MATHEMATICA SCANDINAVICA

Lifts, transfers, and degrees of univariate maps

2022-10-28T12:56:59+02:00

One can compute the local $\mathbb{A}^1$-degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that for endomorphisms of the affine line, one can compute the local $\mathbb{A}^1$-degree at points with inseparable residue field by taking a suitable lift of the polynomial and transferring its local degree. We also discuss the general set-up and strategy in terms of the six functor formalism. As an application, we show that trace forms of number fields are local $\mathbb{A}^1$-degrees.

On the regularity of small symbolic powers of edge ideals of graphs

2022-09-29T13:50:02+02:00

Assume that $G$ is a graph with edge ideal $I(G)$ and let $I(G)^{(s)}$ denote the $s$-th symbolic power of $I(G)$. It is proved that for every integer $s\geq 1$, $$ \mathrm{reg} (I(G)^{(s+1)})\leq \max \bigl \{\mathrm{reg} (I(G))$$ $$+2s, \mathrm{reg} \bigl (I(G)^{(s+1)}+I(G)^s\bigr )\bigr \}. $$ As a consequence, we conclude that $\mathrm{reg} (I(G)^{(2)})\leq \mathrm{reg} (I(G))+2$, and $\mathrm{reg} (I(G)^{(3)})\leq \mathrm{reg} (I(G))+4$. Moreover, it is shown that if for some integer $k\geq 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then $\mathrm{reg} (I(G)^{(s)})\leq 2s+\mathrm{reg} (I(G))-2$, for every integer $s\leq k+1$. Finally, it is proven that $\mathrm{reg} (I(G)^{(s)})=2s$, for $s\in \{2, 3, 4\}$, provided that the complementary graph $\overline {G}$ is chordal.

Topologically stable and persistent points of group actions

2022-09-27T10:12:25+02:00

In this paper, we introduce topologically stable points, persistent points, persistent property, persistent measures and almost persistent measures for first countable Hausdorff group actions of compact metric spaces. We prove that the set of all persistent points is measurable and it is closed if the action is equicontinuous. We also prove that the set of all persistent measures is a convex set and every almost persistent measure is a persistent measure. Finally, we prove that every equicontinuous pointwise topologically stable first countable Hausdorff group action of a compact metric space is persistent. In particular, every equicontinuous pointwise topologically stable flow is persistent.

Boundaries for Gelfand transform images of Banach algebras of holomorphic functions

2022-10-19T16:24:16+02:00

In this paper, we study boundaries for the Gelfand transform image of infinite dimensional analogues of the classical disk algebras. More precisely, given a certain Banach algebra $\mathcal{A}$ of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$, we show that the Shilov boundary for the Gelfand transform image of $\mathcal{A}$ can be explicitly described for a large class of Banach spaces. Some possible application of our result to the famous Corona theorem is also briefly discussed.

Generalized Bernstein functions

2022-10-18T13:59:58+02:00

A class of functions called generalized Bernstein functions is studied. The fundamental properties of this class are given and its relation to generalized Stieltjes functions via the Laplace transform is investigated. The subclass of generalized Thorin-Bernstein functions is characterized in different ways. Examples of generalized Bernstein functions include incomplete gamma functions, Lerch's transcendent and some hypergeometric functions.

Topological boundaries of covariant representations

2023-01-26T13:09:46+01:00

We associate a boundary $\mathcal B_{\pi ,u}$ to each covariant representation $(\pi ,u,H)$ of a $C^*$-dynamical system $(G,A,\alpha )$ and study the action of $G$ on $\mathcal B_{\pi ,u}$ and its amenability properties. We relate rigidity properties of traces on the associated crossed product $C^*$-algebra to faithfulness of the action of the group on this boundary.

Conductivity reconstruction from power density data in limited view

2023-01-29T10:44:58+01:00

In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.

Vol 129 no 1

2023-02-20T08:54:23+01:00

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