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 Annales Mathematicae SilesianaeNumber of Followers: 0     Open Access journal ISSN (Print) 0860-2107 - ISSN (Online) 2391-4238 Published by Sciendo  [371 journals]
• Refinements of Some Classical Inequalities Involving Sinc and Hyperbolic
Sinc Functions

• Abstract: Several bounds of trigonometric-exponential and hyperbolic-exponential type for sinc and hyperbolic sinc functions are presented. In an attempt to generalize the results, some known inequalities are sharpened and extended. Hyperbolic versions are also established, along with extensions.
PubDate: Wed, 23 Nov 2022 00:00:00 GMT

• Some Observations on the Greatest Prime Factor of an Integer

• Abstract: We examine the multiplicity of the greatest prime factor in k-full numbers and k-free numbers. We generalize a well-known result on greatest prime factors and obtain formulas related with the Riemann zeta function.
PubDate: Wed, 23 Nov 2022 00:00:00 GMT

• On Weights Which Admit Harmonic Bergman Kernel and Minimal Solutions of
Laplace’s Equation

• Abstract: In this paper we consider spaces of weight square-integrable and harmonic functions L2H(Ω, µ). Weights µ for which there exists reproducing kernel of L2H(Ω, µ) are named ’admissible weights’ and such kernels are named ’harmonic Bergman kernels’. We prove that if only weight of integration is integrable in some negative power, then it is admissible. Next we construct a weight µ on the unit circle which is non-admissible and using Bell-Ligocka theorem we show that such weights exist for a large class of domains in ℝ2. Later we conclude from the classical result of reproducing kernel Hilbert spaces theory that if the set {f ∈ L2H(Ω, µ) f(z) = c} for admissible weight µ is non-empty, then there is exactly one element with minimal norm. Such an element in this paper is called ’a minimal (z, c)-solution in weight µ of Laplace’s equation on Ω’ and upper estimates for it are given.
PubDate: Thu, 15 Sep 2022 00:00:00 GMT

• Gleason–Kahane–Żelazko Theorem for Bilinear Maps

• Abstract: Let A and B be two unital Banach algebras and 𝔘 = A × B. We prove that the bilinear mapping φ: 𝔘 → ℂ is a bi-Jordan homomorphism if and only if φ is unital, invertibility preserving and jointly continuous. Additionally, if A is commutative, then φ is a bi-homomorphism.
PubDate: Thu, 15 Sep 2022 00:00:00 GMT

• General Limit Formulae Involving Prime Numbers

• Abstract: Let pn be the n th prime number. In this note, we study strictly increasing sequences of positive integers An such that the limit limn→∞ (A1A2 · · · An)1/pn = e holds. This limit formula is in fact a generalization of some previously known results. Furthermore, some other generalizations are established.
PubDate: Thu, 08 Sep 2022 00:00:00 GMT

• Deepest Nodes in Marked Ordered Trees

• Abstract: A variation of ordered trees, where each rightmost edge might be marked or not, if it does not lead to an endnode, is investigated. These marked ordered trees were introduced by E. Deutsch et al. to model skew Dyck paths. We study the number of deepest nodes in such trees. Explicit generating functions are established and the average number of deepest nodes, which approaches 53{5 \over 3} when the number of nodes gets large. This is to be compared to standard ordered trees where the average number of deepest nodes approaches 2.
PubDate: Thu, 08 Sep 2022 00:00:00 GMT

• On Quaternion Gaussian Bronze Fibonacci Numbers

• Abstract: In the present work, a new sequence of quaternions related to the Gaussian Bronze numbers is defined and studied. Binet’s formula, generating function and certain properties and identities are provided. Tridiagonal matrices are considered to determine the general term of this sequence.
PubDate: Thu, 08 Sep 2022 00:00:00 GMT

• On the Two-Dimensional Version of the Sperner Lemma and Brouwer’s
Theorem

• Abstract: In this work the Brouwer fixed point theorem for a triangle was proved by two methods based on the Sperner Lemma. One of the two proofs of Sperner’s Lemma given in the paper was carried out using the so-called index.
PubDate: Thu, 08 Sep 2022 00:00:00 GMT

• On Generalized Jacobsthal and Jacobsthal–Lucas Numbers

• Abstract: Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.
PubDate: Fri, 01 Jul 2022 00:00:00 GMT

• Classification of Odometers: A Short Elementary Proof

• Abstract: The paper deals with odometers (i.e. adding machines) of general type. We give a characterization of self-conjugacies of odometers which enables us to present an elementary proof of a classification of odometers given by Buescu and Stewart in [2]. The paper might also serve as a very quick introduction to odometers.
PubDate: Sat, 28 May 2022 00:00:00 GMT

• A Levi–Civita Equation on Monoids, Two Ways

• Abstract: We consider the Levi–Civita equationf(xy)=g1(x)h1(y)+g2(x)h2(y)f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right)for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.
PubDate: Thu, 12 May 2022 00:00:00 GMT

• A Functional Equation with Biadditive Functions

• Abstract: Let S, H, X be groups. For two given biadditive functions A : S2 → X, B : H2 → X and for two unknown mappings T : S → H, g : S → S we will study the functional equationB(T (x), T (y)) = A(x, g(y)), x, y ∈ S,which is a generalization of the orthogonality equation in Hilbert spaces.
PubDate: Mon, 02 May 2022 00:00:00 GMT

• A Further Generalization of

• Abstract: It is well-known, as follows from the Stirling’s approximation n!∼2πn(n/e)nn! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}}, that n!/n→1/en\root n \of {n!/n \to 1/e}. A generalization of this limit is (11s· 22s· · · nns)1/ns+1 · n−1/(s+1) → e−1/(s+1)2 which was established by N. Schaumberger in 1989 (see [8]). The aim of this work is to establish a new generalization that is in fact an improvement of Schaumberger’s formula for a general sequence An of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers.
PubDate: Mon, 18 Apr 2022 00:00:00 GMT

• An Extension of the Abel–Liouville Identity

• Abstract: In this note, we present an extension of the celebrated Abel– Liouville identity in terms of noncommutative complete Bell polynomials for generalized Wronskians. We also characterize the range equivalence of n-dimensional vector-valued functions in the subclass of n-times differentiable functions with a nonvanishing Wronskian.
PubDate: Mon, 18 Apr 2022 00:00:00 GMT

• Construction of Regular Non-Atomic Strictly-Positive Measures in
Second-Countable Non-Atomic Locally Compact Hausdorff Spaces

• Abstract: This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a set function limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a meticulous way to ensure that the set function limit is σ-additive.
PubDate: Tue, 22 Mar 2022 00:00:00 GMT

• A Variant of D’Alembert’s Functional Equation on Semigroups
with Endomorphisms

• Abstract: Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equationf(xϕ(y))+f(ψ(y)x)=2f(x)f(y),      x,y ∈ S,f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S,where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.
PubDate: Tue, 22 Mar 2022 00:00:00 GMT

• Report of Meeting: The Twenty-first Katowice–Debrecen Winter Seminar on
Functional Equations and Inequalities Brenna (Poland), February 2–5,
2022

• PubDate: Tue, 22 Mar 2022 00:00:00 GMT

• A Parametric Functional Equation Originating from Number Theory

• Abstract: Let S be a semigroup and α, β ∈ ℝ. The purpose of this paper is to determine the general solution f : ℝ2 → S of the following parametric functional equation f(x1+x2+αy1y2,x1y2+x2y1+βy1y2)=f(x1,y1)f(x2,y2),f\left( {{x_1} + {x_2} + \alpha {y_1}{y_2},{x_1}{y_2} + {x_2}{y_1} + \beta {y_1}{y_2}} \right) = f\left( {{x_1},{y_1}} \right)f\left( {{x_2},{y_2}} \right), for all (x1, y1), (x2, y2) ∈ ℝ2, that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices.
PubDate: Mon, 17 Jan 2022 00:00:00 GMT

• Gauss Congruences in Algebraic Number Fields

• Abstract: In this miniature note we generalize the classical Gauss congruences for integers to rings of integers in algebraic number fields.
PubDate: Mon, 17 Jan 2022 00:00:00 GMT

• Jacobsthal Representation Hybrinomials

• Abstract: Jacobsthal numbers are a special case of numbers defined recursively by the second order linear relation and for these reasons they are also named as numbers of the Fibonacci type. They have many interpretations, representations and applications in distinct areas of mathematics. In this paper we present the Jacobsthal representation hybrinomials, i.e. polynomials, which are a generalization of Jacobsthal hybrid numbers.
PubDate: Sat, 27 Nov 2021 00:00:00 GMT

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