Quaestiones Mathematicae

On a cubic family of Thue equations involving Fibonacci numbers and powers of two

Ingrid Vukusic — 2023-03-07

In this paper we completely solve the family of parametrised Thue equations
X(X - F_nY )(X - 2ⁿY ) - Y³ = ± 1,
where F_n is the n-th Fibonacci number. In particular, for any integer n 3 the
Thue equation has only the trivial solutions (±1, 0), (0, ∓1), ∓(F_n,1), ∓(2ⁿ,1).

Approximate local isometries of derivative hardy spaces

A. Jiménez-Vargas — 2023-03-07

For any 1 ≤ p ≤ ∞, let S^p(ⅅ) be the space of holomorphic functions ƒ on ⅅ such that ƒ′ belongs to the Hardy space H^p(ⅅ), with the norm ∥ƒ∥_∑ = ∥f∥_∞+∥ƒ′∥_p. We prove that every approximate local isometry of S^p(ⅅ) is a surjective isometry and that every approximate 2-local isometry of S^p(ⅅ) is a surjective linear isometry. As a consequence, we deduce that the sets of isometric re ections and generalized bi-circular projections on S^p(ⅅ) are also topologically re exive and 2-topologically reflexive.

A new lower bound on the total domination number of a graph

Majid Henning Henning — 2023-03-07

A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The total domination number, γ_t(G), is the minimum cardinality of a total dominating set of G. Chellali and Haynes [J. Combin. Math. Combin. Comput. 58 (2006), 189-193] showed that if T is a nontrivial tree of order n, with ℓ leaves, then γ_t(T) = ⌈(n - ℓ+2)/2⌉. In this paper, we first characterize all trees T of order n with ℓ leaves satisfying γ_t(T) = ⌈(n - ℓ+2)=2⌉. We then generalize this result to connected graphs and show that if G is a connected graph of order n ≥ 2 with k ≥ 0 cycles and ℓ leaves, then γ_](G) ≥ ⌈(n-ℓ+2)=2⌉ - k. We also characterize the graphs G achieving equality for this new bound.

Weak limited sets and operators on Banach lattices

Farid Afkir — 2023-03-07

In this paper, we prove that an operator T : E → F, between two Banach lattices, maps order intervals onto weak limited sets if and only if the modulus jSTj exists and is Dunford-Pettis for every Dunford-Pettis operator S : F → c₀. Next, we establish that a Banach lattice E does not contain any isomorphic copy of ℓ¹ if and only if the order intervals of E are weak limited and the norm of E′ is order continuous. We also investigate the domination problem of the class of weak limited operators.

Multidimensional Kantorovich modifications of exponential sampling series

Tuncer Acar — 2023-03-07

This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section of the function involved. Approximation behaviour of newly constructed operators is in-vestigated at continuity points for log-uniformly continuous functions. The rate of convergence of the series is presented for the same functions by means of logarithmic modulus of continuity. A Voronovskaja type theorem is also presented by means of Mellin derivatives.

Remarks on star weakly Lindelöf spaces

Yan-Kui Song — 2023-03-07

Let P be a topological property. A space X is said to be star P if whenever U is an open cover of X, there exists a subspace A ⊆ X with property P such that X = St(A, 𝒰), where St(A, 𝒰) = ⋃ {U ∈ 𝒰 : U ∩ A ≠ ∅}. In this paper, we investigate the relationships among star Lindelöf spaces, star almost Lindelöf spaces and star weakly Lindelöf spaces, and also study topological properties of star weakly Lindelöf spaces.

Digital Jordan curves and surfaces with respect to a graph connectedness

Josef Šlapal — 2023-03-08

We introduce a graph connectedness induced by a given set of paths of the same length. We focus on the 2-adjacency graph on the digital line ℤ with a certain set of paths of length n for every positive integer n. The connectedness in the strong product of two and three copies of the graph is used to define digital Jordan curves and digital Jordan surfaces, respectively. Such definitions build on an edge-to-edge tiling with triangles in the digital plane and a face-to-face tiling by cubes, prisms and pyramids in the (3D) digital space, respectively.

On S-1-absorbing prime and weakly S-1-absorbing prime ideals

Najib Mahdou — 2023-03-08

Let A be a commutative ring with identity and S be a multiplicative subset of A. In this paper, we introduce and study the notions of S-1-absorbing and weakly S-1-absorbing prime ideals as generalizations of the notion of prime ideals. We define a proper ideal I disjoint with S to be an S-1-absorbing (resp. a weakly S-1-absorbing) prime ideal if there exists s ∈ S such that for all nonunit elements a, b, c ∈ A such that abc ∈ I (resp. 0 ≠ abc ∈ I), we have sab ∈ I or sc ∈ I. Several properties and characterizations of S-1-absorbing prime and weakly S-1-absorbing prime ideals are given. Moreover, we study the transfer of the above properties to some constructions of rings such as trivial ring extensions and amalgamation of rings along an ideal.

On q-analogs of Struve functions

Karima M. Oraby — 2023-03-08

In this paper, we introduce q-analogs of the Struve-Bessel functions in terms of q-sine functions. We study the reality of the zeros of the q-Struve Bessel functions and their properties. Also, the interlacing of the zeros of the q-Struve
Bessel functions and the zeros of corresponding q-Bessel functions are proved under certain conditions on q. We investigate Turán type inequalities for the second and third q-Struve Bessel functions.

Equivalent almost periodic functions in terms of the new property of almost equality

Juan Matías Sepulcre — 2023-03-08

In this paper we introduce the notion of almost equality (or, more specifically, almost equality by translations) of complex functions of an unrestricted real variable in terms of the new concept of ε-translation number of a function with respect to other one, which is inspired by Bohr's notion of ε-translation number associated with an almost periodic function. We develop the main properties of this new class of functions and obtain a characterization through a very important equivalence relation which we introduced in previous papers in the context of the almost periodicity.

The multiplicity of left-to-right maxima in geometrically distributed words

A. Knopfmacher — 2023-03-08

For fixed m ≥ 1, we study the number of weak left-to-right maxima which occur exactly m times in words whose letters satisfy a geometric distribution. First, we find the generating function and two exact expressions for the mean. Thereafter we use Rice's integrals to derive an asymptotic formula as n tends to infinity for the average in random geometric words of length n for each fixed value of m.

Normal ordering in the shift algebra and the dual of Spivey's identity

Matthias Schork — 2023-03-08

Spivey's Bell number identity involving Stirling numbers of the second kind was derived by Spivey considering set partitions. Since its original derivation it has been derived (and generalized) by many authors using different techniques. In particular, Katriel gave a proof using normal ordering in the Weyl algebra. Mező derived the dual of Spivey's identitiy for factorials involving (unsigned) Stirling numbers of the first kind considering permutations and cycles. The latter identity has also been been derived (and generalized) in different fashions. What is lacking in literature is a proof using normal ordering in the spirit of Katriel's proof of Spivey's identity. In the present work, this gap is filled by providing a new proof of the dual of Spivey's identity using normal ordering in the shift algebra.

Geometries with non-commutative joins and their application to near-vector spaces

K-T. Howell — 2023-03-08

In this paper we add to the theory of the geometry of near-structures. More specifically, we define a near-linear space, prove some properties and show that by adding some axioms we arrive at a nearaffine space, as defined by André. As a highlight, we use some of the geometric results to prove an open problem in near-vector space theory, namely that a subset of a near-vector space that is closed under addition and scalar multiplication is a subspace. We end the paper with a first look at the projections of nearaffine spaces.