1861. Strong edge geodetic problem on complete multipartite graphs and some extremal graphs for the problemSandi Klavžar, Eva Zmazek, 2024, izvirni znanstveni članek Povzetek: A set of vertices $X$ of a graph $G$ is a strong edge geodetic set if to any pair of vertices from $X$ we can assign one (or zero) shortest path between them such that every edge of $G$ is contained in at least one on these paths. The cardinality of a smallest strong edge geodetic set of $G$ is the strong edge geodetic number ${\rm sg_e}(G)$ of $G$. In this paper, the strong edge geodetic number of complete multipartite graphs is determined. Graphs $G$ with ${\rm sg_e}(G) = n(G)$ are characterized and ${\rm sg_e}$ is determined for Cartesian products $P_n\,\square\, K_m$. The latter result in particular corrects an error from the literature. Ključne besede: strong edge geodetic problem, complete multipartite graph, edge-coloring, Cartesian product of graphs Objavljeno v DiRROS: 19.02.2024; Ogledov: 361; Prenosov: 151
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1862. Complete nonsingular holomorphic foliations on Stein manifoldsAntonio Alarcón, Franc Forstnerič, 2024, izvirni znanstveni članek Povzetek: Let $X$ be a Stein manifold of complex dimension $n \ge 1$ endowed with a Riemannian metric ${\mathfrak g}$. We show that for every integer $k$ with $\left[\frac{n}{2}\right] \le k \le n-1$ there is a nonsingular holomorphic foliation of dimension $k$ on $X$ all of whose leaves are topologically closed and ${\mathfrak g}$-complete. The same is true if $1\le k \left[\frac{n}{2}\right]$ provided that there is a complex vector bundle epimorphism $TX\to X \times \mathbb{C}^{n-k}$. We also show that if $\mathcal{F}$ is a proper holomorphic foliation on $\mathbb{C}^n$ $(n > 1)$ then for any Riemannian metric ${\mathfrak g}$ on $\mathbb{C}^n$ there is a holomorphic automorphism $\Phi$ of $\mathbb{C}^n$ such that the image foliation $\Phi_*\mathcal{F}$ is ${\mathfrak g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property. Ključne besede: Stein manifolds, complete holomorphic foliations, density property Objavljeno v DiRROS: 19.02.2024; Ogledov: 380; Prenosov: 165
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1863. Lower (total) mutual-visibility number in graphsBoštjan Brešar, Ismael G. Yero, 2024, izvirni znanstveni članek Povzetek: Given a graph $G$, a set $X$ of vertices in $G$ satisfying that between every two vertices in $X$ (respectively, in $G$) there is a shortest path whose internal vertices are not in $X$ is a mutual-visibility (respectively, total mutual-visibility) set in $G$. The cardinality of a largest (total) mutual-visibility set in $G$ is known under the name (total) mutual-visibility number, and has been studied in several recent works. In this paper, we propose two lower variants of these concepts, defined as the smallest possible cardinality among all maximal (total) mutual-visibility sets in $G$, and denote them by $\mu^{-}(G)$ and $\mu_t^{-}(G)$, respectively. While the total mutual-visibility number is never larger than the mutual-visibility number in a graph $G$, we prove that both differences $\mu^{-}(G)-\mu_t^{-}(G)$ and $\mu_t^{-}(G)-\mu^{-}(G)$ can be arbitrarily large. We characterize graphs $G$ with some small values of $\mu^{-}(G)$ and $\mu_t^{-}(G)$, and prove a useful tool called the Neighborhood Lemma, which enables us to find upper bounds on the lower mutual-visibility number in several classes of graphs. We compare the lower mutual-visibility number with the lower general position number, and find a close relationship with the Bollobás-Wessel theorem when this number is considered in Cartesian products of complete graphs. Finally, we also prove the NP-completeness of the decision problem related to $\mu_t^{-}(G)$. Ključne besede: mutual-visibility set, mutual-visibility number, total mutual-visibility set, computational complexity Objavljeno v DiRROS: 19.02.2024; Ogledov: 357; Prenosov: 157
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1864. Fractional Sobolev spaces with kernel function on compact Riemannian manifoldsAhmed Aberqi, Abdesslam Ouaziz, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional $p(\cdot, \cdot)$-Laplacian operator. As one of the main tools, topological degree theory is applied. Ključne besede: nonlinear elliptic problem, fractional Sobolev spaces, kernel function, Lévy-integrability condition, compact Riemannian manifolds, existence of solutions, topological degree theory Objavljeno v DiRROS: 19.02.2024; Ogledov: 377; Prenosov: 152
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1866. Paratesticular adenocarcinoma : unusual presentation of metastasis of pancreatic cancerJanja Ocvirk, Boštjan Šeruga, 2007, drugi znanstveni članki Povzetek: Background. Metastatic paratesticular adenocarcinoma from the pancreatic cancer is very rare. To our knowledge, there are less than 20 cases published in the literature. Case report. We experienced a case of paratesticular adenocarcinoma from the primary pancreatic cancer. A 42-year-old man was presented with locoregionally advanced carcinoma of the tail of the pancreas with intraoperatively found liver metastases and with a tumour in the right hemi-scrotum. Ultrasound of the scrotum revealed a paratesticular tumour. A fine needle aspiration biopsy (FNAB) confirmed a poorly differentiated adenocarcinoma and it was in concordance with the diagnosis of the primary tumour. The patient started treatment with chemotherapy with gemcitabine. Unfortunately, he progressed one month later and the treatment was discontinued. Conclusions. Outcome in the adenocarcinoma of the pancreas is dismal. The only possible treatment option for metastatic disease is systemic therapy but the results are disappointing, as in the present case. Objavljeno v DiRROS: 19.02.2024; Ogledov: 312; Prenosov: 81
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1868. On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearityMin Zhao, Yueqiang Song, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: In this article, we deal with the following $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $ M\left([u]_{s,A}^{p}\right)(-\Delta)_{p, A}^{s} u+V(x)|u|^{p-2} u=\lambda\left(\int_\limits{\mathbb{R}^{N}} \frac{|u|^{p_{\mu, s}^{*}}}{|x-y|^{\mu}} \mathrm{d}y\right)|u|^{p_{\mu, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$ where $0 < s < 1 < p$, $ps < N$, $p < q < 2p^{*}_{s,\mu}$, $0 < \mu < N$, $\lambda$ and $k$ are some positive parameters, $p^{*}_{s,\mu}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions $V$, $M$ satisfy the suitable conditions. By proving the compactness results using the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem. Ključne besede: Hardy-Littlewood-Sobolev nonlinearity, Schrödinger-Kirchhoff equations, variational methods, electromagnetic fields Objavljeno v DiRROS: 16.02.2024; Ogledov: 384; Prenosov: 165
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1869. Nodal solutions for Neumann systems with gradient dependenceKamel Saoudi, Eadah Alzahrani, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: We consider the following convective Neumann systems: $\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-\Delta_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+\delta_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) \text{in}\;\Omega,\\ -\Delta _{p_2}u_2+\frac{|\nabla u_2|^{p_2}}{u_2+\delta_2}=f_2(x,u_1,u_2,\nabla u_1,\nabla u_2) \text{in}\;\Omega, \\ |\nabla u_1|^{p_1-2}\frac{\partial u_1}{\partial \eta }=0=|\nabla u_2|^{p_2-2}\frac{\partial u_2}{\partial \eta} \text{on}\;\partial\,\Omega,\end{array}\right.\end{equation*}$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$) with a smooth boundary $\partial\,\Omega, \delta_1, \delta_2 > 0$ are small parameters, $\eta$ is the outward unit vector normal to $\partial\,\Omega, f_1, f_2: \Omega \times \mathbb{R}^2 \times \mathbb{R}^{2N} \rightarrow \mathbb{R}$ are Carathéodory functions that satisfy certain growth conditions, and $\Delta _{p_i}$ ($1< p_i < N,$ for $i=1,2$) are the $p$-Laplace operators $\Delta _{p_i}u_i=\mathrm{div}(|\nabla u_i|^{p_i-2}\nabla u_i)$, for $u_i \in W^{1,p_i}(\Omega).$ In order to prove the existence of solutions to such systems, we use a sub-supersolution method. We also obtain nodal solutions by constructing appropriate sub-solution and super-solution pairs. To the best of our knowledge, such systems have not been studied yet. Ključne besede: Neumann elliptic systems, gradient dependence, subsolution method, supersolution method, nodal solutions Objavljeno v DiRROS: 16.02.2024; Ogledov: 437; Prenosov: 164
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1870. Quasi-copulas as linear combinations of copulasGregor Dolinar, Bojan Kuzma, Nik Stopar, 2024, izvirni znanstveni članek Povzetek: We prove that every quasi-copula can be written as a uniformly converging infinite sum of multiples of copulas. Furthermore, we characterize those quasi-copulas which can be written as a finite sum of multiples of copulas, i.e., that are a linear combination of two copulas. This generalizes a recent result of Fernández-Sánchez, Quesada-Molina, and Úbeda-Flores who considered linear combinations of discrete copulas. Ključne besede: quasi-copulas, copulas, linear combination, affine combination, Minkowski norm Objavljeno v DiRROS: 16.02.2024; Ogledov: 368; Prenosov: 135
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