<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>In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.</abstract>

Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

<abstract><title style='display:none'>Abstract</title>Inspired by the papers [2, 10] we will study, on 2-divisible groups that need not be abelian, the alienation problem between Drygas’ and the exponential Cauchy functional equations, which is expressed by the equation
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0002_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>-</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math><tex-math>f\left( {x + y} \right) + g\left( {x + y} \right)g\left( {x - y} \right) = f\left( x \right)f\left( y \right) + 2g\left( x \right) + g\left( y \right) + g\left( { - y} \right).</tex-math></alternatives></disp-formula>We also consider an analogous problem for Drygas’ and the additive Cauchy functional equations as well as for Drygas’ and the logarithmic Cauchy functional equations. Interesting consequences of these results are presented.</abstract>

Alienation of Drygas’ and Cauchy’s Functional Equations

Report of Meeting. The Nineteenth Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Zakopane (Poland), January 30–February 2, 2019

Report of Meeting: The Twenty-second Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), February 1–4, 2023

<abstract><title style='display:none'>Abstract</title> In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.</abstract>

A Simple Proof of the Polar Decomposition Theorem

<abstract>
<title style='display:none'>Abstract</title>
We give a proof of the uniform boundedness principle for linear continuous maps from <italic>F</italic> -spaces into topological vector spaces which is elementary and also quite simple.
</abstract>

A Really Simple Elementary Proof of the Uniform Boundedness Principle in <italic>F</italic> -Spaces

<abstract>
<title style='display:none'>Abstract</title>
Non-nil abelian groups are classified on which every ring, different from the zero-ring, is unital. It is shown that the assumption on the associativity of the considered rings does not influence the obtained classification. A significant mistake made by other authors studied this topic is corrected.
</abstract>

A Note on Abelian Groups Supporting Unital Rings

<abstract>
<title style='display:none'>Abstract</title>
This paper presents the concept of a partial idempotent valued <italic>S</italic>-metric space, abbreviated as PIV-<italic>S</italic>-metric space, as a generalization of both the PIV-metric space and <italic>S</italic>-metric space. The study utilizes this new frame-work to establish a fixed point theorem and a best proximity point theorem. Additionally, the paper proves the existence and uniqueness of the best proximity point within this context. Several illustrative examples are provided to demonstrate the practical applications of the main findings.
</abstract>

Fixed Point and Best Proximity Point Results in PIV-<italic>S</italic>-Metric Spaces

<abstract>
<title style='display:none'>Abstract</title>
We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.
</abstract>

Fibonacci Sums Modulo 5

<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 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

<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> 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>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> Some inequalities of Hermite-Hadamard type for GA-convex functions defined on positive intervals are given.</abstract>

Inequalities of Hermite–Hadamard Type for GA-Convex Functions

<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

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