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Povzetek: The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the ▫$\delta$▫-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.
Ključne besede: singular pencil, singular generalized eigenvalue problem, eigenvalue condition number, randomized numerical method, random matrices
Objavljeno v DiRROS: 26.08.2024; Ogledov: 241; Prenosov: 105
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Povzetek: We consider Noether’s problem on the noncommutative rational functions invariant under a linear action of a finite group. For abelian groups the invariant skew-fields are always rational, for solvable group they are rational if the action is well-behaved – given by a so-called complete representation. We determine the groups that admit such representations and call them totally pseudo-unramified. We show that for a solvable group the invariant skew-field is finitely generated. Finally we study totally pseudo-unramified groups and classify totally pseudo-unramified ▫$p$▫-groups of rank at most ▫$5$▫.
Ključne besede: Clifford theory, multiplicity free restrictions, noncommutative Noether’s problem, noncommutative rational invariant, totally unramified groups
Objavljeno v DiRROS: 03.07.2024; Ogledov: 342; Prenosov: 213
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Povzetek: We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in ${\mathbb R}^3$ with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least $2$, for holomorphic null immersions into ${\mathbb C}^n$ with $n \ge 3$, for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any selfdual or anti-self-dual Einstein four-manifold.
Ključne besede: minimal surfaces, Calabi–Yau problem, null curve, Legendrian curve
Objavljeno v DiRROS: 08.04.2024; Ogledov: 444; Prenosov: 195
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Ključne besede: inverse eigenvalue problem, minimum number of distinct eigenvalues, borderings, joins of graphs, paths, cycles, hypercubes
Objavljeno v DiRROS: 18.03.2024; Ogledov: 567; Prenosov: 230
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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: 507; Prenosov: 237
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