Annales Mathematicae Silesianae https://journals.us.edu.pl/index.php/AMSIL <p><em>Annales Mathematicae Silesianae</em>&nbsp;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. Submissions are strictly refereed, and only articles of the highest quality are accepted for publication.</p> <p>The journal does not have article processing charges or article submission charges.</p> <p><em>Annales Mathematicae Silesianae</em>&nbsp;is published by the University of Silesia Press (Poland). The first number of the journal appeared in 1985. Formerly (since 1969) it was published under the title&nbsp;<em>Prace Naukowe Uniwersytetu Śląskiego w Katowicach. Prace Matematyczne.</em></p> <p>The Editorial Board participates in a growing community of Similarity Check System's users to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscript screening to eliminate plagiarism and provide a high standard and quality peer-review process.</p> University of Silesia Press en-US Annales Mathematicae Silesianae 0860-2107 <p><strong>The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.</strong></p> <ol> <li class="show">License<br> This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (<a href="http://creativecommons.org/licenses/by/4.0/">http://creativecommons.org/licenses/by/4.0/</a>). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.</li> <li class="show">Author’s Warranties<br> The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.</li> <li class="show">User Rights<br> Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.</li> <li class="show">Co-Authorship<br> If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.</li> </ol> With Andrzej Lasota there and back again https://journals.us.edu.pl/index.php/AMSIL/article/view/17705 <p>The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota’s results on the author’s research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differentia equations.</p> Ryszard Rudnicki ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2024-07-15 2024-07-15 38 2 134 154 Cosine and sine addition and subtraction law with an automorphism https://journals.us.edu.pl/index.php/AMSIL/article/view/16429 <p>Let <em>S</em> be a semigroup. Our main result is that we describe the complex-valued solutions of the following functional equations<br><em>g</em>(<em>xσ</em>(<em>y</em>)) = <em>g</em>(<em>x</em>)<em>g</em>(<em>y</em>) + <em>f</em>(<em>x</em>)<em>f</em>(<em>y</em>), <em>x</em>, <em>y</em> ∈ <em>S</em>,<br><em>f</em>(<em>xσ</em>(<em>y</em>)) = <em>f</em>(<em>x</em>)<em>g</em>(<em>y</em>) + <em>f</em>(<em>y</em>)<em>g</em>(<em>x</em>), <em>x</em>, <em>y</em> ∈ <em>S</em>,<br>and<br><em>f</em>(<em>xσ</em>(<em>y</em>)) = <em>f</em>(<em>x</em>)<em>g</em>(<em>y</em>) - <em>f</em>(<em>y</em>)<em>g</em>(<em>x</em>), <em>x</em>, <em>y</em> ∈ <em>S</em>,<br>where <em>σ</em> : <em>S</em> → <em>S</em> is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants.<br>We also give some applications.</p> Youssef Aserrar Elhoucien Elqorachi ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2023-11-29 2023-11-29 38 2 155 176 On transcendental entire solution of Fermat-type trinomial and binomial equations under restricted hyper-order https://journals.us.edu.pl/index.php/AMSIL/article/view/16394 <p>In this paper we are focusing on finding the transcendental entire solution of Fermat-type trinomial and binomial equations, by restricting the hyper-order to be less than one. As the hyper-order is a crucial parameter that characterizes the growth of entire functions, it will be interesting to investigate this unexplored domain, as far as practible, with certain restriction on hyper order. Our results are the improvements of previous results reported in recent papers [12], [13]. We have provided a series of examples to demonstrate and validate the effectiveness of our proposed solutions.</p> Abhijit Banerjee Jhuma Sarkar ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2023-11-22 2023-11-22 38 2 177 194 Strongly M_φM_ψ-convex functions, the Hermite-Hadamard-Fejér inequality and related results https://journals.us.edu.pl/index.php/AMSIL/article/view/15453 <p>We present Hermite-Hadamard-Fejér type inequalities for strongly M<sub>φ</sub>M<sub>ψ</sub>-convex functions. Some refinements of them and bounds for the integral mean of the product of two functions are also obtained.</p> Mea Bombardelli Sanja Varošanec ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2023-11-22 2023-11-22 38 2 195 213 An alternative equation for generalized polynomials of degree two https://journals.us.edu.pl/index.php/AMSIL/article/view/15499 <p>In this paper we consider a generalized polynomial <em>f</em> : ℝ → ℝ of degree two that satisfies the additional equation f(x)f(y) = 0 for the pairs (<em>x</em>,<em>y</em>) ∈ <em>D</em>, where <em>D</em> ⊆ ℝ<sup>2</sup> is given by some algebraic condition. In the particular cases when there exists a positive rational m fulfilling<br><em>D</em> = { (<em>x</em>,<em>y</em>) ∈ ℝ<sup>2</sup> | <em>x</em><sup>2</sup> - <em>my</em><sup>2</sup> = 1 },<br>we prove that <em>f</em>(<em>x</em>) = 0 for all <em>x</em> ∈ ℝ.</p> Zoltán Gábor Boros Rayene Menzer ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2023-10-26 2023-10-26 38 2 214 220 On a new generalization of Pell hybrid numbers https://journals.us.edu.pl/index.php/AMSIL/article/view/17429 <p>In this paper, we define and study a new one-parameter generalization of the Pell hybrid numbers. Based on the definition of <em>r</em>-Pell numbers, we define the <em>r</em>-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 <em>r</em>-Pell hybrid numbers.</p> Dorota Bród Anetta Szynal-Liana Iwona Włoch ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2024-04-27 2024-04-27 38 2 221 240 Bidimensional extensions of cobalancing and Lucas-cobalancing numbers https://journals.us.edu.pl/index.php/AMSIL/article/view/16430 <p>A new bidimensional version of cobalancing numbers and Lucasbalancing numbers are introduced. Some properties and identities satisfied by these new bidimensional sequences are studied.</p> J. Chimpanzo M. V. Otero-Espinar A. Borges P. Vasco P. Catarino ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2023-11-29 2023-11-29 38 2 241 262 The subset-strong product of graphs https://journals.us.edu.pl/index.php/AMSIL/article/view/16730 <p>In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks.</p> Mehdi Eliasi ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2024-01-10 2024-01-10 38 2 263 283 Determinants of Toeplitz-Hessenberg matrices with generalized Leonardo number entries https://journals.us.edu.pl/index.php/AMSIL/article/view/16728 <p>Let <em>u</em><sub>n</sub> = <em>u<sub>n</sub></em>(<em>k</em>) denote the generalized Leonardo number defined recursively by <em>u<sub>n</sub></em> = <em>u</em><sub><em>n</em>-1</sub>&nbsp;+ <em>u</em><sub><em>n</em>-2</sub>&nbsp;+ <em>k</em> for <em>n</em>≥2, where <em>u</em><sub>0</sub> = <em>u</em><sub>1</sub> = 1. Terms of the sequence <em>u<sub>n</sub></em>(1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1≤<em>k</em>≤7, where several connections are made to entries in the <em>On-Line Encyclopedia of Integer Sequences</em>. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case <em>k</em>=1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.</p> Taras Goy Mark Shattuck ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2024-01-10 2024-01-10 38 2 284 313 Some general theorems about a class of sets of numbers https://journals.us.edu.pl/index.php/AMSIL/article/view/16499 <p>We prove a theorem which unifies some formulas, for example the counting function of some sets of numbers including all positive integers, <em>h</em>-free numbers, <em>h</em>-full numbers, etc. We also establish a conjecture and give some examples where the conjecture holds.</p> Rafael Jakimczuk ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2023-12-13 2023-12-13 38 2 314 335 On doubled and quadrupled Fibonacci type sequences https://journals.us.edu.pl/index.php/AMSIL/article/view/16395 <p>In this paper we study a family of doubled and quadrupled Fibonacci type sequences obtained by distance generalization of Fibonacci sequence. In particular we obtain doubled Fibonacci sequence, doubled and quadrupled Padovan sequence and quadrupled Narayana’s sequence. We give a binomial direct formula for these sequences using graph methods, and also we derive a number of identities. Moreover, we study matrix generators of these sequences and determine connections with the Pascal’s triangle.</p> Nur Şeyma Yilmaz Andrzej Włoch Engin Özkan Dominik Strzałka ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2023-11-22 2023-11-22 38 2 336 350 Closure operations on Intuitionistic Linear algebras https://journals.us.edu.pl/index.php/AMSIL/article/view/17252 <p>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 <strong>L</strong> is used to study the unifying properties of some subclasses of the lattice of filters of <strong>L</strong>. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-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.</p> Y. L. Tenkeu Jeufack E. R. Alomo Temgoua O. A. Heubo-Kwegna ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2024-03-20 2024-03-20 38 2 351 380 Report of Meeting. The Twenty-third Katowice-Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 - February 3, 2024 https://journals.us.edu.pl/index.php/AMSIL/article/view/17428 <p>Report of Meeting. The Twenty-third Katowice-Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 - February 3, 2024.</p> Redakcja AMSil ##submission.copyrightStatement## http://creativecommons.org/licenses/by/4.0 2024-04-27 2024-04-27 38 2 381 399