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.

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.

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.

We consider the Levi–Civita equation
f(xy) = g₁(x)h₁(y) + g₂(x)h₂(y)
for unknown functions f, g_1, g_2, h_1, h₂ : S→ℂ, where S 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 f 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 f 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.

It is well-known, as follows from the Stirling's approximation n! ~ \sqrt{2πn}(n/e)ⁿ, that \sqrt[n]{n!}/n→1/e. A generalization of this limit is (1^{1^s} · 2^{2^s} ··· n^{n^s})^{1/n^{s+1}} · n^-1/(s+1)→e^{-1/(s+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 A_n 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.

Let p_n be the nth prime number. In this note, we study strictly increasing sequences of positive integers A_n such that the limit lim_n→∞(A₁A₂···A_n)^1/p_n = e holds. This limit formula is in fact a generalization of some previously known results. Furthermore, some other generalizations are established.

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.

Let S, H, X be groups. For two given biadditive functions A:S²→X, B:H²→X and for two unknown mappings T:S→H, g:S→S we will study the functional equation
B(T(x),T(y)) = A(x,g(y)), x,y∈S,
which is a generalization of the orthogonality equation in Hilbert spaces.

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 n-dimensional vector-valued functions in the subclass of n-times differentiable functions with a nonvanishing Wronskian.

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 5/3 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.

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

In this paper we consider spaces of weight square-integrable and harmonic functions L²H(Ω,μ). Weights μ for which there exists reproducing kernel of L²H(Ω,μ) 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 μ 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 {f∈L²H(Ω,μ)| f(z) = c} for admissible weight μ is non-empty, then there is exactly one element with minimal norm. Such an element in this paper is called ’a minimal (z,c)-solution in weight μ of Laplace’s equation on Ω’ and upper estimates for it are given.