MATHEMATICA SCANDINAVICA https://www.mscand.dk/ en-US <p>Submission of manuscripts implies that the work described has not been published before (except in the form of an abstract or as part of a published lecture, review or thesis), that it is not under consideration for publication elsewhere and that, if and when the manuscript is accepted for publication, the authors agree to automatic transfer of the copyright to the publisher.&nbsp;</p> mscand@math.au.dk (Andrew Swann) mscand@math.au.dk (Anne Mette Christiansen) Sun, 08 Apr 2018 09:40:50 +0200 OJS 3.1.1.2 http://blogs.law.harvard.edu/tech/rss 60 $k$-shellable simplicial complexes and graphs https://www.mscand.dk/article/view/102975 <p>In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying an expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture.</p><p>Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.</p> Rahim Rahmati-Asghar ##submission.copyrightStatement## https://www.mscand.dk/article/view/102975 Sun, 08 Apr 2018 09:40:49 +0200 Topological rigidity of quasitoric manifolds https://www.mscand.dk/article/view/97303 <p>Quasitoric manifolds are manifolds that admit an action of the torus that is locally the same as the standard action of $T^n$ on $\mathbb{C}^n$. It is known that the quotients of such actions are nice manifolds with corners. We prove that a class of locally standard manifolds, that contains the quasitoric manifolds, is equivariantly rigid, i.e., that any manifold that is $T^n$-homotopy equivalent to a quasitoric manifold is $T^n$-homeomorphic to it.</p> Vassilis Metaftsis, Stratos Prassidis ##submission.copyrightStatement## https://www.mscand.dk/article/view/97303 Sun, 08 Apr 2018 09:40:50 +0200 Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces https://www.mscand.dk/article/view/97308 <p>Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O} _S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld-Mukai bundles to prove that $A$ is cut out by the global sections of a rank $1$ torsion-free sheaf $\mathcal{G} $ on $S$. Furthermore, we show that $c_1(\mathcal{G} )$ with one exception is adapted to $\mathcal{O} _S(C)$ and satisfies $\operatorname{Cliff} (c_1(\mathcal{G} )_{|C})\leq \operatorname{Cliff} (A)$, thereby confirming a conjecture posed by Donagi and Morrison. We also show that the same methods can be used to give a simple proof of the conjecture in the $g^1_d$ case.</p><p>In the final section, we give an example of the mentioned exception where $h^0(C,c_1(\mathcal{G} )_{|C})$ is dependent on the curve $C$ in its linear system, thereby failing to be adapted to $\mathcal{O} _S(C)$.</p> Nils Henry Rasmussen ##submission.copyrightStatement## https://www.mscand.dk/article/view/97308 Sun, 08 Apr 2018 09:40:50 +0200 Nearest points on toric varieties https://www.mscand.dk/article/view/101478 <p>We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.</p> Martin Helmer, Bernd Sturmfels ##submission.copyrightStatement## https://www.mscand.dk/article/view/101478 Sun, 08 Apr 2018 09:40:50 +0200 Stability of rank two Ulrich bundles on projective $K3$ surfaces https://www.mscand.dk/article/view/101999 <p>Let $F\subseteq \mathbb{P}^{N}$ be a $K3$ surface of degree $2a$, where $a\ge 2$. In this paper we deal with Ulrich bundles on $F$ of rank $2$. We deal with their stability and we construct $K3$ surfaces endowed with families of non-special Ulrich bundles of rank $2$ for each $a\ge 2$.</p> Gianfranco Casnati, Federica Galluzzi ##submission.copyrightStatement## https://www.mscand.dk/article/view/101999 Sun, 08 Apr 2018 09:40:50 +0200 Mappings preserving approximate orthogonality in Hilbert $C^*$-modules https://www.mscand.dk/article/view/102945 <p>We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta , \varepsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta , \varepsilon )$-orthogonality preserving. In particular, if $\mathscr {E}$ is a full Hilbert $\mathscr {A}$-module with $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T, S\colon \mathscr {E}\to \mathscr {E}$ are two linear mappings satisfying $|\langle Sx, Sy\rangle | = \|S\|^2|\langle x, y\rangle |$ for all $x, y\in \mathscr {E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta , \varepsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T\colon \mathscr {E} \to \mathscr {F}$ is a nonzero $\mathscr {A}$-linear $(\delta , \varepsilon )$-orthogonality preserving mapping between $\mathscr {A}$-modules, then \[ \bigl \|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle \bigr \|\leq \frac {4(\varepsilon - \delta )}{(1 - \delta )(1 + \varepsilon )} \|Tx\|\|Ty\|\qquad (x, y\in \mathscr {E}). \] As a result, we present some characterizations of the orthogonality preserving mappings.</p> Mohammad Sal Moslehian, Ali Zamani ##submission.copyrightStatement## https://www.mscand.dk/article/view/102945 Sun, 08 Apr 2018 09:40:50 +0200 Clark measures and a theorem of Ritt https://www.mscand.dk/article/view/104444 <p>We determine when a finite Blaschke product $B$ can be written, in a non-trivial way, as a composition of two finite Blaschke products (Ritt's problem) in terms of the Clark measure for $B$. Our tools involve the numerical range of compressed shift operators and the geometry of certain polygons circumscribing the numerical range of the relevant operator. As a consequence of our results, we can determine, in terms of Clark measures, when two finite Blaschke products commute.</p> Isabelle Chalendar, Pamela Gorkin, Jonathan R. Partington, William T. Ross ##submission.copyrightStatement## https://www.mscand.dk/article/view/104444 Sun, 08 Apr 2018 09:40:50 +0200 The bounded approximation property of variable Lebesgue spaces and nuclearity https://www.mscand.dk/article/view/102962 <p>In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the Grothendieck-Lidskii formula. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb{R}^n$ in terms of global symbols.</p> Julio Delgado, Michael Ruzhansky ##submission.copyrightStatement## https://www.mscand.dk/article/view/102962 Sun, 08 Apr 2018 09:40:50 +0200 Volume index https://www.mscand.dk/article/view/105219 Volume index Mathematica Scandinavica ##submission.copyrightStatement## https://www.mscand.dk/article/view/105219 Sun, 08 Apr 2018 09:40:50 +0200 Issue covers https://www.mscand.dk/article/view/105220 Issure covers Mathematica Scandinavica ##submission.copyrightStatement## https://www.mscand.dk/article/view/105220 Sun, 08 Apr 2018 09:40:50 +0200