https://www.mscand.dk/issue/feed MATHEMATICA SCANDINAVICA 2018-07-04T11:46:15+02:00 Andrew Swann mscand@math.au.dk Open Journal Systems https://www.mscand.dk/article/view/102975 $k$-shellable simplicial complexes and graphs 2018-07-04T11:46:15+02:00 Rahim Rahmati-Asghar rahmatiasgahr.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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/97303 Topological rigidity of quasitoric manifolds 2018-07-04T11:46:11+02:00 Vassilis Metaftsis vmet@aegean.gr Stratos Prassidis prasside@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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/97308 Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces 2018-07-04T11:46:08+02:00 Nils Henry Rasmussen nils.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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/101478 Nearest points on toric varieties 2018-07-04T11:46:05+02:00 Martin Helmer m.helmer@math.ku.dk Bernd Sturmfels bernd@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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/101999 Stability of rank two Ulrich bundles on projective $K3$ surfaces 2018-07-04T11:46:01+02:00 Gianfranco Casnati gianfranco.casnati@polito.it Federica Galluzzi federica.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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/102945 Mappings preserving approximate orthogonality in Hilbert $C^*$-modules 2018-07-04T11:45:57+02:00 Mohammad Sal Moslehian moslehian@um.ac.ir Ali Zamani Zamani.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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/104444 Clark measures and a theorem of Ritt 2018-07-04T11:45:50+02:00 Isabelle Chalendar isabelle.chalendar@u-pem.fr Pamela Gorkin pgorkin@bucknell.edu Jonathan R. Partington J.R.Partington@leeds.ac.uk William T. Ross wross@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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/102962 The bounded approximation property of variable Lebesgue spaces and nuclearity 2018-07-04T11:45:54+02:00 Julio Delgado j.delgado@imperial.ac.uk Michael Ruzhansky m.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+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/105219 Volume index 2018-04-08T09:40:50+02:00 Mathematica Scandinavica mscand@math.au.dk Volume index 2018-04-08T09:40:50+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/105220 Issue covers 2018-04-08T09:40:50+02:00 Mathematica Scandinavica mscand@math.au.dk Issure covers 2018-04-08T09:40:50+02:00 ##submission.copyrightStatement##