Abstract: We construct a separately continuous function f : ℚ × ℚ → [0; 1] and a dense subset D ⊆ ℚ × ℚ such that f[D] is not dense in f[ℚ × ℚ], in other words, f is separately continuous and not somewhat (feebly) continuous. PubDate: Mon, 11 Oct 2021 00:00:00 GMT

Abstract: The primary object of study is the “cosine-sine” functional equation f(xy) = f(x)g(y)+g(x)f(y)+h(x)h(y) for unknown functions f, g, h : S → ℂ, where S is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles.We also discuss the special case f(xy) = f(x)g(y) + g(x)f(y) − g(x)g(y) separately, since it has an independent direct solution on a general semigroup.We give the continuous solutions on topological semigroups for both equations. PubDate: Tue, 05 Oct 2021 00:00:00 GMT

Abstract: Ring properties of amalgamated products are investigated. We offer new, elementary arguments which extend results from [5] and [12] to noncommutative setting and also give new properties of amalgamated rings. PubDate: Mon, 30 Aug 2021 00:00:00 GMT

Abstract: The convergence of sequences and non-unique fixed points are established in ℳ-orbitally complete cone metric spaces over the strongly minihedral cone, and scalar weighted cone assuming the cone to be strongly minihedral. Appropriate examples and applications validate the established theory. Further, we provide one more answer to the question of the existence of the contractive condition in Cone metric spaces so that the fixed point is at the point of discontinuity of a map. Also, we provide a negative answer to a natural question of whether the contractive conditions in the obtained results can be replaced by its metric versions. PubDate: Mon, 30 Aug 2021 00:00:00 GMT

Abstract: In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a statedependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model. PubDate: Mon, 30 Aug 2021 00:00:00 GMT

Abstract: In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known recursive sequences, as the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, etc. We establish some general limit formulas, where the product of the first n terms of these sequences appears. Furthermore, we prove some general limits that connect these sequences to the number e(≈ 2.71828...). PubDate: Tue, 27 Jul 2021 00:00:00 GMT

Abstract: We consider the MHD system in a bounded domain Ω ⊂ ℝN , N = 2, 3, with Dirichlet boundary conditions. Using Dan Henry’s semigroup approach and Giga–Miyakawa estimates we construct global in time, unique solutions to fractional approximations of the MHD system in the base space (L2(Ω))N × (L2(Ω))N. Solutions to MHD system are obtained next as a limits of that fractional approximations. PubDate: Tue, 27 Jul 2021 00:00:00 GMT

Abstract: For a continuous and positive function w (λ), λ> 0 and µ a positive measure on [0, ∞) we consider the following 𝒟-logarithmic integral transform𝒟ℒog(w,μ)(T):=∫0∞w(λ)1n(λ+Tλ)dμ(λ),\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),}where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.We show among others that, if A, B > 0 with BA + AB ≥ 0, then𝒟ℒog(w,μ)(A)+𝒟ℒog(w,μ)(B)≥𝒟ℒog(w,μ)(A+B).\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right).In particular we have16π2+dilog(A+B)≥dilog(A)+dilog(B),{1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right),where the dilogarithmic function dilog : [0, ∞) → ℝ is defined bydilog(t):=∫1t1ns1-sds, t≥0.{\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0.Some examples for integr... PubDate: Wed, 26 May 2021 00:00:00 GMT

Abstract: We establish necessary and sufficient conditions allowing separation of pair of real functions by an m-convex and by an m-affine function. Some examples and a geometric interpretation of m-convexity of a function is exhibited, as well as a Jensen’s inequality for this kind of function. PubDate: Wed, 26 May 2021 00:00:00 GMT

Abstract: In this work it was proved Matkowski’s fixed point theorem. The consequences of this theorem are also presented. PubDate: Wed, 26 May 2021 00:00:00 GMT

Abstract: We introduce the notion of an (α, β, γ) triple system, which generalizes the familiar generalized Jordan triple system related to a construction of simple Lie algebras. We then discuss its realization by considering some bilinear algebras and vice versa. Next, as a new concept, we study triality relations (a triality group and a triality derivation) associated with these triple systems; the relations are a generalization of the automorphisms and derivations of the triple systems. Also, we provide examples of several involutive triple systems with a tripotent element. PubDate: Tue, 13 Apr 2021 00:00:00 GMT

Abstract: In this paper we introduce some new types of contractive mappings by combining Caristi contraction, Ćirić-quasi contraction and weak contraction in the framework of a metric space. We prove some fixed point theorems for such type of mappings over complete metric spaces with the help of φ-diminishing property. Some examples are given in strengthening the hypothesis of our established theorems. PubDate: Tue, 13 Apr 2021 00:00:00 GMT

Abstract: The aim of this note is to study the distribution function of certain sequences of positive integers, including, for example, Bell numbers, factorials and primorials. In fact, we establish some general asymptotic formulas in this regard. We also prove some limits that connect these sequences with the number e. Furthermore, we present a generalization of the primorial. PubDate: Tue, 26 Jan 2021 00:00:00 GMT

Abstract: A generalization of the Hermite–Hadamard (HH) inequality for a positive convex stochastic process, by means of a newly proposed fractional integral operator, is hereby established. Results involving the Riemann– Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl and Liouville fractional integrals are deduced as particular cases of our main result. In addition, we also apply some known HH results to obtain some estimates for the expectations of integrals of convex and p-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral, Revista Integración, temas de matemáticas 36 (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction. PubDate: Thu, 17 Dec 2020 00:00:00 GMT

Abstract: A new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P𝓁n is expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times. PubDate: Mon, 14 Dec 2020 00:00:00 GMT

Abstract: The aim of this paper is to characterize the solutions Φ : G → M2(ℂ) of the following matrix functional equationsΦ(xy)+Φ(σ(y)x)2=Φ(x)Φ(y), x,y,∈G,{{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G,andΦ(xy)−Φ(σ(y)x)2=Φ(x)Φ(y), x,y,∈G,{{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G,where G is a group that need not be abelian, and σ : G → G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined. PubDate: Mon, 14 Dec 2020 00:00:00 GMT

Abstract: In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F (x) = σx + . . . is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation. PubDate: Tue, 06 Oct 2020 00:00:00 GMT

Abstract: In this paper, we introduce the generalized Tetranacci hybrid numbers and, as special cases, Tetranacci and Tetranacci-Lucas hybrid numbers. Moreover, we present Binet’s formulas, generating functions, and the summation formulas for those hybrid numbers. PubDate: Tue, 06 Oct 2020 00:00:00 GMT

Abstract: If (μn)n=1∞\left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and (vn)n=1∞\left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that ∑n=1∞‖vn‖μn(X)<∞\sum\nolimits_{n = 1}^\infty {\left\ {{v_n}} \right\ {\mu _n}\left( X \right)} < \infty, then ω(S)=∑n=1∞vnμn(S)\omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)} defines a vector measure of bounded variation on (X, Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (X, Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on 𝔼-valued measures for any Banach space 𝔼 that has the Radon-Nikodym property. PubDate: Tue, 06 Oct 2020 00:00:00 GMT