http://www.mscand.dk/issue/feedMATHEMATICA SCANDINAVICA2018-04-08T09:40:50+02:00Andrew Swannmscand@math.au.dkOpen Journal Systemshttp://www.mscand.dk/article/view/102975$k$-shellable simplicial complexes and graphs2018-04-08T09:40:49+02:00Rahim Rahmati-Asgharrahmatiasgahr.r@gmail.com<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>2018-04-08T09:40:49+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/97303Topological rigidity of quasitoric manifolds2018-04-08T09:40:50+02:00Vassilis Metaftsisvmet@aegean.grStratos Prassidisprasside@aegean.gr<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>2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/97308Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces2018-04-08T09:40:50+02:00Nils Henry Rasmussennils.h.rasmussen@usn.no<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>2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/101478Nearest points on toric varieties2018-04-08T09:40:50+02:00Martin Helmerm.helmer@math.ku.dkBernd Sturmfelsbernd@berkeley.edu<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>2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/101999Stability of rank two Ulrich bundles on projective $K3$ surfaces2018-04-08T09:40:50+02:00Gianfranco Casnatigianfranco.casnati@polito.itFederica Galluzzifederica.galluzzi@unito.it<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>2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/102945Mappings preserving approximate orthogonality in Hilbert $C^*$-modules2018-04-08T09:40:50+02:00Mohammad Sal Moslehianmoslehian@um.ac.irAli ZamaniZamani.ali85@yahoo.com<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>2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/104444Clark measures and a theorem of Ritt2018-04-08T09:40:50+02:00Isabelle Chalendarisabelle.chalendar@u-pem.frPamela Gorkinpgorkin@bucknell.eduJonathan R. PartingtonJ.R.Partington@leeds.ac.ukWilliam T. Rosswross@richmond.edu<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>2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/102962The bounded approximation property of variable Lebesgue spaces and nuclearity2018-04-08T09:40:50+02:00Julio Delgadoj.delgado@imperial.ac.ukMichael Ruzhanskym.ruzhansky@imperial.ac.uk<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>2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/105219Volume index2018-04-08T09:40:50+02:00Mathematica Scandinavicamscand@math.au.dkVolume index2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICAhttp://www.mscand.dk/article/view/105220Issue covers2018-04-08T09:40:50+02:00Mathematica Scandinavicamscand@math.au.dkIssure covers2018-04-08T09:40:50+02:00Copyright (c) 2018 MATHEMATICA SCANDINAVICA