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Abstract: We give a short proof of a weighted version of the discrete Hardy inequality. This includes the known case of classical monomial weights with optimal constant. The proof is based on the ideas of the short direct proof given recently in [P. Lefèvre, Arch. Math. (Basel), 114, No. 2, 195–198 (2020)]. PubDate: 2023-11-28

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Abstract: We study the representation of real numbers by Perron series (P-representation) given by $$\left(\left.0;1\right]\ni x=\sum_{n=0}^{\infty }\frac{{r}_{0}{r}_{1}\dots {r}_{n}}{\left({p}_{1}-1\right){p}_{1}\dots \left({p}_{n}-1\right){p}_{n}{p}_{n+1}}={\Delta }_{{p}_{1}{p}_{2}\dots }^{P}\right.,$$ where rn, pn ∈ ℕ, pn+1 ≥ rn + 1, and its transcoding ( \(\overline{P }\) -representation) $${x=\Delta }_{{g}_{1}{g}_{2}\dots }^{\overline{P} },$$ where gn = pn − rn−1. We establish the properties of \(\overline{P }\) -representations typical of almost all numbers with respect to the Lebesgue measure (normal properties of the representations of numbers). We also examine the conditions of existence of the frequency of a digit i in the \(\overline{P }\) -representation of a number \({x=\Delta }_{{g}_{1}{g}_{2}\dots {g}_{2}\dots }^{\overline{P} }\) defined by the equality $${\nu }_{i}^{\overline{P} }\left(x\right)=\underset{k\to \infty }{\mathrm{lim}}\frac{{N}_{i}^{\overline{P} }\left(x,k\right)}{k},$$ where \({N}_{i}^{\overline{P} }\left(x,k\right)\) denotes the amount of numbers n such that gn = i and n ≤ k. In particular, we establish conditions under which the frequency \({\nu }_{i}^{\overline{P} }\left(x\right)\) exists and is constant for almost all x ∈ (0; 1]. In addition, we also determine the conditions under which the digits in \(\overline{P }\) -representations are encountered finitely or infinitely many times for almost all numbers from (0; 1]. PubDate: 2023-11-28

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Abstract: By using the q-Jackson integral and some elements of the q-harmonic analysis associated with the q-Hankel transform, we introduce and study a q-analog of the Hankel–Stockwell transform. We present some properties from harmonic analysis (Plancherel formula, inversion formula, reproducing kernel, etc.). Furthermore, we establish a version of Heisenberg’s uncertainty principles. Finally, we study the q-Hankel–Stockwell transform on a subset of finite measure. PubDate: 2023-11-28

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Abstract: Investigation of the theory of complex functions is one of the most fascinating aspects of the theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Numerous mathematical concepts are explained when viewed through the theory of complex functions. Let \(f\left(z\right)\in A, f\left(z\right)=z+{\sum }_{n\ge 2}^{\infty }{a}_{n}{z}^{n},\) be an analytic function in an open unit disc U = {z : z < 1, z ∈ ℂ} normalized by f(0) = 0 and f′(0) = 1. For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where r is a positive integer of order \({2}^{-r}\left(0<{2}^{-r}\le \frac{1}{2}\right).\) By using subordination, we propose a criterion for f(z) ∈ S*[ar, br]. The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity are examined for f(z) ∈ A. PubDate: 2023-11-28

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Abstract: We introduce a new Diophantine inequality with prime numbers. Let \(1<c<\frac{10}{9}.\) We show that, for any fixed θ > 1, every sufficiently large positive number N, and a small constant ε > 0, the tangent inequality $$\left {p}_{1}^{c} {\mathrm{tan}}^{\theta }\left(\mathrm{log}{p}_{1}\right)+{p}_{2}^{c} {\mathrm{tan}}^{\theta }\left(\mathrm{log}{p}_{2}\right)+{p}_{3}^{c} {\mathrm{tan}}^{\theta }\left(\mathrm{log}{p}_{3}\right)-N\right <\varepsilon $$ has a solution in prime numbers p1, p2, and p3. PubDate: 2023-11-28

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Abstract: We examine the existence of fixed points of generalized α-nonexpansive mappings on CATp(0) spaces. We establish various convergence results for a newly defined algorithm associated with α-nonexpansive mappings and present some illustrative examples to show the efficiency of the proposed algorithm and to support the above-mentioned results. We also define monotone generalized α-nonexpansive mappings and prove some existence and convergence results for these mappings. PubDate: 2023-11-25

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Abstract: We prove and discuss some new weak-type (1,1) inequalities for the maximal operators of T means with respect to the Vilenkin system generated by monotonic coefficients. We also apply the accumulated results to prove that these T means are almost everywhere convergent. As applications, we present both some well-known and new results. PubDate: 2023-11-25

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Abstract: We consider a Schrödinger equation \(i{\partial }_{t}^{\rho }u\left(x,t\right)-{u}_{xx}\left(x,t\right)=p\left(t\right)q\left(x\right)+f\left(x,t\right),0<t\le T,0<\rho <1,\) with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with u(x, t), a time-dependent factor p(t) of the source function is also unknown. To solve this inverse problem, we use an additional condition B[u(∙, t)] =ψ(t) with an arbitrary bounded linear functional B. The existence and uniqueness theorem for the solution to the problem under consideration is proved. The stability inequalities are obtained. The applied method makes it possible to study a similar problem by taking, instead of d2/dx2, an arbitrary elliptic differential operator A(x,D) with compact inverse. PubDate: 2023-11-25

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Abstract: We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities for 2π-periodic functions f that can be represented in the form of generalized Poisson integrals of functions φ from the space Lp, 1 ≤ p ≤ ∞. In these inequalities, the moduli of deviations of the interpolation Lagrange polynomials \(\left f\left(x\right)-{\widetilde{S}}_{n-1}\left(f;x\right)\right \) for every x ∈ ℝ are expressed via the best approximations \({E}_{n}{\left(\varphi \right)}_{{L}_{p}}\) of the functions φ by trigonometric polynomials in the Lp-metrics. We also deduce asymptotic equalities for the exact upper bounds of pointwise approximations of the generalized Poisson integrals of functions that belong to the unit balls in the spaces Lp, 1 ≤ p ≤ ∞, by interpolating trigonometric polynomials on the classes \({C}_{\beta ,p}^{\alpha ,r}\) . PubDate: 2023-11-25

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Abstract: Suppose that p(z) = 1 + zϕ″(z)/ϕ′(z), where ϕ(z) is a locally univalent analytic function in the unit disk D with ϕ(0) = ϕ′(1) − 1 = 0. We establish the lower and upper bounds for the best constants σ0 and σ1 such that \({e}^{{-\sigma }_{0}/2}<\left p\left(z\right)\right <{e}^{{\sigma }_{0}/2}\) and p(w)/p(z) < \({e}^{{\sigma }_{1}}\) for z, w ∈ D, respectively, imply the univalence of ϕ(z) in D. PubDate: 2023-11-25

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Abstract: We study a hybrid mean-value problem related to the generalized Dedekind sum, certain generalized Hardy sums, and Kloosterman sum and obtain several meaningful conclusions with the help of the analytic method and the properties of the sum of characters and the Gauss sum. PubDate: 2023-11-11

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Abstract: By using the definition of modified (h, m, s)-convex functions of the second type, we present various refinements of the classical Hermite–Hadamard inequality obtained within the framework of weighted integrals. Throughout the paper, we show that various known results available from the literature can be obtained as particular cases of our results. PubDate: 2023-11-11

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Abstract: We study an evolutionary equation with an operator (i∂/∂x), where φ is a smooth function satisfying certain conditions. As special cases of this equation, we get a partial differential equation of parabolic type with derivatives of finite and infinite orders and an equation with certain operators of fractional differentiation. It is shown that the restriction of the operator (i∂/∂x) to some spaces of type S coincides with a pseudodifferential operator constructed according to the function φ regarded as a symbol. We establish the correct solvability of a nonlocal multipoint (in time) problem for an equation of this kind with an initial function, which is an element of the space of generalized functions of ultradistribution type. The properties of the fundamental solution of this problem are analyzed. PubDate: 2023-11-11

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Abstract: We consider an impulsive Dirac system on Sturmian time scales and present an existence theorem for this system. Maximal, minimal, and self-adjoint operators generated by the impulsive dynamic Dirac system are constructed. We also construct the Green function for this problem. Finally, an eigenfunction expansion is obtained. PubDate: 2023-11-11

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Abstract: We study the existence of a solution to a general type of complex Hessian equation on some Cegrell classes. For a given measure μ defined on an m-hyperconvex domain Ω ⊂ ℂn, under suitable conditions, we prove that the equation χ(.)Hm(.) = μ has a solution that belongs to the class ℰm,χ(Ω). PubDate: 2023-11-10

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Abstract: Since the general definition of topology is based on the characteristics of the standard Euclidean topology, the relationships between the ordering on real numbers and its topology have been generalized over time and studied in numerous aspects. The compatibility of partially ordered sets with the topology on these sets was studied by many researchers. On the other hand, well-orderedness is an important concept of the set theory. We define the concept of topological well-orderedness, which can be regarded as a topological generalization of well-orderedness in the set theory and analyze its basic properties. In this way, the relationship between well-orderedness and topology is established from a different point of view. Finally, some basic applications of the concept of topological well-orderedness to the graph theory are investigated. PubDate: 2023-11-10

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Abstract: We derive some results concerning the quaternionic Davis–Wielandt shell for a bounded right linear operator in a right quaternionic Hilbert space. The relations between the geometric properties of the quaternionic Davis–Wielandt shells and the algebraic properties of quaternionic operators are obtained. PubDate: 2023-11-09

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Abstract: We determine periodic and weakly periodic ground states with subgroups of index three for the Ising model on the Cayley tree of order three. PubDate: 2023-11-09

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Abstract: We prove the generalized stability of the functional equations $$\Vert f\left(x+y\right)\Vert =\Vert f\left(x\right)+f\left(y\right)\Vert \mathrm{ and }\Vert f\left(x-y\right)\Vert =\Vert f\left(x\right)-f\left(y\right)\Vert $$ in p-uniformly convex spaces with p ≥ 1. PubDate: 2023-11-08

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Abstract: Let R be a commutative ring with identity, which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R − {0} such that Ir = (0). The total graph of nonzero annihilating ideals of R, denoted by Ω(R), is a graph with the vertex set A(R)* and the set of all nonzero annihilating ideals of R in which two distinct vertices I and J are joined if and only if I + J is also an annihilating ideal of R. We study the forcing metric dimension of Ω(R) and determine the forcing metric dimension of Ω(R). It is shown that the forcing metric dimension of Ω(R) is equal either to zero or to the metric dimension. PubDate: 2023-11-08