http://www.mscand.dk/issue/feedMATHEMATICA SCANDINAVICA2017-05-27T15:30:23+02:00Andrew Swannmscand@math.au.dkOpen Journal Systemshttp://www.mscand.dk/article/view/25728A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory2017-05-27T15:30:20+02:00Tony J. Puthenpurakaltputhen@math.iitb.ac.in<p>Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\operatorname{\underline{CM}}(A)$ be the stable category of maximal Cohen-Macauley $A$-modules and let $\operatorname{\underline{ICM}}(A)$ denote the set of isomorphism classes in $\operatorname{\underline{CM}}(A)$. We define a function $\xi \colon \operatorname{\underline{ICM}}(A) \to \mathbb{Z}$ which behaves well with respect to exact triangles in $\operatorname{\underline{CM}}(A)$. We then apply this to (Gorenstein) liaison theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liaison classes of $\{\,\mathfrak{m}^n \mid n\geq 1 \,\}$ is an infinite set. We also prove that if $A$ is Henselian with finite representation type with $A/\mathfrak{m}$ uncountable then for each $m \geq 1$ the set $\mathcal {C}_m = \{\, I \mid I \text { is a codim $2$ CM-ideal with } e_0(A/I) \leq m \,\}$ is contained in finitely many even liaison classes $L_1,\dots ,L_r$ (here $r$ may depend on $m$).</p>2017-05-27T15:30:20+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25850The irreducibility of power compositional sextic polynomials and their Galois groups2017-05-27T15:30:21+02:00Joshua HarringtonJoshua.Harrington@cedarcrest.eduLenny Joneslkjone@ship.edu<p>We define a <em>power compositional sextic polynomial</em> to be a monic sextic polynomial $f(x):=h(x^d)\in \mathbb{Z} [x]$, where $h(x)$ is an irreducible quadratic or cubic polynomial, and $d=3$ or $d=2$, respectively. In this article, we use a theorem of Capelli to give necessary and sufficient conditions for the reducibility of $f(x)$, and also a description of the factorization of $f(x)$ into irreducibles when $f(x)$ is reducible. In certain situations, when $f(x)$ is irreducible, we also give a simple algorithm to determine the Galois group of $f(x)$ without the calculation of resolvents. The algorithm requires only the use of the Rational Root Test and the calculation of a single discriminant. In addition, in each of these situations, we give infinite families of polynomials having the possible Galois groups.</p>2017-05-27T15:30:21+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25613Finite-rank Bratteli-Vershik diagrams are expansive—a new proof2017-05-27T15:30:21+02:00Siri-Malén Høynessiri.m.hoynes@ntnu.no<p><span>Downarowicz and Maass (Ergod. Th. and Dynam. Sys. 28 (2008), 739–747) proved that the Cantor minimal system associated to a properly ordered Bratteli diagram of finite rank is either an odometer system or an expansive system. We give a new proof of this truly remarkable result which we think is more transparent and easier to understand. We also address the question (Question 1) raised by Downarowicz and Maass and we find a better (i.e. lower) bound. In fact, we conjecture that the bound we have found is optimal.</span></p>2017-05-27T15:30:21+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25693Introduction to the Ekedahl Invariants2017-05-27T15:30:22+02:00Ivan Martinoi.martino@northeastern.eduIn 2009, T. Ekedahl introduced certain cohomological invariants for finite groups. In this work we present these invariants and we give an equivalent definition that does not involve the notion of algebraic stacks. Moreover we show certain properties for the class of the classifying stack of a finite group in the Kontsevich value ring.2017-05-27T15:30:22+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25612A note on holomorphic functions and the Fourier-Laplace transform2017-05-27T15:30:22+02:00Marcus Carlssonmarcus.carlsson@math.lu.seJens Wittstenjensw@maths.lth.se<p><span>We revisit the classical problem of when a given function, which is analytic in the upper half plane $\mathbb{C} _+$, can be written as the Fourier transform of a function or distribution with support on a half axis $(-\infty ,b]$, $b\in \mathbb{R} $. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as $H^p(\mathbb{C} _+)$.</span></p>2017-05-27T15:30:22+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25625On a class of operators in the hyperfinite $\mathrm{II}_1$ factor2017-05-27T15:30:22+02:00Zhangsheng Zhuzhuzhangsheng2010@163.comJunsheng Fangjunshengfang@hotmail.comRui Shiruishi@dlut.edu.cn<p>Let $R$ be the hyperfinite $\mathrm {II}_1$ factor and let $u$, $v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2\pi i\theta } uv$ for an irrational number $\theta$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue measurable function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra.</p>2017-05-27T15:30:22+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25624Fourier algebras of parabolic subgroups2017-05-27T15:30:23+02:00Søren Knudbyknudby@math.ku.dk<p>We study the following question: given a locally compact group when does its Fourier algebra coincide with the subalgebra of the Fourier-Stieltjes algebra consisting of functions vanishing at infinity? We provide sufficient conditions for this to be the case.</p><p>As an application, we show that when $P$ is the minimal parabolic subgroup in one of the classical simple Lie groups of real rank one or the exceptional such group, then the Fourier algebra of $P$ coincides with the subalgebra of the Fourier-Stieltjes algebra of $P$ consisting of functions vanishing at infinity. In particular, the regular representation of $P$ decomposes as a direct sum of irreducible representations although $P$ is not compact.</p>2017-05-27T15:30:23+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25832On the asymptotic expansion of the logarithm of Barnes triple gamma function II2017-05-27T15:30:23+02:00Stamatis Koumandosskoumand@ucy.ac.cyHenrik L. Pedersenhenrikp@math.ku.dk<p>The remainders in an asymptotic expansion of the logarithm of Barnes triple gamma function give rise to completely monotonic functions of positive order.</p>2017-05-27T15:30:23+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25793Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$2017-05-27T15:30:23+02:00Željko ČučkovićZeljko.Cuckovic@utoledo.eduSönmez ŞahutoğluSonmez.Sahutoglu@utoledo.edu<p>Let $\Omega \subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\varphi \in C^1(\overline{\Omega})$ such that $\varphi $ is harmonic on the non-trivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi }$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ “along” the non-trivial disks in the boundary.</p>2017-05-27T15:30:23+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/25626A simple proof of the existence of Haar measure on amenable groups2017-05-27T15:30:23+02:00Alexander J. Izzoaizzo@bgsu.eduA simple proof of the existence of Haar measure on amenable groups is given.2017-05-27T15:30:23+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/26247Issue covers2017-05-27T15:30:23+02:00Mathematica Scandinavicamscand@math.au.dkIssue covers.2017-05-27T15:30:23+02:00Copyright (c) 2017 MATHEMATICA SCANDINAVICA