1861. Fractional Sobolev spaces with kernel function on compact Riemannian manifoldsAhmed Aberqi, Abdesslam Ouaziz, Dušan Repovš, 2024, original scientific article Abstract: 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. Keywords: nonlinear elliptic problem, fractional Sobolev spaces, kernel function, Lévy-integrability condition, compact Riemannian manifolds, existence of solutions, topological degree theory Published in DiRROS: 19.02.2024; Views: 359; Downloads: 144
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1863. Paratesticular adenocarcinoma : unusual presentation of metastasis of pancreatic cancerJanja Ocvirk, Boštjan Šeruga, 2007, other scientific articles Abstract: 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. Published in DiRROS: 19.02.2024; Views: 296; Downloads: 80
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1865. On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearityMin Zhao, Yueqiang Song, Dušan Repovš, 2024, original scientific article Abstract: 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. Keywords: Hardy-Littlewood-Sobolev nonlinearity, Schrödinger-Kirchhoff equations, variational methods, electromagnetic fields Published in DiRROS: 16.02.2024; Views: 371; Downloads: 157
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1866. Nodal solutions for Neumann systems with gradient dependenceKamel Saoudi, Eadah Alzahrani, Dušan Repovš, 2024, original scientific article Abstract: 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. Keywords: Neumann elliptic systems, gradient dependence, subsolution method, supersolution method, nodal solutions Published in DiRROS: 16.02.2024; Views: 421; Downloads: 158
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1867. Quasi-copulas as linear combinations of copulasGregor Dolinar, Bojan Kuzma, Nik Stopar, 2024, original scientific article Abstract: 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. Keywords: quasi-copulas, copulas, linear combination, affine combination, Minkowski norm Published in DiRROS: 16.02.2024; Views: 350; Downloads: 128
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1869. Resolvability and convexity properties in the Sierpiński product of graphsMichael A. Henning, Sandi Klavžar, Ismael G. Yero, 2024, original scientific article Abstract: Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpiński product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. The Sierpiński metric dimension and the upper Sierpiński metric dimension of two graphs are determined. Closed formulas are determined for Sierpiński products of trees, and for Sierpiński products of two cycles where the second factor is a triangle. We also prove that the layers with respect to the second factor in a Sierpiński product graph are convex. Keywords: Sierpiński product of graphs, metric dimension, trees, convex subgraph Published in DiRROS: 16.02.2024; Views: 382; Downloads: 171
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1870. Tumor vaccinesMojca Frank Bertoncelj, Alojz Ihan, 2006, review article Abstract: Tumor vaccines have several potential advantages over standard anticancer regirrcents. They represent highly specific anticancer therapy. Inducing tumor-specific memory T-lymphocytes, they have potential for long-lived antitumor effects. However, clinical trials, in which cancer patients were vaccinated with tccmor aaccines, have been so far mainly disappointing. There are many reasons for the inefficiency of tumor vaccines. Most cancer antigens are normal self-molecules to which imrrtune tolerance exists. That is why the population of tumor-specific lymphocytes is represented by a small number of low-affinity T-lymphocytes that induce weak antitumor immune response. Simultaneously, tumors evolve many mechanisms to actively evade immune system,what makes them poorly immunogenic or even tolerogenic. Novel irrtmunotherapeutic strategies are directed toward breaking immune tolerance to tumor antigens, enhancing immunogenicity of tumor vaccines and overcoming mechanisms of tumor escape. There are several approaches, unfortunately, all of them still far away from an ideal tumor vaccine that would reject a tumor. Difficulties in the activation of antitumor immune response by tumor vaccines have led to the development of alternative immunotherapeutic strategies that directly focus on effector mechanisms of immune system (adoptive tumor-specific T-lymphocyte transfer and tumor specific monoclonal antibodies). Published in DiRROS: 15.02.2024; Views: 304; Downloads: 83
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