Quaestiones Mathematicae

Words over a finite alphabet avoiding 1243

Toufik Mansour — 2024-09-20

In this paper, we establish a system of recurrence relations and an algorithm for finding the generating function A_k(x) for the number of words over alphabet [k] of length n that avoid 1243, for arbitrary k. In particular, we present an explicit formula for the generating function _Ak(x) for 1 ≤ k ≤ 10.

On conjugacy of additive actions in the affine Cremona group

Ivan Arzhantsev — 2024-09-20

An additive action on an irreducible algebraic variety X is an effective action Gn/a × X → X with an open orbit of the vector group Gn/a. Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra.

Uniqueness of second-order elliptic operators with unbounded and degenerate coefficients in L₁-spaces

Tan Duc Do — 2024-09-20

Let d ∈ N. Let C = (c_kl)_1≤k,l≤d ∈ W^1,∞_loc (R^d,R^d×d), W = (w_k)_1≤k≤d ∈ W^1,∞_loc (R^d,R^d) and V ∈ _L∞,loc(R^d,R). Consider the formal second-order differential operator

Au = −div (C ∇u) +W · ∇u + V u

in L₁(R^d). We show that the closure of (A,C^∞_c (R^d)) is quasi-m-accretive under certain conditions on the coefficients.

A study on approximate controllability of linear impulsive equations in Hilbert spaces

Nazim I. Mahmudov — 2024-09-20

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in
terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results.

A lemma on C₀-semigroups on time scales and approximate controllability of the heat dynamic equation

Martin Bohner — 2024-09-20

In this paper, we present a lemma that allows us to characterize a broad class of C₀-semigroups on time scales, which can be applied to prove existence and uniqueness of solutions of systems of partial differential equations where the time domain is a time scale. The result obtained is applied to study the controllability of the heat equation on time scales. The lemma presented in this paper can be seen as a unification of the one proved by H. Leiva in [28], for semigroups in [0,∞).

Chen-Chvátal’s conjecture for graphs with restricted girth

Luis Pedro Montejano — 2024-09-20

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chv´atal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chv´atal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

Ambrosetti-Prodi type results for elliptic equations with nonlinear gradient terms on an exterior domain

Jiao Zhao — 2024-09-20

The aim of this paper is to establish an Ambrosetti-Prodi type result for the problem

−Δu = K(|x|)f(|x|, u, |∇u|) + sφ, x ∈ Ω,
αu + β ∂u/∂n |∂Ω = 0,
lim_|x|→∞ u(x) = 0,

where s ∈ R is a parameter, Ω = {x ∈ R^N : |x| > r0}, N ≥ 3, K : [r0,∞) → [0,+∞) is continuous and satisfies ꭍ^∞/_r0r^N−1K(r)dr < ∞. f : [r0,∞) × R × [0,+∞) → R is continuous. φ ∈ C(Ω) with φ ≩ 0 in Ω.

The isomorphism problem for rational group algebras of finite metacyclic nilpotent groups

Angel García-Blázquez — 2024-09-20

We prove that if G and H are finite metacyclic groups with isomorphic rational group algebras and one of them is nilpotent, then G and H are isomorphic.

A generalization of Riesz* homomorphisms on order unit spaces

Florian Boisen — 2024-09-20

Riesz homomorphisms on vector lattices have been generalized to Riesz* homomorphisms on ordered vector spaces by van Haandel using a condition on sets of finitely many elements. Van Haandel attempted to prove that it suffices to take sets of two elements. We show that this is not true, in general. The description by two elements motivates to introduce mild Riesz* homomorphisms. We investigate their properties on order unit spaces, where the geometry of the dual cone plays a crucial role. Hereby, we mostly focus on the finite-dimensional case.

A fully discrete scheme on piecewise-equidistant mesh for singularly perturbed delay integro-differential equations

Musa Cakir — 2024-09-20

This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems.

On mid (p, r)-compact operators

Salthiel Malesela Maepa — 2024-09-20

Let 1 ≤ p ≤ ∞ and 1 ≤ r ≤ p^∗ where p^∗ is the conjugate index of p. We introduce and study mid (p, r)-compact sets and operators. We begin by introducing and defining the mid (p, r)-compact subsets of a Banach space X and the mid (p, r)- compact operators between Banach spaces X and Y . The set of mid (p, r)-compact operators between Banach spaces X and Y is denoted by K^mid/ _(p,r)(X, Y ). We prove that the ideal (K^mid/_(p,r)(X,Y),κ^mid/ _(p,r)(·)) is a quasi-Banach operator ideal. We also introduce and study (p, r)-limited subsets in Banach spaces. We prove that every mid (p, r)-compact subset of X is (p, r)-limited and that the set K^mid/ _(p,r)(X, Y ) consists of (p, r)-limited sets.