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Abstract: In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $\Gamma$, $\Gamma \not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove that in this case either $\Gamma$ belongs to a well understood family of exceptional graphs or at least $5/12$ of the vertices of $\Gamma$ belong to a regular orbit of $g$. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms $C$ of a connected $3$-valent vertex-transitive graph $\Gamma$ in terms of the number of vertices of $\Gamma$ and the length of a longest orbit of $C$.
Keywords: graphs, automorphism groups, vertex-transitive, regular orbit, cubic, tetravalent
Published in DiRROS: 19.02.2024; Views: 291; Downloads: 109
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Abstract: 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.
Keywords: Stein manifolds, complete holomorphic foliations, density property
Published in DiRROS: 19.02.2024; Views: 254; Downloads: 98
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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: 229; Downloads: 91
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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: 189; Downloads: 58
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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: 290; Downloads: 102
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