At present, the approximation properties of harmonic Poisson integrals for the upper half-plane, which satisfy the Laplace equation in Cartesian coordinates, have not been sufficiently studied. This paper studies the approximation properties of conjugate harmonic Poisson integrals for the upper half-plane on classes of Hölder functions. An exact equality is found for the upper bound of the deviation of the conjugate harmonic Poisson integral from its boundary value in a uniform metric.
Let $\mathscr{F}$ be an $\omega$-closed family of the subsets of the set of all non-negative integers $\omega$ and $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ be the bicyclic extension generated by $\mathscr{F}$, which are defined in [Visnyk Lviv Univ. Ser. Mech.-Mat. 2020, 90, 5-19]. In [Visnyk Lviv Univ. Ser. Mech.-Mat. 2023, 95, 28-45] we describe injective monoid endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ with the family $\mathscr{F}^3=\{[0),[1),[2)\}$ and show that for every injective monoid endomorphism $\varepsilon$ of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ there exists a positive integer $k$ such that $\varepsilon=\alpha_{[k]}$ and the mapping $\alpha_{[k]}\colon \boldsymbol{B}_{\omega}^{\mathscr{F}^3}\to\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ is defined by the formula \[(i,j,[p))\alpha_{[k]}=\begin{cases}(ki,kj,[p)), & \text{if}\; p\in\{0,1\},\\(k(i+1)-1,k(j+1)-1,[2)), & \text{if}\; p=2,\end{cases}\] for all $i,j\in\omega$. In this paper, we describe injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$. In particular, we construct injective endomorphisms $\varpi_3$ and $\lambda$ of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ and describe the subsemigroup $\left\langle\lambda,\varpi_3\right\rangle$ of the semigroup $\boldsymbol{End}_{\textsf{inj}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^3})$ of all injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$, generated by $\varpi_3$ and $\lambda$. We show that for every injective endomorphism $\varepsilon$ of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ there exists an injective endomorphism $\iota\in\left\langle\lambda,\varpi_3\right\rangle$ such that $\varepsilon=\alpha_{[k]}\circ\iota$ for some positive integer $k$, where $\alpha_{[k]}$ is an injective monoid endomorphism of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$. Also, we prove that the submonoid of $\boldsymbol{End}_{\textsf{inj}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^3})$, which is generated by the set $\big\{\alpha_{[k]}\colon k\in\mathbb{N}\big\}$ and by the endomorphism $\lambda$, is isomorphic to the semidirect product $(\mathbb{N},\cdot)\ltimes_{\mathfrak{h}}(\omega,+)$.
We study mappings of the Sobolev classes defined in some plane domain. We have obtained estimates of the distortion of the distance under these mappings at the boundary. In particular, we have proved that if the integral averages of the characteristic of mappings are finite, then these mappings are Hölder continuous.
In this article, we investigate majorization results for a class of meromorphic univalent functions of complex order. By utilizing differential operators, various subclasses of analytic and meromorphic functions have been previously defined and studied. Our research focuses on majorization properties within this specific class. Because the $q$-calculus (quantum calculus) has applications in many mathematical fields, the present study aims to improve and generalize majorization results for the class of meromorphic functions combined by $q$-differential operator while presenting a valid form of previously reported results. Additionally, advances in the area under study were achieved, and various novel implications of the main conclusion presented as corollaries were provided. We also highlight some new or known consequences of our results in the form of corollaries. In particular, some of early known results are improved and refined.
The main objective of this study is to construct nonlinear Lototsky-Durrmeyer type operators and to investigate the approximation properties of these newly constructed operators. Research on Lototsky operators began with the work [Canad. J. Math. 1966, 18, 89-91] by J.P. King and approximation with nonlinear integral operators of convolution type was introduced by J. Musielak in 1983. In recent years, both theories have gained popularity through the work of many scientists. In this context, our primary motivation in this study is to introduce the nonlinear Lototsky-Durrmeyer type operators arising from the combination of these two theories and subsequently examine the approximation properties of these operators, particularly for functions with bounded variation and derivatives with bounded variation.
In this article, we define a generalized concept of pairwise eventually $H$-restrictive multi-valued mappings which is based on the restrictive conditions described in [Symmetry 2020, 12 (1), 127]. We also utilize the concept of $D(\epsilon)$-restrictive mappings to explore the coincidence points of the pair of a continuous single-valued map and an $H$-continuous multi-valued map. Our main results are further applied to deduce their corresponding fuzzy fixed point theorems. The latter technique is in appreciation of the fact that many results from the existing literature on contractive or nonexpansive multifunctions have their fuzzy counterparts. Finally, for the validity of our results, some useful examples are also added.
It is well known that an irrational number is quadratic if and only if its regular continued fraction expansion is ultimately periodic. However, no such characterization is known for other real irrational numbers. In 1949, A.Ya. Khinchin conjectured that partial denominators of the regular continued fractions of real algebraic numbers of degree higher than 2 are unbounded. In other words, if partial denominators of the regular continued fractions is bounded, then it is a quadratic number or a transcendental number. In this paper, we observe the regular continued fractions of real algebraic numbers of degree higher than 2. More precisely, we give the minimal polynomials of the real algebraic numbers appearing in the regular continued fractions and establish their properties.
We prove that every strictly convex abelian metric group has a canonical structure of a normed space over the field of real numbers. We deduce this fact from the $\mathbb R$-normability of strictly convex metric groups. Moreover, we prove that a strictly convex (more generaly, $\mathbb R$-normable) metric group is a finite-dimensional normed space if and only if it is locally compact if and only if it is (compactly) finite-dimensional. Also we prove that every strictly convex metric space is geodesic.
Exact order estimates of the best $m$-term trigonometric approximation and the best orthogonal trigonometric approximation of functions from the Nikol'skii-Besov-type classes $B^{\Omega}_{p,\theta}$ in the Lebesgue subspaces $B_{q,1}$ for certain relations between the parameters $p$ and $q$ are obtained. It is shown that in the considered cases the mentioned approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ in the spaces $B_{q,1}$ and $L_q$ differ in order. In addition, it was found that for $1
In the rectangular domain we study a problem with integral conditions with respect to one of the variables for a partial differential equation with singular Bessel operators. A criterion for the unique solvability of this problem and sufficient conditions for the existence of its solution are established. To solve the problem of small denominators arising in some cases of the considered problem, we used the metric approach.
We show that the spectrum of the Sturm-Liouville problem on a connected simple equilateral graph with the Dirichlet boundary conditions at the pendant vertices is related with the spectrum of the discrete Laplacian of the corresponding combinatorial graph. It enables us to compare the spectra of discrete Laplacians to find co-spectral combinatorial graphs and finally co-spectral quantum graphs. Using this method we prove that there are no co-spectral (in our sense) graphs with the number of edges less or equal 7. Thus, in this case the inverse problem of recovering the shape of a quantum graph possesses a unique solution.
The aim of this study is to introduce the concepts of deferred $(H,1)$-summability of order $\nu $, deferred strongly harmonically summability of order $\nu $ and deferred statistical logarithmic convergence of order $\nu $ of sequences of real numbers. Besides we give some inclusion relations related to these concepts.
A local convergence analysis is developed for an eight-order method to solve Banach space defined nonlinear equation under $\omega$-continuity. Earlier efforts require the existence of the ninth derivative to show the convergence on the finite Euclidean space $\mathbb R^k$. However, high order derivatives do not appear in the method. Moreover, no error estimates are available. Therefore, the previous efforts cannot assure the convergence if these derivatives do not exist although the method may converge. The present article addresses these problems. In particular, the new convergence conditions require only the existence of the first derivative appearing in the method. Moreover, error estimates become available. Furthermore, a region is determined containing only one solution of the equation. The novelty of the developed process allows its usage on other methods, since it is independent of the method. The numerical example complements the theory.
The purpose of this article is to investigate the class of weakly $p$-nuclear bilinear operators between Banach spaces. This notion extends the classical theory of nuclear operators introduced by A. Grothendieck, as well as its multilinear generalizations developed by A. Pietsch and others. In particular, we characterize weakly $p$-nuclear bilinear operators through appropriate tensor norms, showing that the space of such operators forms a Banach ideal and analyzing some of its structural properties. Moreover, in the context of duality theory for these operator spaces, we introduce the class of quasi-Cohen $p$-nuclear bilinear operators and establish a Pietsch-type domination theorem.
We explore the properties of the paranormed sequence spaces $c_0(p,\mathscr{D}^\alpha)$, $c(p,\mathscr{D}^\alpha)$, and $\ell_\infty(p,\mathscr{D}^\alpha)$, which are generated by an infinite matrix $\mathscr{D}^\alpha$ involving a generalized divisor sum function $\sigma^{(\alpha)}$ to classical Maddox spaces $c_0(p)$, $c(p)$, and $\ell_\infty(p)$, respectively. The matrix $\mathscr{D}^\alpha=(d^\alpha_{m,r})$ is defined such that $d^\alpha_{m,r} = \dfrac{r^\alpha}{\sigma^{(\alpha)}(m)}$ if $r$ is a divisor of $m$, and $0$, otherwise. Our analysis includes the determination of the Schauder basis and the computation of dual spaces ($\alpha$-, $\beta$-, and $\gamma$-duals) for these newly defined paranormed spaces. Additionally, we characterize matrix transformations from $\ell_{\infty}(p,\mathscr{D}^\alpha)$ into several known sequence spaces, and present related matrix characterizations as direct consequences.
In this note, we introduce explicit formulas for the solution of the Poisson problem in a ball for the logarithmic Laplacian by means of semigroup theory and the Fourier transform. In particular, the solution for such problem is closely related to Volterra functions, which arise, for instance, in some convolution-type integral equations with logarithmic kernels.
For a Dirichlet series $F(s)=\sum\limits_{n=0}^{\infty} a_n\exp\{s\lambda_n\},\, s=\sigma+it$, with the abscissa of absolute convergence $\sigma_a=A\in(-\infty,\,+\infty]$, let $ M(\sigma, F)=\sup\{|F(\sigma+it)|:\,t\in {\mathbb R}\}$ for $\sigma1$, $q>1$, $\alpha \in L$, $\beta \in L$, $\ln\,\beta(x+O(1))=(1+o(1))\ln\,\beta(x)$ as $x\to+\infty$ and $\varlimsup\limits_{n\to\infty}\frac{\ln\,\ln\,\alpha(\lambda_n)}{\ln\,\beta\left(\frac{1}{\lambda_n}\ln\,\frac{1}{|a_n|}\right)}=\eta^*>0$, then $\varlimsup\limits_{\sigma\to+\infty}\frac{\alpha(\ln\,M(\beta^{-1}(q\beta(\sigma)),F))}{\alpha^p(\ln\,M(\sigma,F))}=+\infty$ for each $q>p^{1/\eta^*}$. Similar result is obtained for Dirichlet series with zero abscissa absolute convergence.
We investigate relations between the group of symmetries of a subspace of the symmetric tensor product of a separable Hilbert space and a representation of the partition function of a quantum entangled family of particles associated with this subspace. Consider a system of $N$ noninteracting identical bosons. In general, the system is described by the $N$-fold symmetric tensor product $\mathcal{E}^{\odot N},$ where $\mathcal{E}$ is the Hilbert space that describes the one-particle system. In some cases, the system can be described by some subspace of $\mathcal{E}^{\odot N}.$ We consider the case in which this subspace is the closure of the linear span of some subset of the eigenbasis of the Hamiltonian of the system. We describe the group of symmetries $S$ such that the partition function of the system is $S$-symmetric.
The geometric characterization of curves with zero curvature points presents inherent limitations when employing the classical Frenet frame. This research investigates the mathematical properties of normal curves in $E^3_1$ (Minkowski 3-space) through the application of the Bishop frame. The study systematically analyzes the geometric and topological properties of both spacelike and timelike normal curves, establishing rigorous mathematical conditions necessary and sufficient for their classification within the Bishop frame formalism. We derive fundamental theorems characterizing these curves and examine their differential geometric invariants. The investigation extends to the analysis of curvature relationships, parallel transport properties, and the behavior of normal curves under the Bishop frame parametrization. Our findings contribute to the theoretical framework of curve theory in pseudo-Riemannian geometry, particularly in spaces with indefinite metrics, demonstrating the robustness of the Bishop frame approach for characterizing curves where traditional Frenet analysis becomes singular.
This article focuses on the controllability results for fuzzy Hilfer fractional differential equations through the measure of noncompactness. The main results are established using Mönch's fixed point theorem, along with essential tools such as semigroup theory, fractional calculus, and the measure of noncompactness. Finally, the theoretical results are applied by providing an interesting example.
