Volume 31, Issue 5

The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 \mathbb{T}^{1} . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.

Let R be a prime ring, let 0 ≠ b ∈ R {0\neq b\in R} , and let α and β be two automorphisms of R . Suppose that F : R → R {F:R\rightarrow R} , F 1 : R → R {F_{1}:R\rightarrow R} are two b -generalized ( α , β ) {(\alpha,\beta)} -derivations of R associated with the same ( α , β ) {(\alpha,\beta)} -derivation d : R → R d:R\rightarrow R , and let G : R → R G:R\rightarrow R be a b -generalized ( α , β ) (\alpha,\beta) -derivation of R associated with ( α , β ) (\alpha,\beta) -derivation g : R → R g:R\rightarrow R . The main objective of this paper is to investigate the following algebraic identities: (1) F ⁢ ( x ⁢ y ) + α ⁢ ( x ⁢ y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+\alpha(xy)+\alpha(yx)=0} , (2) F ⁢ ( x ⁢ y ) + G ⁢ ( x ) ⁢ α ⁢ ( y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+G(x)\alpha(y)+\alpha(yx)=0} , (3) F ⁢ ( x ⁢ y ) + G ⁢ ( y ⁢ x ) + α ⁢ ( x ⁢ y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+G(yx)+\alpha(xy)+\alpha(yx)=0} , (4) F ⁢ ( x ) ⁢ F ⁢ ( y ) + G ⁢ ( x ) ⁢ α ⁢ ( y ) + α ⁢ ( y ⁢ x ) = 0 {F(x)F(y)+G(x)\alpha(y)+\alpha(yx)=0} , (5) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( x ⁢ y ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(xy)=0} , (6) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) = 0 {F(xy)+d(x)F_{1}(y)=0} , (7) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(yx)=0} , (8) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( x ⁢ y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(xy)+\alpha(yx)=0} , (9) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( y ⁢ x ) - α ⁢ ( x ⁢ y ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(yx)-\alpha(xy)=0} , (10) [ F ⁢ ( x ) , x ] α , β = 0 {[F(x),x]_{\alpha,\beta}=0} , (11) ( F ⁢ ( x ) ∘ x ) α , β = 0 {(F(x)\circ x)_{\alpha,\beta}=0} , (12) F ⁢ ( [ x , y ] ) = [ x , y ] α , β {F([x,y])=[x,y]_{\alpha,\beta}} , (13) F ⁢ ( x ∘ y ) = ( x ∘ y ) α , β {F(x\circ y)=(x\circ y)_{\alpha,\beta}} for all x , y {x,y} in some suitable subset of R .

It is shown how to approximate the solution of functional differential equations in terms of Bell’s polynomials. Some numerical checks are shown, by using the computer algebra system Mathematica © {{}^{\copyright}} .

A version of the Floquet theory for first order delay differential equations is proposed. Formula of solutions representation is obtained. On this basis, the stability of first order delay differential equations is studied. An analogue of the classical integral Lyapunov–Zhukovskii test of stability is proved. New, in comparison with all known, tests of the exponential stability are obtained on the basis of the Floquet theory. A possibility to achieve the exponential stability is connected with oscillation of solutions.

In this paper, we prove that, for β ∈ ℂ {\beta\in{\mathbb{C}}} , every α ∈ ℂ {\alpha\in{\mathbb{C}}} has at most finitely many (possibly none at all) representations of the form α = d n ⁢ β n + d n - 1 ⁢ β n - 1 + … + d 0 {\alpha=d_{n}\beta^{n}+d_{n-1}\beta^{n-1}+\dots+d_{0}} with nonnegative integers n , d n , d n - 1 , … , d 0 {n,d_{n},d_{n-1},\dots,d_{0}} if and only if β is a transcendental number or an algebraic number which has a conjugate over ℚ {{\mathbb{Q}}} (possibly β itself) in the real interval ( 1 , ∞ ) {(1,\infty)} . The nontrivial part here is to show that for every algebraic number β lying with its all conjugates in ℂ ∖ ( 1 , ∞ ) {{\mathbb{C}}\setminus(1,\infty)} , there is α ∈ ℚ ⁢ ( β ) {\alpha\in{\mathbb{Q}}(\beta)} with infinitely many such representations. In a particular case, when β is a quadratic algebraic number, this was recently established by Kala and Zindulka.

In this paper, we deal with the notion of V a {V_{a}} -deformed free convolution, introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power of V a {V_{a}} -deformed free convolution. Then we provide an approximation of elements of the CSK family generated by V a {V_{a}} -deformed free Poisson distribution.

In this paper, we explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space X . The results are formulated in terms of some topological conditions made on X or on its dual X * {X^{*}} . In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided.

In the present paper, we examine the perturbation of continuous frames and Riesz-type frames in Hilbert C * {C^{*}} -modules. We extend the Casazza–Christensen general perturbation theorem for Hilbert space frames to continuous frames in Hilbert C * {C^{*}} -modules. We obtain a necessary condition under which the perturbation of a Riesz-type frame of Hilbert C * {C^{*}} -modules remains to be a Riesz-type frame. Also, we examine the effect of duality on the perturbation of continuous frames in Hilbert C * {C^{*}} -modules, and we prove that if the operator frame of a continuous frame F is near to the combination of the synthesis operator of a continuous Bessel mapping G and the analysis operator of F , then G is a continuous frame.

The notion of (real-valued) almost measurable functions on probability spaces is introduced and some of their properties are considered. It is shown that any almost measurable function may be treated as a quasi-random variable in the sense of [A. Kharazishvili, On some version of random variables, Trans. A. Razmadze Math. Inst. 177 2023, 1, 143–146].

This paper introduces and studies the ϕ-u- S -flat (resp., nonnil-u- S -injective) modules, which are a generalization of both ϕ-flat modules and u- S -flat modules (resp., both nonnil-injective modules and u- S -injective modules). We give the Cartan–Eilenberg–Bass theorem for nonnil-u- S -Noetherian rings. Finally, we offer some new characterizations of the ϕ-von Neumann regular ring.

The Busemann–Petty problem of arbitrary measure for symmetric star bodies is proposed and studied by Zvavitch, which is a generalization of the classical Busemann–Petty problem. In this paper, we study the Busemann–Petty-type problem for homogeneous measure for general star bodies.

The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed.

In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors ℭ {\mathfrak{C}} , and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form ∥ ℭ ⁢ x k ∥ t , 1 ≤ U ⁢ ∥ x ∥ l p k \|\mathfrak{C}x^{k}\|_{t,1}\leq U\|x\|_{l_{p}}^{k} ( k = 1 , 2 k=1,2 ), where x is a sequence, ℭ ⁢ x k {\mathfrak{C}x^{k}} is a tensor, and ∥ ⋅ ∥ t , 1 {\|\cdot\|_{t,1}} , ∥ ⋅ ∥ l p {\|\cdot\|_{l_{p}}} are the tensor and sequence norms, respectively. The constant U is independent of x , and we seek the smallest possible value of U .

We study the local maximal oscillatory integral operator T α , β ∗ ⁢ ( f ) ⁢ ( x ) = sup 0 < t < 1 ⁡ | ∫ ℝ n e i ⁢ | t ⁢ ξ | α | t ⁢ ξ | β ⁢ Ψ ⁢ ( | t ⁢ ξ | ) ⁢ f ^ ⁢ ( ξ ) ⁢ e 2 ⁢ π ⁢ i ⁢ 〈 x , ξ 〉 ⁢ 𝑑 ξ | , \displaystyle T_{\alpha,\beta}^{\ast}(f)(x)=\sup_{0<t<1}\Bigg{|}\int_{\mathbb{% R}^{n}}\frac{e^{i|t\xi|^{\alpha}}}{|t\xi|^{\beta}}\Psi(|t\xi|)\widehat{f}(\xi)% e^{2\pi i\langle x,\xi\rangle}\,d\xi\Bigg{|}, where α ∈ ( 0 , 1 ) {\alpha\in(0,1)} , β > 0 {\beta>0} , and Ψ is a cutoff function that vanishes in a neighborhood of the origin. First, in the case 0 < p < 1 {0<p<1} , we obtain the H p ⁢ ( ℝ n ) → L p ⁢ ( ℝ n ) {{{H^{p}}({{\mathbb{R}^{n}}})}\rightarrow{{L^{p}({{\mathbb{R}^{n}}})}}} boundedness of T α , β ∗ {T_{\alpha,\beta}^{\ast}} with the sharp relation among α , β {\alpha,\beta} and p . Then, using interpolation, we obtain the L p ⁢ ( ℝ n ) {{{L^{p}({{\mathbb{R}^{n}}})}}} boundedness on T α , β ∗ {T_{\alpha,\beta}^{\ast}} when p > 1 {p>1} , which is an improvement of the recent result by Kenig and Staubach. At the critical case p = 1 {p=1} and β = n ⁢ α 2 {\beta=\frac{n\alpha}{2}} , we show T α , β ∗ : B q ⁢ ( ℝ n ) → L 1 , ∞ ⁢ ( ℝ n ) {T_{\alpha,\beta}^{\ast}:B_{q}({\mathbb{R}^{n}})\rightarrow L^{1,\infty}({% \mathbb{R}^{n}})} , where B q ⁢ ( ℝ n ) {B_{q}({\mathbb{R}^{n}})} is the block space introduced by Lu, Taibleson and Weiss in order to study the almost every convergence of the Bochner–Riesz means at the critical index. As a further application, we obtain the convergence speed of a combination to the fractional Schrödinger operators { e i ⁢ t ⁢ k ⁢ | △ | α } {\{e^{itk|\triangle|^{\alpha}}\}} .

Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.

This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.