Opuscula Mathematica
Loading...
ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2025
Volume
Vol. 45
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 45 (2025)
Projects
Pages
Articles
Magnetic Dirichlet Laplacian in curved waveguides
(Wydawnictwa AGH, 2025) Barseghyan, Diana; Bernstein, Swanhild; Schneider, Baruch; Zimmermann, Martha Lina
For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable with respect to waveguide deformations. This means that if the waveguide is a straight strip then the spectrum of the Dirichlet Laplacian is purely essential. From the other hand, the perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, the Dirichlet–Laplace operator with a magnetic field is considered. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small but non-local deformations of the waveguide.
Calculation of explicit expressions for the Hopf bifurcation limit cycles in delay-differential equations
(Wydawnictwa AGH, 2025) Gabeiras, José Enríquez; Padial Molina, Juan Francisco
This paper introduces a methodology to derive explicit power series approximations for the limit cycle periodic solutions of the Hopf bifurcation in autonomous discrete delay differential equations (DDE). The procedure extends the methodology introduced by Casal and Freedman in 1980, by providing a detailed algorithm that iteratively performs systematic calculations up to any desired order of approximation, ensuring a specific error tolerance for any nonlinear DDE presenting a Hopf bifurcation. The methodology is applied to three relevant delay-differential models to illustrate its features: a recently introduced car-following mobility model that explains oscillations in road traffic, a SIR epidemic model for propagation of diseases with temporary immunity, and a simplified macroeconomic system to model business cycles.
Upper distance-two domination
(Wydawnictwa AGH, 2025) Hedetniemi, Jason T.; Hedetniemi, Stephen T.; Lewis, Thomas M.
Let $G = (V, E)$ be a graph with vertex set $V$ and edge set $E$. A set $S \subset V$ is a $2$-packing in $G$ if for any two vertices $u,v \in S$, the distance between them satisfies $d(u,v) \gt 2$. The upper $2$-packing number $P_2(G)$ is the maximum cardinality of a $2$-packing in $G$. A set $S \subset V$ is a dominating set for $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ is the minimum cardinality of a dominating set in $G$. A set $S \subset V$ is a distance-$2$ dominating set if for every vertex $v \in V - S$ there exists a vertex $u \in S$ such that $d(u,v) \leq 2$. The upper distance-$2$ domination number $\Gamma_{\leq 2}(G)$ is the maximum cardinality of a minimal distance-$2$ dominating set in $G$. In this paper we establish two families of graphs $G$ for which $P_2(G) = \gamma(G) = \Gamma_{\leq 2}(G)$, which extend several well-known equalities of the form $P_2(G) = \gamma(G)$.
Block Jacobi matrices and Titchmarsh-Weyl function
(Wydawnictwa AGH, 2025) Moszyński, Marcin; Świderski, Grzegorz
We collect some results and notions concerning generalizations for block Jacobi matrices of several concepts, which have been important for spectral studies of the simpler and better known scalar Jacobi case. We focus here on some issues related to the matrix Titchmarsh-Weyl function, but we also consider generalizations of some other tools used by subordinacy theory, including the matrix orthogonal polynomials, the notion of finite cyclicity, a variant of a notion of nonsubordinacy, as well as Jitomirskaya-Last type semi-norms. The article brings together some issues already known, our new concepts, and also improvements and strengthening of some results already existing. We give simpler proofs of some known facts, or we add details usually omitted in the existing literature. The introduction contains a separate part devoted to a brief review of the main spectral analysis methods used so far for block Jacobi operators.
Integral representation of solutions to Dirac systems
(Wydawnictwa AGH, 2025) Rzepnicki, Łukasz
We introduce a novel integral form for a fundamental set of solutions to one-dimensional Dirac systems with an integrable potential and spectral parameter $\mu \in \mathbb{C}$. This method enables the construction of solutions that are analytic in $\mu$ within the half-plane $\operatorname{Im} \mu\gt -r$, $r\geq 0$ and $|\mu| \to \infty$. Consequently, we derive estimates for the solutions that remain valid not just within a horizontal strip but throughout the entire half-plane.