MATHEMATICA SCANDINAVICA https://www.mscand.dk/ en-US mscand@math.au.dk (Arne Jensen) mscand@math.au.dk (Anne Mette Christiansen) Tue, 31 Aug 2021 12:50:05 +0200 OJS 3.2.1.4 http://blogs.law.harvard.edu/tech/rss 60 Divisors of expected Jacobian type https://www.mscand.dk/article/view/126042 <p>Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of $D$-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some $D$-module theoretic invariant given by the degree of the Kashiwara operator.</p> Josep Àlvarez Montaner, Francesc Planas-Vilanova Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126042 Tue, 31 Aug 2021 00:00:00 +0200 A groupoid picture of the Elek algebras https://www.mscand.dk/article/view/122419 <p>We reformulate a construction by Gábor Elek, which associates $C^{\ast} $-algebras with uniformly recurrent subgroups, in the language of groupoid $C^{\ast} $-algebras. This allows us to simplify several proofs from the original paper and add the converse direction to Elek's characterisation of nuclearity, showing that his sufficient condition is in fact necessary. We furthermore relate our groupoids to the dynamics of the group acting on its uniformly recurrent subgroup.</p> Clemens Borys Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/122419 Tue, 31 Aug 2021 00:00:00 +0200 Asymptotic behavior of $j$-multiplicities https://www.mscand.dk/article/view/126029 <p>Let $R= \oplus_{n\in \mathbb{N}_0}R_n$ be a Noetherian homogeneous ring with local base ring $(R_0,\mathfrak{m}_0)$. Let $R_+= \oplus_{n\in \mathbb{N}}R_n$ denote the irrelevant ideal of $R$ and let $M=\oplus_{n\in \mathbb{Z}}M_n$ be a finitely generated graded $R$-module. When $\dim(R_0)\leq 2$ and $\mathfrak{q}_0$ is an arbitrary ideal of $R_0$, we show that the $j$-multiplicity of the graded local cohomology module $j_0({\mathfrak{q}_0},H_{R_+}^i(M)_n)$ has a polynomial behavior for all $n\ll0$.</p> Thiago Henrique de Freitas, Victor Hugo Jorge Pérez, Pedro Henrique Lima Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126029 Tue, 31 Aug 2021 00:00:00 +0200 Invariants of linkage of modules https://www.mscand.dk/article/view/125992 <p>Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$, $N$ be two Cohen-Macaulay $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $\mathrm{Ext}^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Tor}^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is a Cohen-Macaulay module then we show that $\mathrm{Ext}^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(D^\dagger , N) = 0$ for all $i \gg 0$, where $D^\dagger = \mathrm{Ext}^r_A(D,A)$ and $r = \mathrm{codim}(D)$. As a consequence we get that $\mathrm{Ext}^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(N, N) = 0$ for all $i \gg 0$. We also show that $\mathrm{End}_A(M)/\mathrm{rad}\,\mathrm{End}_A(M) \cong (\mathrm{End}_A(N)/\mathrm{rad}\,\mathrm{End}_A(N))^{\mathrm{op}}$. We also give a negative answer to a question of Martsinkovsky and Strooker.</p> Tony J. Puthenpurakal Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/125992 Tue, 31 Aug 2021 00:00:00 +0200 On $\mathcal{M}$-normal embedded subgroups and the structure of finite groups https://www.mscand.dk/article/view/126034 <p>Let $G$ be a group and $H$ be a subgroup of $G$. $H$ is said to be $\mathcal{M}$-normal supplemented in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H_1K&lt;G$ for every maximal subgroup $H_1$ of $H$. Furthermore, $H$ is said to be $\mathcal{M}$-normal embedded in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K=1$ or $H\cap K$ is $\mathcal{M}$-normal supplemented in $G$. In this paper, some new criteria for a group to be nilpotent and $p$-supersolvable for some prime $p$ are obtained.</p> Ruifang Chen, Xianhe Zhao, Rui Li Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126034 Tue, 31 Aug 2021 00:00:00 +0200 Area-perimeter duality in polygon spaces https://www.mscand.dk/article/view/126041 <p>Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixed area and perimeter are also determined. Besides, we extend the classical isoperimetric duality to all critical points. In conclusion a few general remarks on dual extremal problems in polygon spaces and beyond are given.</p> Giorgi Khimshiashvili, Gaiane Panina, Dirk Siersma Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126041 Tue, 31 Aug 2021 00:00:00 +0200 Strongly elliptic operators and exponentiation of operator Lie algebras https://www.mscand.dk/article/view/126020 <p>An intriguing feature which is often present in theorems regarding<br>the exponentiation of Lie algebras of unbounded linear operators on<br>Banach spaces is the assumption of hypotheses on the Laplacian<br>operator associated with a basis of the operator Lie algebra.<br>The main objective of this work is to show that one can substitute<br>the Laplacian by an arbitrary operator in the enveloping algebra and<br>still obtain exponentiation, as long as its closure generates a<br>strongly continuous one-parameter semigroup satisfying certain norm<br>estimates, which are typical in the theory of strongly elliptic<br>operators.</p> Rodrigo A. H. M. Cabral Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126020 Tue, 31 Aug 2021 00:00:00 +0200 The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$ https://www.mscand.dk/article/view/125994 <p>Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.</p> Ayoub El Gasmi Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/125994 Tue, 31 Aug 2021 00:00:00 +0200 Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group https://www.mscand.dk/article/view/125997 <p>We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.</p> Kazunori Kodaka Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/125997 Tue, 31 Aug 2021 00:00:00 +0200 On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree https://www.mscand.dk/article/view/126053 <p>We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.</p> Norman Levenberg, Franck Wielonsky Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126053 Tue, 31 Aug 2021 00:00:00 +0200 Universal $C^∗$-algebras with the local lifting property https://www.mscand.dk/article/view/126018 <p>The Local Lifting Property (LLP) is a localized version of projectivity for completely positive maps between $\mathrm{C}^*$-algebras. Outside of the nuclear case, very few $\mathrm{C}^*$-algebras are known to have the LLP\@. In this article, we show that the LLP holds for the algebraic contraction $\mathrm{C}^*$-algebras introduced by Hadwin and further studied by Loring and Shulman. We also show that the universal Pythagorean $\mathrm{C}^*$-algebras introduced by Brothier and Jones have the Lifting Property.</p> Kristin E. Courtney Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126018 Tue, 31 Aug 2021 00:00:00 +0200 Some two-point boundary value problems for systems of higher order functional differential equations https://www.mscand.dk/article/view/126021 <p>In the paper we study the question of the solvability and unique solvability of systems of the higher order differential equations with the argument deviations <br />\begin{equation*} u_i^{(m_i)}(t)=p_i(t)u_{i+1}(\tau _{i}(t))+ q_i(t), (i=\overline {1, n}), \text {for $t\in I:=[a, b]$}, <br />\end{equation*} <br />and <br />\begin{equation*}u_i^{(m_i)} (t)=F_{i}(u)(t)+q_{0i}(t), (i = \overline {1, n}), \text {for $ t\in I$}, <br />\end{equation*} <br />under the conjugate <br />$u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {1, m_i-k_i}$, $i=\overline {1, n}$, <br />and the right-focal <br />$u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {k_i+1,m_i}$, $i=\overline {1, n}$, <br />boundary conditions, where $u_{n+1}=u_1, $ $n\geq 2, $ $m_i\geq 2, $ $p_i \in L_{\infty }(I; R), $ $q_i, q_{0i}\in L(I; R), $ $\tau _i\colon I\to I$ are the measurable functions, $F_i$ are the local Caratheodory's class operators, and $k_i$ is the integer part of the number $m_i/2$.<br /><br />In the paper are obtained the efficient sufficient conditions that guarantee the unique solvability of the linear problems and take into the account explicitly the effect of argument deviations, and on the basis of these results are proved new conditions of the solvability and unique solvability for the nonlinear problems.</p> Sulkhan Mukhigulashvili Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/126021 Tue, 31 Aug 2021 00:00:00 +0200 Issue covers https://www.mscand.dk/article/view/128389 <p>Issue covers</p> Mathematica Scandinavica Copyright (c) 2021 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/128389 Tue, 31 Aug 2021 00:00:00 +0200