https://www.mscand.dk/issue/feed MATHEMATICA SCANDINAVICA 2020-10-23T03:35:06+02:00 Andrew Swann mscand@math.au.dk Open Journal Systems https://www.mscand.dk/article/view/120488 Forms over fields and Witt's lemma 2020-10-23T03:34:00+02:00 David Sprehn mscand@math.au.dk Nathalie Wahl mscand@math.au.dk <p>We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.</p> 2020-09-03T14:08:45+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121106 Asymptotics of some generalized Mathieu series 2020-10-23T03:33:54+02:00 Stefan Gerhold mscand@math.au.dk Friedrich Hubalek mscand@math.au.dk Živorad Tomovski mscand@math.au.dk <p>We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.</p> 2020-09-03T14:10:46+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121462 Inhomogeneous Diophantine approximation over fields of formal power series 2020-10-23T03:33:48+02:00 Yann Bugeaud mscand@math.au.dk L. Singhal mscand@math.au.dk Zhenliang Zhang mscand@math.au.dk <p>We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb {F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point&nbsp;$\underline {y}$ in $\mathbb {F}_q((T^{-1}))^2$ by the $\mathrm {SL}_2(\mathbb {F}_q[T])$-orbit of a given point $\underline {x}$ in $\mathbb {F}_q((T^{-1}))^2$.</p> 2020-09-03T14:12:01+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121167 Symmetric Riemann surfaces with no points fixed by orientation preserving automorphisms 2020-10-23T03:33:42+02:00 Ewa Kozłowska-Walania mscand@math.au.dk <p>We study the symmetric Riemann surfaces for which the group of orientation preserving automorphisms acts without fixed points. We show that any finite group can give rise to such an action, determine the maximal number of non-conjugate symmetries for such surfaces and find a sharp upper bound on maximal total number of ovals for a set of $k$ symmetries with ovals. We also solve the minimal genus problem for dihedral groups acting on the surfaces described above, for odd genera.</p> 2020-09-03T14:13:23+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121452 A direct proof that toric rank $2$ bundles on projective space split 2020-10-23T03:33:36+02:00 David Stapleton mscand@math.au.dk <p>The point of this paper is to give a short, direct proof that rank&nbsp;$2$ toric vector bundles on $n$-dimensional projective space split once $n$ is at least $3$. This result is originally due to Bertin and Elencwajg, and there is also related work by Kaneyama, Klyachko, and Ilten-Süss. The idea is that, after possibly twisting the vector bundle, there is a section which is a complete intersection.</p> 2020-09-03T14:15:33+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/119708 The Dirichlet problem for $m$-subharmonic functions on compact sets 2020-10-23T03:33:29+02:00 Per Åhag mscand@math.au.dk Rafał Czyż mscand@math.au.dk Lisa Hed mscand@math.au.dk <p>We characterize those compact sets for which the Dirichlet problem has a solution within the class of continuous $m$-subharmonic functions defined on a compact set, and then within the class of $m$-harmonic functions.</p> 2020-09-03T14:17:04+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/119844 On octahedrality and Müntz spaces 2020-10-23T03:33:24+02:00 André Martiny mscand@math.au.dk <p>We show that every Müntz space can be written as a direct sum of Banach spaces $X$ and $Y$, where $Y$ is almost isometric to a subspace of $c$ and $X$ is finite dimensional. We apply this to show that no Müntz space is locally octahedral or almost square.</p> 2020-09-03T14:25:20+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/119741 Weighted composition operators on weighted Bergman spaces induced by doubling weights 2020-10-23T03:35:06+02:00 Juntao Du mscand@math.au.dk Songxiao Li mscand@math.au.dk Yecheng Shi mscand@math.au.dk <p>In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi$ on Bergman type spaces $A_\omega ^p$ induced by a doubling weight ω. Let $X=\{u\in H(\mathbb{D} ): uC_\varphi \colon A_\omega ^p\to A_\omega ^p\ \text {is bounded}\}$. For some regular weights ω, we obtain that $X=H^\infty$ if and only if ϕ is a finite Blaschke product.</p> 2020-09-03T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/120288 On $C^*$-algebras associated to actions of discrete subgroups of $\operatorname{SL}(2,\mathbb{R})$ on the punctured plane 2020-10-23T03:33:18+02:00 Jacopo Bassi mscand@math.au.dk <p>Dynamical conditions that guarantee stability for discrete transformation group $C^*$-algebras are determined. The results are applied to the case of some discrete subgroups of $\operatorname{SL} (2,\mathbb{R} )$ acting on the punctured plane by means of matrix multiplication of vectors. In the case of cocompact subgroups, further properties of such crossed products are deduced from properties of the $C^*$-algebra associated to the horocycle flow on the corresponding compact homogeneous space of $\operatorname{SL} (2,\mathbb{R} )$.</p> 2020-09-03T14:28:10+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/120579 Strengthened convexity of positive operator monotone decreasing functions 2020-10-23T03:33:12+02:00 Megumi Kirihata mscand@math.au.dk Makoto Yamashita mscand@math.au.dk <p>We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map), and functional calculus by operator monotone functions defined on the positive real numbers instead of the logarithmic function.</p> 2020-09-03T14:29:46+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/120889 Exact Green's formula for the fractional Laplacian and perturbations 2020-10-23T03:33:06+02:00 Gerd Grubb mscand@math.au.dk <p>Let Ω be an open, smooth, bounded subset of $\mathbb{R}^n$. In connection with the fractional Laplacian $(-\Delta )^a$ ($a&gt;0$), and more generally for a $2a$-order classical pseudodifferential operator (ψdo) $P$ with even symbol, one can define the Dirichlet value $\gamma _0^{a-1}u$, resp. Neumann value $\gamma _1^{a-1}u$ of $u(x)$, as the trace, resp. normal derivative, of $u/d^{a-1}$ on $\partial \Omega$, where $d(x)$ is the distance from $x\in \Omega$ to $\partial \Omega$; they define well-posed boundary value problems for $P$.</p> <p>A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B\gamma _0^{a-1}u,\gamma _0^{a-1}v)_{\partial \Omega }$, where $B$ is a first-order ψdo on $\partial \Omega$. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta$, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta$ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.</p> 2020-09-03T14:31:16+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/120920 Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces 2020-10-23T03:33:00+02:00 Aamena Al-Qabani mscand@math.au.dk Titus Hilberdink mscand@math.au.dk Jani A. Virtanen mscand@math.au.dk <p>We study the Fredholm properties of Toeplitz operators acting on doubling Fock Hilbert spaces, and describe their essential spectra for bounded symbols of vanishing oscillation. We also compute the index of these Toeplitz operators in the special case when $\varphi (z) = \lvert {z}\rvert^{\beta }$ with $\beta &gt;0$. Our work extends the recent results on Toeplitz operators on the standard weighted Fock spaces to the setting of doubling Fock spaces.</p> 2020-09-03T14:32:27+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121069 The block Schur product is a Hadamard product 2020-10-23T03:32:54+02:00 Erik Christensen mscand@math.au.dk <p>Given two $n \times n$ matrices $A = (a_{ij})$ and $B=(b_{ij})$ with entries in $B(H)$ for some Hilbert space $H$, their <em>block Schur product</em> is the $n \times n$ matrix $A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and&nbsp;$g$ on the torus with Fourier coefficients $(f_n)$ and&nbsp;$(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the <em>Schur product</em> on scalar matrices is also known as the <em>Hadamard product</em>.</p> <p>We show that for a C*-algebra $\mathcal{A}$, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A}$ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named <em>the Hadamard product</em>.</p> <p>We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.</p> 2020-09-03T14:33:33+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121092 Stable homotopy, $1$-dimensional NCCW complexes, and Property (H) 2020-10-23T03:32:48+02:00 Qingnan An mscand@math.au.dk George A. Elliott mscand@math.au.dk Zhichao Liu mscand@math.au.dk Yuanhang Zhang mscand@math.au.dk <p>In this paper, we show that the homomorphisms between two unital one-dimensional NCCW complexes with the same KK-class are stably homotopic, that is, after adding on a common homomorphism (with finite dimensional image), they are homotopic. As a consequence, any one-dimensional NCCW complex has the Property&nbsp;(H).</p> 2020-09-03T14:34:41+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121826 Issue covers 2020-09-03T14:39:13+02:00 Mathematica Scandinavica mscand@math.au.dk <p>Issue covers</p> 2020-09-03T00:00:00+02:00 ##submission.copyrightStatement## https://www.mscand.dk/article/view/121827 Volume index 2020-09-03T14:40:55+02:00 Mathematica Scandinavica mscand@math.au.dk <p>Volume index</p> 2020-09-03T00:00:00+02:00 ##submission.copyrightStatement##