MATHEMATICA SCANDINAVICA https://www.mscand.dk/ en-US mscand@math.au.dk (Arne Jensen) mscand@math.au.dk (Anne Mette Christiansen) Thu, 24 Feb 2022 11:35:06 +0100 OJS 3.2.1.4 http://blogs.law.harvard.edu/tech/rss 60 $\lambda$-quiddité sur $\mathbb{Z}[\alpha]$ avec $\alpha$ transcendant https://www.mscand.dk/article/view/128972 <p>Dans le cadre de l'étude des frises de Coxeter, M.&nbsp;Cuntz a introduit la notion de $\lambda$-quiddité irréductible. L'objectif de cette note est de lister toutes les $\lambda$-quiddités irréductibles sur l'anneau $\mathbb{Z}[\alpha]$ dans le cas où $\alpha$ est un nombre complexe transcendant.</p> Flavien Mabilat Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/128972 Thu, 24 Feb 2022 00:00:00 +0100 Density of $f$-ideals and $f$-ideals in mixed small degrees https://www.mscand.dk/article/view/129244 <p>A squarefree monomial ideal is called an $f$-ideal if its Stanley–Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct $f$-ideals generated in small degrees.</p> Huy T`ai H`a, Graham Keiper, Hasan Mahmood, Jonathan L. O'Rourke Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129244 Thu, 24 Feb 2022 00:00:00 +0100 Affine and formal abelian group schemes on $p$-polar rings https://www.mscand.dk/article/view/129704 <p>We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ of characteristic $p$ is defined on, and in fact only depends on, a weaker structure than that of a $k$-algebra. We call this structure a $p$-polar $k$-algebra. By extension, the functors of points for any $p$-adic affine commutative group scheme and for any formal group are defined on, and only depend on, $p$-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any $p$-polar $k$-algebra $P$, and it agrees with the cofree commutative Hopf algebra on a commutative $k$-algebra $A$ if $P$ is the $p$-polar algebra underlying $A$; a dual result holds for free commutative Hopf algebras on finite $k$-coalgebras.</p> Tilman Bauer Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129704 Thu, 24 Feb 2022 00:00:00 +0100 Representability of the local motivic Brouwer degree https://www.mscand.dk/article/view/129287 <p>We study which quadratic forms are representable as the local degree of a map $f \colon \mathbb{A}^n \to \mathbb{A}^n$ with an isolated zero at $0$, following the work of Kass and Wickelgren who established the connection to the quadratic form of Eisenbud, Khimshiashvili, and Levine. Our main observation is that over some base fields $k$, not all quadratic forms are representable as a local degree. Empirically the local degree of a map $f \colon \mathbb{A}^n \to \mathbb{A}^n$ has many hyperbolic summands, and we prove that in fact this is the case for local degrees of low rank. We establish a complete classification of the quadratic forms of rank at most $7$ that are representable as the local degree of a map over all base fields of characteristic different from $2$. The number of hyperbolic summands was also studied by Eisenbud and Levine, where they establish general bounds on the number of hyperbolic forms that must appear in a quadratic form that is representable as a local degree. Our proof method is elementary and constructive in the case of rank 5 local degrees, while the work of Eisenbud and Levine is more general. We provide further families of examples that verify that the bounds of Eisenbud and Levine are tight in several cases.</p> Gereon Quick, Therese Strand, Glen Matthew Wilson Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129287 Thu, 24 Feb 2022 00:00:00 +0100 Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form https://www.mscand.dk/article/view/129245 <p>Our aim in this paper is to give Trudinger-type inequalities for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our result are new even for the doubling metric measure setting. As a corollary, we give Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form in the framework of double phase functions with variable exponents.</p> Takao Ohno, Tetsu Shimomura Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129245 Thu, 24 Feb 2022 00:00:00 +0100 Weighted holomorphic Dirichlet series and composition operators with polynomial symbols https://www.mscand.dk/article/view/129686 <p>In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a characterization so that the direct sum of the identity plus a weighted forward shift operator on $\ell^2$ is cyclic.</p> Emmanuel Fricain, Camille Mau Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129686 Thu, 24 Feb 2022 00:00:00 +0100 Dimension of images of large level sets https://www.mscand.dk/article/view/129246 <p>Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\colon E\to \mathbb{R} $, where $E$ is some interval on the line having positive length. For $0&lt;\alpha &lt;1$ let $I_\alpha (f)$ denote the set of values $y\in \mathbb{R} $ whose preimage $f^{-1}(y)$ has Hausdorff dimension at least $\alpha$. We consider how large can be the Hausdorff dimension of $I_\alpha (f)$, as $f$ ranges over the set of all $k$-times continuously differentiable functions from $E$ into $\mathbb{R} $. We show that the sharp upper bound on $\dim I_\alpha (f)$ is $(1-\alpha )/k$.</p> Gavin Armstrong, Anthony G. O'Farrell Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129246 Thu, 24 Feb 2022 00:00:00 +0100 Cover1 https://www.mscand.dk/article/view/131745 <p>Cover</p> Mathematica Scandinavica Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/131745 Thu, 24 Feb 2022 00:00:00 +0100