Thermodynamic Formalism Methods In the Theory of Iteration of Mappings in Dimension One, Real and Complex

<abstract xmlns="http://www.w3.org/1999/xhtml">The aim of this paper is to characterize the solutions Φ : <italic>G → M</italic>2(ℂ) of the following matrix functional equations
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0025_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>σ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac><mo>=</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mo>∈</mo><mi>G</mi><mo>,</mo></mrow></math><tex-math>{{\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,</tex-math></alternatives></disp-formula>
and
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0025_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>σ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac><mo>=</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mo>∈</mo><mi>G</mi><mo>,</mo></mrow></math><tex-math>{{\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,</tex-math></alternatives></disp-formula>
where <italic>G</italic> is a group that need not be abelian, and <italic>σ</italic> : <italic>G → G</italic> is an involutive automorphism of <italic>G</italic>. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.</abstract>

A Variant of D’alembert’s Matrix Functional Equation

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>e</italic>. Furthermore, we present a generalization of the primorial.</abstract>

A Note on the Asymptotic Behavior of the Distribution Function of a General Sequence

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of <italic>formal functional</italic> equations. We give a characterization of such families where the set of multipliers (first coefficients) <italic>σ</italic> of its members <italic>F</italic> (<italic>x</italic>) = <italic>σx</italic> + . . . is infinite, in particular of such families which are maximal with respect to inclusion, so called families of <italic>type I</italic>. 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.</abstract>

Families of Commuting Formal Power Series and Formal Functional Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">If <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>(</mo><mrow><msub><mrow><mi>μ</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup></mrow></math><tex-math>\left( {{\mu _n}} \right)_{n = 1}^\infty</tex-math></alternatives></inline-formula> are positive measures on a measurable space (<italic>X,</italic> Σ) and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>(</mo><mrow><msub><mrow><mi>v</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup></mrow></math><tex-math>\left( {{v_n}} \right)_{n = 1}^\infty</tex-math></alternatives></inline-formula> are elements of a Banach space 𝔼 such that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mrow><mrow><mo>‖</mo><mrow><msub><mrow><mi>v</mi></mrow><mi>n</mi></msub></mrow><mo>‖</mo></mrow><msub><mrow><mi>μ</mi></mrow><mi>n</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow><mo>&lt;</mo><mo>∞</mo></mrow></math><tex-math>\sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} &lt; \infty</tex-math></alternatives></inline-formula>, then <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ω</mi><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mrow><msub><mrow><mi>v</mi></mrow><mi>n</mi></msub><msub><mrow><mi>μ</mi></mrow><mi>n</mi></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></mrow></math><tex-math>\omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)}</tex-math></alternatives></inline-formula> defines a vector measure of bounded variation on (<italic>X,</italic> Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (<italic>X,</italic> 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.</abstract>

On the Radon-Nikodym Property for Vector Measures and Extensions of Transfunctions

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>p</italic>-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [<italic>Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral</italic>, Revista Integración, temas de matemáticas <bold>36</bold> (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction.</abstract>

Department of Mathematics and Computer Science

Generalized Fractional Inequalities of the Hermite–Hadamard Type for Convex Stochastic Processes

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>P</italic>𝓁<italic>n</italic> is expressed as a linear combination of <italic>P</italic><italic>mn</italic>. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter <italic>R</italic> = 2<italic>r</italic>, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.</abstract>

Summing a Family of Generalized Pell Numbers

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

Department of Mathematics, Art and Science Faculty

Generalized Tetranacci Hybrid Numbers

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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 