https://www.mscand.dk/issue/feed MATHEMATICA SCANDINAVICA 2018-12-14T02:39:09+01:00 Andrew Swann mscand@math.au.dk Open Journal Systems https://www.mscand.dk/article/view/105278 Non-Koszul quadratic Gorenstein toric rings 2018-12-14T02:39:01+01:00 Kazunori Matsuda kaz-matsuda@ist.osaka-u.ac.jp <p>Koszulness of Gorenstein quadratic algebras of small socle degree is studied. In this paper, we construct non-Koszul Gorenstein quadratic toric ring such that its socle degree is more than $3$ by using stable set polytopes.</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/105307 Zero-divisor graphs of amalgamations 2018-12-14T02:38:57+01:00 Salah-Eddine Kabbaj kabbaj@kfupm.edu.sa Abdeslam Mimouni amimouni@kfupm.edu.sa <p>Let $f\colon A\rightarrow B$ be a homomorphism of commutative rings and let $J$ be an ideal of $B$. The <em>amalgamation</em> of $A$ with $B$ along $J$ with respect to $f$ is the subring of $A\times B$ given by \[ A\bowtie ^{f}J:=\{(a,f(a)+j) \mid a\in A, j\in J\}. \] This paper investigates the zero-divisor graph of amalgamations. Our aim is to characterize when the graph is complete and compute its diameter and girth for various contexts of amalgamations. The new results recover well-known results on duplications, and yield new and original examples issued from amalgamations.</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/106223 Projections of Mukai varieties 2018-12-14T02:38:46+01:00 Michał Kapustka mscand@math.au.dk This note is an answer to a problem proposed by Iliev and Ranestad. We prove that the projections of general nodal linear sections of suitable dimension of Mukai varieties $M_g$ are linear sections of $M_{g-1}$. 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/106920 The depth and LS category of a topological space 2018-12-14T02:38:42+01:00 Yves Félix mscand@math.au.dk Steve Halperin mscand@math.au.dk <p>The depth of an augmented ring $\varepsilon \colon A\to k $ is the least $p$, or ∞, such that \begin {equation*} \Ext _A^p(k , A)\neq 0. \end {equation*} When $X$ is a simply connected finite type CW complex, $H_*(\Omega X;\mathbb {Q})$ is a Hopf algebra and the universal enveloping algebra of the Lie algebra $L_X$ of primitive elements. It is known that $\depth H_*(\Omega X;\mathbb {Q}) \leq \cat X$, the Lusternik-Schnirelmann category of $X$.</p> <p>For any connected CW complex we construct a completion $\widehat {H}(\Omega X)$ of $H_*(\Omega X;\mathbb {Q})$ as a complete Hopf algebra with primitive sub Lie algebra $L_X$, and define $\depth X$ to be the least $p$ or ∞ such that \[ \Ext ^p_{UL_X}(\mathbb {Q}, \widehat {H}(\Omega X))\neq 0. \] Theorem: for any connected CW complex, $\depth X\leq \cat X$.</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/105087 Orbit equivalence of graphs and isomorphism of graph groupoids 2018-12-14T02:39:09+01:00 Toke Meier Carlsen toke.carlsen@gmail.com Marius Lie Winger marius.l.winger@ntnu.no <p>We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the (topological) isolated points of the boundary path space of a graph. As a result, we are able to show that the groupoids of two directed graphs with finitely many vertices and no sinks are isomorphic if and only if the two graphs are orbit equivalent, and that the groupoids of the stabilisations of two such graphs are isomorphic if and only if the stabilisations of the graphs are orbit equivalent.</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/105465 Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$ 2018-12-14T02:38:53+01:00 Carsten Elsner carsten.elsner@fhdw.de Yohei Tachiya tachiya@hirosaki-u.ac.jp <p>In its most elaborate form, the Jacobi theta function is defined for two complex variables $z$ and τ by $\theta (z|\tau ) =\sum _{\nu =-\infty }^{\infty } e^{\pi i\nu ^2\tau + 2\pi i\nu z}$, which converges for all complex number $z$, and τ in the upper half-plane. The special case \[ \theta _3(\tau ):=\theta (0|\tau )= 1+2\sum _{\nu =1}^{\infty } e^{\pi i\nu ^2 \tau } \] is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function $\theta (z|\tau )$. In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant $\theta _3(\tau )$. For example, the three values $\theta _3(\tau )$, $\theta _3(n\tau )$, and $D\theta _3(\tau )$ are algebraically independent over $\mathbb{Q} $ for any τ such that $q=e^{\pi i\tau }$ is an algebraic number, where $n\geq 2$ is an integer and $D:=(\pi i)^{-1}{d}/{d\tau }$ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values $\theta _3(\tau )$ and $\theta _3(2^m\tau )$ for $m\geq 1$. As an application of our main theorem, the algebraic dependence over $\mathbb{Q} $ of the three values $\theta _3(\ell \tau )$, $\theta _3(m\tau )$, and $\theta _3(n\tau )$ for integers $\ell ,m,n\geq 1$ is also presented.</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/105662 On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators 2018-12-14T02:38:50+01:00 Dah-Chin Luor dclour@isu.edu.tw <p>Necessary and sufficient conditions for the weight function $u$ are obtained, which provide the boundedness for a class of averaging operators from $L_p^+$ to $L_{q,u}^+$. These operators include the multiple Hardy-Littlewood averages and related maximal operators, geometric mean operators, and geometric maximal operators. We show that, under suitable conditions, the boundedness of these operators are equivalent. Our theorems extend several one-dimensional results to multi-dimensional cases and to operators with multiple kernels. We also show that in the case $p&lt;q$, some one-dimensional results do not carry over to the multi-dimensional cases, and the boundedness of $T$ from $L_p^+$ to $L_{q,u}^+$ holds only if $u=0$ almost everywhere.</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/105124 Quantitative factorization of weakly compact, Rosenthal, and $\xi$-Banach-Saks operators 2018-12-14T02:39:05+01:00 Kevin Beanland beanlandk@wlu.edu Ryan M. Causey causeyrm@miamioh.edu <p>We prove quantitative factorization results for several classes of operators, including weakly compact, Rosenthal, and ξ-Banach-Saks operators.</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/107648 Issue covers 2018-12-14T02:38:38+01:00 Mathematica Scandinavica mscand@math.au.dk <p>Issue covers</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/107649 Volume index 2018-12-14T02:38:33+01:00 Mathematica Scandinavica mscand@math.au.dk <p>Volume index</p> 2018-09-05T00:00:00+02:00 ##submission.copyrightStatement##