1. Domains without parabolic minimal submanifolds and weakly hyperbolic domainsFranc Forstnerič, 2023, izvirni znanstveni članek Povzetek: We show that if $\Omega$ is an $m$-convex domain in $\mathbb{R}^n$ for some $2 \le m < n$ whose boundary $b\Omega$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $\Omega$ does not contain any immersed parabolic minimal submanifolds of dimension $\ge m$. In particular, if $M$ is a properly embedded non-flat minimal hypersurface in $\mathbb{R}^n$ with a tubular neighbourhood of positive radius, then every immersed parabolic hypersurface in $\mathbb{R}^n$ intersects $M$. In dimension $n=3$, this holds if $M$ has bounded Gaussian curvature function. We also introduce the class of weakly hyperbolic domains $\Omega$ in $\mathbb{R}^n$, characterised by the property that every conformal harmonic map $\mathbb{C} \to \Omega$ is constant, and we elucidate their relationship with hyperbolic domains, and domains without parabolic minimal surfaces. Ključne besede: minimal surfaces, m-plurisubharmonic functions, hyperbolic domain Objavljeno v DiRROS: 10.04.2024; Ogledov: 68; Prenosov: 29 Celotno besedilo (242,50 KB) Gradivo ima več datotek! Več... |
2. The Waring problem for matrix algebras, IIMatej Brešar, Peter Šemrl, 2023, izvirni znanstveni članek Povzetek: Let $f$ be a noncommutative polynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$, then every trace zero $n\times n$ matrix over $F$ can be written as $\alpha_1 A_1+\alpha_2A_2+\alpha_3A_3$ for some $A_i$ in the image of $f$ in $M_n(F)$. Ključne besede: Waring problem, noncommutatative polynomials, matrix algebras Objavljeno v DiRROS: 10.04.2024; Ogledov: 58; Prenosov: 22 Celotno besedilo (133,03 KB) Gradivo ima več datotek! Več... |
3. Variety of mutual-visibility problems in graphsSerafino Cicerone, Gabriele Di Stefano, Lara Drožđek, Jaka Hedžet, Sandi Klavžar, Ismael G. Yero, 2023, izvirni znanstveni članek Povzetek: If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families. Ključne besede: mutual-visibility, total mutual-visibility, dual mutual-visibility number, outer mutual-visibility, grid graphs, torus graphs, computational complexity Objavljeno v DiRROS: 10.04.2024; Ogledov: 72; Prenosov: 33 Celotno besedilo (456,36 KB) Gradivo ima več datotek! Več... |
4. On a definition of logarithm of quaternionic functionsGraziano Gentili, Jasna Prezelj, Fabio Vlacci, 2023, izvirni znanstveni članek Povzetek: For a slice-regular quaternionic function $f$, the classical exponential function ${\mathrm exp} f$ is not slice-regular in general. An alternative definition of an exponential function, the $\ast$-exponential ${\mathrm exp}_\ast$, was given in the work by Altavilla and de Fabritiis (2019): if $f$ is a slice-regular function, then ${\mathrm exp}_\ast f$ is a slice-regular function as well. The study of a $\ast$-logarithm ${\mathrm log}_\ast f$ of a slice-regular function $f$ becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a ${\mathrm log}_\ast f$ depends only on the structure of the zero set of the vectorial part $f_v$ of the slice-regular function $f = f_0 + f_v$, besides the topology of its domain of definition. We also show that, locally, every slice-regular nonvanishing function has a $\ast$-logarithm and, at the end, we present an example of a nonvanishing slice-regular function on a ball which does not admit a $\ast$-logarithm on that ball. Ključne besede: regular functions over quaternions, quaternionic logarithm of slice-regular functions Objavljeno v DiRROS: 10.04.2024; Ogledov: 57; Prenosov: 25 Celotno besedilo (425,44 KB) Gradivo ima več datotek! Več... |
5. Critical edges in Rips complexes and persistencePeter Goričan, Žiga Virk, 2023, izvirni znanstveni članek Povzetek: We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X, d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function $d$ When $d$ attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of $d$. We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of $d$ that induce a change in persistence. We conclude with a proof that each locally isolated minimum of $d$ can be detected through persistent homology with selective Rips complexes. The results of this paper offer the first interpretation of critical scales of persistent homology (obtained via Rips complexes) for general compact metric spaces. Ključne besede: persistent homology, Rips complex, critical simplex, reconstruction result Objavljeno v DiRROS: 10.04.2024; Ogledov: 52; Prenosov: 23 Celotno besedilo (579,74 KB) Gradivo ima več datotek! Več... |
6. Graphs with equal Grundy domination and independence numberGábor Bacsó, Boštjan Brešar, Kirsti Kuenzel, Douglas F. Rall, 2023, izvirni znanstveni članek Povzetek: The Grundy domination number, ${\gamma_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $jKljučne besede: Grundy domination, independence number, upper domination number, bipartite graphs Objavljeno v DiRROS: 09.04.2024; Ogledov: 59; Prenosov: 36 Celotno besedilo (803,91 KB) Gradivo ima več datotek! Več... |
7. Wandering domains arising from Lavaurs maps with Siegel disksMatthieu Astorg, Luka Boc Thaler, Han Peters, 2023, izvirni znanstveni članek Povzetek: The first example of polynomial maps with wandering domains was constructed in 2016 by the first and last authors, together with Buff, Dujardin and Raissy. In this paper, we construct a second example with different dynamics, using a Lavaurs map with a Siegel disk instead of an attracting fixed point. We prove a general necessary and sufficient condition for the existence of a trapping domain for nonautonomous compositions of maps converging parabolically towards a Siegel-type limit map. Constructing a skew-product satisfying this condition requires precise estimates on the convergence to the Lavaurs map, which we obtain by a new approach. We also give a self-contained construction of parabolic curves, which are integral to this new method. Ključne besede: Fatou sets, holomorphic dynamics, parabolic implosion, polynomial mappings, skew-products, wandering Fatou components, parabolic curves, nonautonomous dynamics Objavljeno v DiRROS: 09.04.2024; Ogledov: 63; Prenosov: 27 Celotno besedilo (1,55 MB) Gradivo ima več datotek! Več... |
8. Injective coloring of graphs revisitedBoštjan Brešar, Babak Samadi, Ismael G. Yero, 2023, izvirni znanstveni članek Povzetek: An open packing in a graph $G$ is a set $S$ of vertices in $G$ such that no two vertices in $S$ have a common neighbor in $G$. The injective chromatic number $\chi_i(G)$ of $G$ is the smallest number of colors assigned to vertices of ▫$G$▫ such that each color class is an open packing. Alternatively, the injective chromatic number of $G$ is the chromatic number of the two-step graph of $G$, which is the graph with the same vertex set as $G$ in which two vertices are adjacent if they have a common neighbor. The concept of injective coloring has been studied by many authors, while in the present paper we approach it from two novel perspectives, related to open packings and the two-step graph operation. We prove several general bounds on the injective chromatic number expressed in terms of the open packing number. In particular, we prove that $\chi_i(G) \ge \frac{1}{2}\sqrt{\frac{1}{4}+\frac{2m-n}{\rho^{o}}}$ holds for any connected graph $G$ of order $n\ge 2$, size $m$, and the open packing number ${\rho^{o}}$, and characterize the class of graphs attaining the bound. Regarding the well known bound $\chi_i(G)\ge \Delta(G)$, we describe the family of extremal graphs and prove that deciding when the equality holds (even for regular graphs) is NP-complete, solving an open problem from an earlier paper. Next, we consider the chromatic number of the two-step graph of a graph, and compare it with the clique number and the maximum degree of the graph. We present two large families of graphs in which $\chi_i(G)$ equals the cardinality of a largest clique of the two-step graph of $G$. Finally, we consider classes of graphs that admit an injective coloring in which all color classes are maximal open packings. We give characterizations of three subclasses of these graphs among graphs with diameter 2, and find a partial characterization of hypercubes with this property. Ključne besede: two-step graph of a graph, injective coloring, open packing, hypercubes Objavljeno v DiRROS: 09.04.2024; Ogledov: 61; Prenosov: 43 Celotno besedilo (460,72 KB) Gradivo ima več datotek! Več... |
9. The core of a vertex-transitive complementary prismMarko Orel, 2023, izvirni znanstveni članek Povzetek: The complementary prism $\Gamma \overline{\Gamma}$ is obtained from the union of a graph $\Gamma$ and its complement $\overline{\Gamma}$ where each pair of identical vertices in $\Gamma$ and $\overline{\Gamma}$ is joined by an edge. It generalizes the Petersen graph, which is the complementary prism of the pentagon. The core of a vertex-transitive complementary prism is studied. In particular, it is shown that a vertex-transitive complementary prism $\Gamma \overline{\Gamma}$ is a core, i.e. all its endomorphisms are automorphisms, whenever $\Gamma$ is a core or its core is a complete graph. Ključne besede: graph homomorphism, complementary prism, self-complementary graph, vertex-transitive graph, core Objavljeno v DiRROS: 09.04.2024; Ogledov: 54; Prenosov: 25 Celotno besedilo (309,75 KB) Gradivo ima več datotek! Več... |
10. Domination and independence numbers of large 2-crossing-critical graphsVesna Iršič, Maruša Lekše, Miha Pačnik, Petra Podlogar, Martin Praček, 2023, izvirni znanstveni članek Povzetek: After 2-crossing-critical graphs were characterized in 2016, their most general subfamily, large 3-connected 2-crossing-critical graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for their domination and independence number. Ključne besede: crossing-critical graphs, domination number, independence number Objavljeno v DiRROS: 09.04.2024; Ogledov: 48; Prenosov: 26 Celotno besedilo (393,09 KB) Gradivo ima več datotek! Več... |