Opuscula Mathematica
Loading...
ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2024
Volume
Vol. 44
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 44 (2024)
Projects
Pages
Articles
Uniqueness for a class p-Laplacian problems when a parameter is large
(Wydawnictwa AGH, 2024) Alreshidi, Bandar; Hai, D. D.
We prove uniqueness of positive solutions for the problem $-\Delta_{p}u=\lambda f(u)\text{ in }\Omega,\ u=0\text{ on }\partial \Omega,$ where $1\lt p\lt 2$ and $p$ is close to 2, $\Omega$ is bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$, $f:[0,\infty)\rightarrow [0,\infty )$ with $f(z)\sim z^{\beta }$ at $\infty$ for some $\beta \in (0,1)$, and $\lambda$ is a large parameter. The monotonicity assumption on $f$ is not required even for u large.
Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity
(Wydawnictwa AGH, 2024) Cordeiro, Sebastião; Raposo, Carlos; Ferreira, Jorge; Rocha, Daniel; Shahrouzi, Mohammad
This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma.
Local irregularity conjecture for 2-multigraphs versus cacti
(Wydawnictwa AGH, 2024) Grzelec, Igor; Woźniak, Mariusz
A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. A locally irregular colorable multigraph $G$ is any multigraph which admits a locally irregular coloring. We denote by $\textrm{lir}(G)$ the locally irregular chromatic index of a multigraph $G$, which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph $G$. In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph $G$, which is not isomorphic to $K_2$, multigraph $^{2}G$ obtained from $G$ by doubling each edge satisfies $\textrm{lir}(^2G)\leq 2$. We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph $G$ satisfies $\textrm{lir}(G)\leq 3$. At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved.
Two-weight norm inequalities for rough fractional integral operators on Morrey spaces
(Wydawnictwa AGH, 2024) Ho, Kwok-Pun
We establish the two-weight norm inequalities for the rough fractional integral operators on Morrey spaces.
Conditional mean embedding and optimal feature selection via positive definite kernels
(Wydawnictwa AGH, 2024) Jørgensen, Palle E.T.; Song, Myung-Sin; Tian, James
Motivated by applications, we consider new operator-theoretic approaches to conditional mean embedding (CME). Our present results combine a spectral analysis-based optimization scheme with the use of kernels, stochastic processes, and constructive learning algorithms. For initially given non-linear data, we consider optimization-based feature selections. This entails the use of convex sets of kernels in a construction o foptimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm), each choice of a kernel $K$ in turn yields a variety of Hilbert spaces and realizations of features. A novel aspect of our work is the inclusion of a secondary optimization process over a specified convex set of positive definite kernels, resulting in the determination of »optimal« feature representations.