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

<abstract xmlns="http://www.w3.org/1999/xhtml">

We study the problem of solvability of the equation
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0031_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><msub><mo>∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow><mi>g</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>ω</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mi>P</mi><mrow><mo>(</mo><mrow><mi>d</mi><mi>ω</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow>
</math>
<tex-math>\varphi \left( x \right) = \int_\Omega {g\left( w \right)} \varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right) + F\left( x \right),</tex-math>
</alternatives>
</disp-formula>
where <italic>P</italic> is a probability measure on a <italic>σ</italic>-algebra of subsets of Ω, assuming Hölder continuity of <italic>F</italic> on the range of <italic>f</italic>.
</abstract>

Note on an Iterative Functional Equation

<abstract xmlns="http://www.w3.org/1999/xhtml">

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup <italic>S</italic> of a group <italic>G</italic> generates <italic>G</italic> in the sense that <italic>G</italic> = <italic>S · S</italic>−1, then a generalized polynomial on <italic>S</italic> with values in an Abelian group <italic>H</italic> can be extended to a generalized polynomial on <italic>G</italic> into <italic>H</italic>. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.
</abstract>

Generalized Polynomials on Semigroups

<abstract xmlns="http://www.w3.org/1999/xhtml">
Let <italic>X </italic>be an Abelian group, <italic>Y </italic>be a commutative monoid, <italic>K </italic>⊂<italic>Y </italic> be a submonoid and <italic>F </italic>: <italic>X → </italic>2<italic>Y </italic><italic>\ </italic>{∅} be a set-valued map. Under some additional assumptions on ideals <italic>ℐ</italic>1 in <italic>X </italic>and <italic>ℐ</italic>2 in <italic>X</italic>2, we prove that if <italic>F </italic>is <italic>ℐ</italic>2-almost everywhere <italic>K</italic>-additive, then there exists a unique up to <italic>K K</italic>-additive set-valued map <italic>G </italic>: <italic>X → </italic>2<italic>Y </italic><italic>\</italic>{∅} such that <italic>F </italic>= <italic>G ℐ</italic>1-almost everywhere in <italic>X</italic>. Our considerations refers to the well known de Bruijn’s result [1].
</abstract>

On Almost Everywhere <italic>K</italic>-Additive Set-Valued Maps

<abstract xmlns="http://www.w3.org/1999/xhtml">

New upper bounds for the weighted Chebyshev functional under various conditions, including those of Steffensen type, are given. The obtained results are used to establish some new bounds for the Jensen functional.
</abstract>

New Upper Bounds for the Weighted Chebyshev Functional

<abstract xmlns="http://www.w3.org/1999/xhtml">

We consider sequences of Steklov type operators and an associated functional equation. For a suitable sequence, we establish asymptotic formulas.
</abstract>

Steklov Type Operators and Functional Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">
Cancellation conditions play a central role in the representation theory of measurement for a weak order on a finite two-dimensional Cartesian product set <italic>X</italic>. A weak order has an additive representation if and only if it violates no cancellation conditions. Given <italic>X</italic>, a longstanding open problem is to determine the simplest set of cancellation conditions that is violated by every linear order that is not additively representable. Here, we report that the simplest set of cancellation conditions on a 5 by 5 product <italic>X </italic>is obtained.
</abstract>

A Basic Set of Cancellation Violating Sequences for Finite Two-Dimensional Non-Additive Measurement

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we consider homogeneous quasideviation means generated by real functions (defined on (0, ∞)) which are concave around the point 1 and possess certain upper estimates near 0 and ∞. It turns out that their concave envelopes can be completely determined. Using this description, we establish su˚cient conditions for the Hardy property of the homogeneous quasideviation mean and we also furnish an upper estimates for its Hardy constant.
</abstract>

Estimating the Hardy Constant of Nonconcave Homogeneous Quasideviation Means

<abstract xmlns="http://www.w3.org/1999/xhtml">

Motivated by the Szostok problem on functions with monotonic differences (2005, 2007), we consider <italic>a</italic>-Wright convex functions as a generalization of Wright convex functions. An application of these results to obtain new proofs of known results as well as new results is presented.
</abstract>

On Functions with Monotonic Differences

<abstract xmlns="http://www.w3.org/1999/xhtml">

We analyze in our paper questions of the theory of relativity. We approach this theory from the point of view of velocities and their composition. This is where the functional equations appear. Solving them leads to a world where velocities are bounded from above, the upper bound being exactly the “speed of light”.
</abstract>

Speed of Light or Composition of Velocities

<abstract xmlns="http://www.w3.org/1999/xhtml">

We present an adaptive method of approximate integration of convex (as well as concave) functions based on a certain refinement of the celebrated Hermite–Hadamard inequality. Numerical experiments are performed and the role of harmonic numbers is shown.
</abstract>

Adaptive Integration of Convex Functions of One Real Variable

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Annales Mathematicae Silesianae

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 develop important, new mathematical ideas and results or solve outstanding problems.  Article processing chargesThe journal does not have article processing charges (APCs) nor article submission charges.  Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process.  Peer Review process  Annales Mathematicae Silesianae is a single-blind peer reviewed journal.Normally the papers are peer reviewed by two independent referees who are experts in the appropriate branch of mathematics. According to their opinions the manuscripts are either accepted or rejected. In the case their opinions contradict each other we send the paper to the third reviewer.Reviewers are asked to provide detailed, constructive comments that will help both the editors to make a decision on the publication and the author(s) to improve their manuscript. Preparing the review the reviewers may use the following scheme: review_scheme.pdf. Reviewers are also asked to declare that there is no conflict of interests between them and the authors.  Target Group  - researchers in all branches of pure and applied mathematics 