We also obtain a fixed point theorem.

Ultimate Boundedness in the Sense of Poisson of Solutions to Systems of Differential Equations and Lyapunov Functions

Abstract: The notions of different types of boundedness in the sense of Poisson of solutions to systems of differential equations are introduced. Sufficient conditions are obtained for different types of boundedness of solutions in the sense of Poisson, which are introduced in the paper.