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

In this work the Brouwer fixed point theorem for a triangle was proved by two methods based on the Sperner Lemma. One of the two proofs of Sperner’s Lemma given in the paper was carried out using the so-called index.
</abstract>

On the Two-Dimensional Version of the Sperner Lemma and Brouwer’s Theorem

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

Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.
</abstract>

On Generalized Jacobsthal and Jacobsthal–Lucas Numbers

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

In the present work, a new sequence of quaternions related to the Gaussian Bronze numbers is defined and studied. Binet’s formula, generating function and certain properties and identities are provided. Tridiagonal matrices are considered to determine the general term of this sequence.
</abstract>

University of Trás-os-Montes e Alto Douro

On Quaternion Gaussian Bronze Fibonacci Numbers

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

We consider the Levi–Civita equation
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0009_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>g</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>h</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>g</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>h</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></math>
<tex-math>f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right)</tex-math>
</alternatives>
</disp-formula>
for unknown functions <italic>f, g</italic>1, <italic>g</italic>2, <italic>h</italic>1, <italic>h</italic>2 : <italic>S →</italic> ℂ, where <italic>S</italic> is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that <italic>f</italic> is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function <italic>f</italic> is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.
</abstract>

A Levi–Civita Equation on Monoids, Two Ways

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

It is well-known, as follows from the Stirling’s approximation <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0006_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>n</mi><mo>!</mo><mo>∼</mo><msqrt><mrow><mn>2</mn><mi>π</mi><mi>n</mi><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>/</mo><mi>e</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>n</mi></msup></mrow></msqrt></mrow></math>
<tex-math>n! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}}</tex-math>
</alternatives>
</inline-formula>, that <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0006_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mroot><mrow><mi>n</mi><mo>!</mo><mo>/</mo><mi>n</mi><mo>→</mo><mn>1</mn><mo>/</mo><mi>e</mi></mrow><mi>n</mi></mroot></mrow></math>
<tex-math>\root n \of {n!/n \to 1/e}</tex-math>
</alternatives>
</inline-formula>. A generalization of this limit is (11<italic>s</italic>· 22<italic>s</italic>· · · <italic>nns</italic>)1<italic>/ns</italic>+1 <italic>· n−</italic>1<italic>/</italic>(<italic>s</italic>+1) <italic>→ e−</italic>1<italic>/</italic>(<italic>s</italic>+1)2 which was established by N. Schaumberger in 1989 (see [8]). The aim of this work is to establish a new generalization that is in fact an improvement of Schaumberger’s formula for a general sequence <italic>An </italic>of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers.
</abstract>

A Further Generalization of <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0006_eq_001.png"/>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mrow><mo>lim</mo></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mroot><mrow><mi>n</mi><mo>!</mo><mo>/</mo><mi>n</mi></mrow><mi>n</mi></mroot><mo>=</mo><mn>1</mn><mo>/</mo><mi>e</mi></mrow></math>
<tex-math>{\lim _{n \to \infty }}\root n \of {n!/n} = 1/e</tex-math>
</alternatives>
</inline-formula>

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

Let <italic>p</italic><italic>n</italic> be the <italic>n</italic> th prime number. In this note, we study strictly increasing sequences of positive integers <italic>A</italic><italic>n</italic> such that the limit lim<italic>n</italic>→∞ (<italic>A</italic>1<italic>A</italic>2 <italic>· · · A</italic><italic>n</italic>)1<italic>/pn</italic> = <italic>e</italic> holds. This limit formula is in fact a generalization of some previously known results. Furthermore, some other generalizations are established.
</abstract>

General Limit Formulae Involving Prime Numbers

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

The paper deals with odometers (i.e. adding machines) of general type. We give a characterization of self-conjugacies of odometers which enables us to present an elementary proof of a classification of odometers given by Buescu and Stewart in [2]. The paper might also serve as a very quick introduction to odometers.
</abstract>

Classification of Odometers: A Short Elementary Proof

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

Let <italic>S</italic>, <italic>H</italic>, <italic>X</italic> be groups. For two given biadditive functions <italic>A</italic> : <italic>S</italic>2 → <italic>X</italic>, <italic>B</italic> : <italic>H</italic>2 → <italic>X</italic> and for two unknown mappings <italic>T</italic> : <italic>S</italic> → <italic>H</italic>, <italic>g</italic> : <italic>S</italic> → <italic>S</italic> we will study the functional equation
<italic>B</italic>(<italic>T</italic> (<italic>x</italic>), <italic>T</italic> (<italic>y</italic>)) = <italic>A</italic>(<italic>x, g</italic>(<italic>y</italic>)), <italic>x, y</italic> ∈ <italic>S,</italic>
which is a generalization of the orthogonality equation in Hilbert spaces.
</abstract>

A Functional Equation with Biadditive Functions

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

In this note, we present an extension of the celebrated Abel– Liouville identity in terms of noncommutative complete Bell polynomials for generalized Wronskians. We also characterize the range equivalence of <italic>n</italic>-dimensional vector-valued functions in the subclass of <italic>n</italic>-times differentiable functions with a nonvanishing Wronskian.
</abstract>

Department of Mathematics, Faculty of Education

An Extension of the Abel–Liouville Identity

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

A variation of ordered trees, where each rightmost edge might be marked or not, if it does not lead to an endnode, is investigated. These marked ordered trees were introduced by E. Deutsch et al. to model skew Dyck paths. We study the number of deepest nodes in such trees. Explicit generating functions are established and the average number of deepest nodes, which approaches <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0015_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></math>
<tex-math>{5 \over 3}</tex-math>
</alternatives>
</inline-formula> when the number of nodes gets large. This is to be compared to standard ordered trees where the average number of deepest nodes approaches 2.
</abstract>

Deepest Nodes in Marked Ordered Trees

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

Let <italic>A</italic> and <italic>B</italic> be two unital Banach algebras and 𝔘 = <italic>A</italic> × <italic>B</italic>. We prove that the bilinear mapping <italic>φ</italic>: 𝔘 → ℂ is a bi-Jordan homomorphism if and only if <italic>φ</italic> is unital, invertibility preserving and jointly continuous. Additionally, if A is commutative, then <italic>φ</italic> is a bi-homomorphism.
</abstract>

Gleason–Kahane–Żelazko Theorem for Bilinear Maps

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

In this paper we consider spaces of weight square-integrable and harmonic functions <italic>L</italic>2<italic>H</italic>(Ω, <italic>µ</italic>). Weights <italic>µ</italic> for which there exists reproducing kernel of <italic>L</italic>2<italic>H</italic>(Ω, <italic>µ</italic>) are named ’admissible weights’ and such kernels are named ’harmonic Bergman kernels’. We prove that if only weight of integration is integrable in some negative power, then it is admissible. Next we construct a weight <italic>µ</italic> on the unit circle which is non-admissible and using Bell-Ligocka theorem we show that such weights exist for a large class of domains in ℝ2. Later we conclude from the classical result of reproducing kernel Hilbert spaces theory that if the set {<italic>f</italic> ∈ <italic>L</italic>2<italic>H</italic>(Ω, <italic>µ</italic>)<italic>|f</italic>(<italic>z</italic>) = <italic>c</italic>} for admissible weight <italic>µ</italic> is non-empty, then there is exactly one element with minimal norm. Such an element in this paper is called ’a minimal (<italic>z, c</italic>)-solution in weight <italic>µ</italic> of Laplace’s equation on Ω’ and upper estimates for it are given.
</abstract>

On Weights Which Admit Harmonic Bergman Kernel and Minimal Solutions of Laplace’s Equation

AHEAD OF PRINT

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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 