MATHEMATICA SCANDINAVICA https://www.mscand.dk/ Mathematica Scandinavica en-US MATHEMATICA SCANDINAVICA 0025-5521 Non-Koszul quadratic Gorenstein toric rings https://www.mscand.dk/article/view/105278 <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> Kazunori Matsuda ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 161 173 10.7146/math.scand.a-105278 Zero-divisor graphs of amalgamations https://www.mscand.dk/article/view/105307 <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> Salah-Eddine Kabbaj Abdeslam Mimouni ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 174 190 10.7146/math.scand.a-105307 Projections of Mukai varieties https://www.mscand.dk/article/view/106223 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 $M_{g-1}$. Michał Kapustka ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 191 219 10.7146/math.scand.a-106223 The depth and LS category of a topological space https://www.mscand.dk/article/view/106920 <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> Yves Félix Steve Halperin ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 220 238 10.7146/math.scand.a-106920 Orbit equivalence of graphs and isomorphism of graph groupoids https://www.mscand.dk/article/view/105087 <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> Toke Meier Carlsen Marius Lie Winger ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 239 248 10.7146/math.scand.a-105087 Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$ https://www.mscand.dk/article/view/105465 <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> Carsten Elsner Yohei Tachiya ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 249 272 10.7146/math.scand.a-105465 On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators https://www.mscand.dk/article/view/105662 <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> Dah-Chin Luor ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 273 296 10.7146/math.scand.a-105662 Quantitative factorization of weakly compact, Rosenthal, and $\xi$-Banach-Saks operators https://www.mscand.dk/article/view/105124 <p>We prove quantitative factorization results for several classes of operators, including weakly compact, Rosenthal, and ξ-Banach-Saks operators.</p> Kevin Beanland Ryan M. Causey ##submission.copyrightStatement## 2018-08-13 2018-08-13 123 2 297 319 10.7146/math.scand.a-105124 Issue covers https://www.mscand.dk/article/view/107648 <p>Issue covers</p> Mathematica Scandinavica ##submission.copyrightStatement## 2018-08-25 2018-08-25 123 2 Volume index https://www.mscand.dk/article/view/107649 <p>Volume index</p> Mathematica Scandinavica ##submission.copyrightStatement## 2018-08-25 2018-08-25 123 2 320 320