<abstract><title style='display:none'>Abstract</title>The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider special kinds of hybrid numbers, namely the Jacobsthal and the Jacobsthal-Lucas hybrid numbers and we give some their properties.</abstract>

On Jacobsthal and Jacobsthal-Lucas Hybrid Numbers

<abstract><title style='display:none'>Abstract</title> We consider the simplest parabolic-elliptic model of chemotaxis in the whole space and in several space dimensions. Criteria either for the existence of radial global-in-time solutions or their blowup in terms of suitable Morrey spaces norms are discussed. This is an extended version of the lecture presented at the University of Silesia on January 12, 2018, commemorating Professor Andrzej Lasota-great scholar, master of fine mathematics and applications to real world.</abstract>

Mathematical Challenges in the Theory of Chemotaxis

<abstract><title style='display:none'>Abstract</title>Let <italic>R</italic> be a prime ring with center <italic>Z</italic>(<italic>R</italic>). A map <italic>G</italic> : <italic>R</italic> →<italic>R</italic> is called a multiplicative (generalized) (<italic>α</italic>, <italic>β</italic>)-derivation if <italic>G</italic>(<italic>xy</italic>)= <italic>G</italic>(<italic>x</italic>)<italic>α</italic>(<italic>y</italic>)+<italic>β</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>) is fulfilled for all <italic>x</italic>; <italic>y</italic> ∈ <italic>R</italic>, where <italic>g</italic> : <italic>R</italic> → <italic>R</italic> is any map (not necessarily derivation) and <italic>α</italic>; <italic>β</italic> : <italic>R</italic> → <italic>R</italic> are automorphisms. Suppose that <italic>G</italic> and <italic>H</italic> are two multiplicative (generalized) (<italic>α</italic>, <italic>β</italic>)-derivations associated with the mappings <italic>g</italic> and <italic>h</italic>, respectively, on <italic>R</italic> and <italic>α</italic>, <italic>β</italic> are automorphisms of <italic>R</italic>. The main objective of the present paper is to investigate the following algebraic identities: (<italic>i</italic>) <italic>G</italic>(<italic>xy</italic>) + <italic>α</italic>(<italic>xy</italic>) = 0, (<italic>ii</italic>) <italic>G</italic>(<italic>xy</italic>) + <italic>α</italic>(<italic>yx</italic>) = 0, (<italic>iii</italic>) <italic>G</italic>(<italic>xy</italic>) + <italic>G</italic>(<italic>x</italic>)<italic>G</italic>(<italic>y</italic>) = 0, (<italic>iv</italic>) <italic>G</italic>(<italic>xy</italic>) = <italic>α</italic>(<italic>y</italic>) ○ <italic>H</italic>(<italic>x</italic>) and (<italic>v</italic>) <italic>G</italic>(<italic>xy</italic>) = [<italic>α</italic>(<italic>y</italic>), <italic>H</italic>(<italic>x</italic>)] for all <italic>x</italic>, <italic>y</italic> in an appropriate subset of <italic>R</italic>.</abstract>

A Note on Multiplicative (Generalized) (<italic>α</italic>, <italic>β</italic>)-Derivations in Prime Rings

<abstract><title style='display:none'>Abstract</title>In this paper we present a new one parameter generalization of the classical Pell numbers. We investigate the generalized Binet’s formula, the generating function and some identities for <italic>r</italic>-Pell numbers. Moreover, we give a graph interpretation of these numbers.</abstract>

On a New One Parameter Generalization of Pell Numbers

<abstract><title style='display:none'>Abstract</title>The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation.We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, <italic>u</italic>2<italic>− g</italic>2 = (<italic>u</italic> + <italic>g</italic>)(<italic>u − g</italic>). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.</abstract>

Insert a Root to Extract a Root of Quintic Quickly

<abstract><title style='display:none'>Abstract</title> The main aim of this manuscript is to prove the following result. Let n &gt; 2 be a fixed integer and R be a k-torsion free semiprime ring with identity, where k ∈ {2, n - 1, n}. Let us assume that for the additive mapping T : R→ R 3T(xn) = T(x)xn-1+ xT(xn-2)x + xn-1T(x), x ∈R, is also fulfilled. Then T is a two-sided centralizer.</abstract>

On a Functional Equation Related to Two-Sided Centralizers

<abstract>
<title style='display:none'>Abstract</title>
In this paper, we define and study a new one-parameter generalization of the Pell hybrid numbers. Based on the definition of <italic>r</italic>-Pell numbers, we define the <italic>r</italic>-Pell hybrid numbers. We give their properties: character, Binet formula, summation formula, and generating function. Moreover, we present Catalan, Cassini, d’Ocagne, and Vajda type identities for the <italic>r</italic>-Pell hybrid numbers.
</abstract>

On a New Generalization of Pell Hybrid Numbers

<abstract>
<title style='display:none'>Abstract</title>
In this paper we determine the complex-valued solutions of the Kannappan-cosine functional equation <italic>g</italic>(<italic>xyz</italic>0) = <italic>g</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>) <italic>− f</italic>(<italic>x</italic>)<italic>f</italic>(<italic>y</italic>), <italic>x, y ∈ S</italic>, where <italic>S</italic> is a semigroup and <italic>z</italic>0 is a fixed element in <italic>S.</italic>
</abstract>

A Kannappan-Cosine Functional Equation on Semigroups

Report of Meeting: The Twenty-third Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 – February 3, 2024

<abstract>
<title style='display:none'>Abstract</title>
In this paper, we prove Talenti’s comparison theorem for mixed local/nonlocal elliptic operators and derive the Faber–Krahn inequality for the first eigenvalue of the Dirichlet mixed local/nonlocal problem. Our findings are relevant to the fractional <italic>p</italic>&amp;<italic>q−</italic>Laplacian operator.
</abstract>

Symmetrization for Mixed Operators

<abstract>
<title style='display:none'>Abstract</title>
In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra <bold>L</bold> is used to study the unifying properties of some subclasses of the lattice of filters of <bold>L</bold>. In particular, we obtain that for a stable closure operation <italic>c</italic> on an IL-algebra, the collection of <italic>c</italic>-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored.
</abstract>

Closure Operations on Intuitionistic Linear Algebras

On a Certain Polish Mathematician: A Brief Summary of Scientific Research and Achievements of Kazimierz Nikodem, Professor of Mathematics

<abstract><title style='display:none'>Abstract</title> In this article, quadratic B-spline Galerkin method has been employed to solve the time fractional order Schrödinger equation. Numerical solutions and error norms L2 and L∞ are presented in tables. </abstract>

Numerical Solution of Time Fractional Schrödinger Equation by Using Quadratic B-Spline Finite Elements

<abstract><title style='display:none'>Abstract</title>In this article, we will define Padovan’s hybrid numbers, based on the new noncommutative numbering system studied by Özdemir ([7]). Such a system that is a set involving complex, hyperbolic and dual numbers. In addition, Padovan’s hybrid numbers are created by combining this set, satisfying the relation <italic>ih</italic> = <italic>−hi</italic> = <italic>ɛ</italic> + <italic>i</italic>. Given this, some properties and identities are shown for these numbers, such as Binet’s formula, generating matrix, characteristic equation, norm, and generating function. In addition, these numbers are extended to the integer field and some identities are made.</abstract>

The Hybrid Numbers of Padovan and Some Identities

<abstract><title style='display:none'>Abstract</title>Counterparts of the Ohlin and Levin–Stečkin theorems for strongly convex functions are proved. An application of these results to obtain some known inequalities related with strongly convex functions in an alternative and unified way is presented.</abstract>

Ohlin and Levin–Stečkin-Type Results for Strongly Convex Functions

<abstract><title style='display:none'>Abstract</title> In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be expressed as a sum of a derivation and a center valued map vanishing at commutators. We then apply our results for triangular algebras. Some illuminating examples are also included.</abstract>

Lie Derivations on Trivial Extension Algebras

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Annales Mathematicae Silesianae publishes significant research and survey papers from all branches of pure and applied mathematics, and reports of meetings. It welcomes contributed papers that...

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Annales Mathematicae Silesianae publishes significant research and survey papers from all branches of pure and applied mathematics, and reports of meetings....