https://www.mscand.dk/issue/feedMATHEMATICA SCANDINAVICA2021-01-15T02:38:13+01:00Andrew Swannmscand@math.au.dkOpen Journal Systemshttps://www.mscand.dk/article/view/120488Forms over fields and Witt's lemma2021-01-15T02:38:07+01:00David Sprehnmscand@math.au.dkNathalie Wahlmscand@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/121106Asymptotics of some generalized Mathieu series2021-01-15T02:38:01+01:00Stefan Gerholdmscand@math.au.dkFriedrich Hubalekmscand@math.au.dkŽivorad Tomovskimscand@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/121462Inhomogeneous Diophantine approximation over fields of formal power series2021-01-15T02:37:54+01:00Yann Bugeaudmscand@math.au.dkL. Singhalmscand@math.au.dkZhenliang Zhangmscand@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 $\underline {y}$ 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/121167Symmetric Riemann surfaces with no points fixed by orientation preserving automorphisms2021-01-15T02:37:49+01:00Ewa Kozłowska-Walaniamscand@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/121452A direct proof that toric rank $2$ bundles on projective space split2021-01-15T02:37:43+01:00David Stapletonmscand@math.au.dk<p>The point of this paper is to give a short, direct proof that rank $2$ 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/119708The Dirichlet problem for $m$-subharmonic functions on compact sets2021-01-15T02:37:38+01:00Per Åhagmscand@math.au.dkRafał Czyżmscand@math.au.dkLisa Hedmscand@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/119844On octahedrality and Müntz spaces2021-01-15T02:37:32+01:00André Martinymscand@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/119741Weighted composition operators on weighted Bergman spaces induced by doubling weights2021-01-15T02:38:13+01:00Juntao Dumscand@math.au.dkSongxiao Limscand@math.au.dkYecheng Shimscand@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/120288On $C^*$-algebras associated to actions of discrete subgroups of $\operatorname{SL}(2,\mathbb{R})$ on the punctured plane2021-01-15T02:37:26+01:00Jacopo Bassimscand@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/120579Strengthened convexity of positive operator monotone decreasing functions2021-01-15T02:37:20+01:00Megumi Kirihatamscand@math.au.dkMakoto Yamashitamscand@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/120889Exact Green's formula for the fractional Laplacian and perturbations2021-01-15T02:37:14+01:00Gerd Grubbmscand@math.au.dk<p>Let Ω be an open, smooth, bounded subset of $ \mathbb{R}^n $. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), 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/120920Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces2021-01-15T02:37:08+01:00Aamena Al-Qabanimscand@math.au.dkTitus Hilberdinkmscand@math.au.dkJani A. Virtanenmscand@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 $\beta >0$. 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/121069The block Schur product is a Hadamard product2021-01-15T02:37:02+01:00Erik Christensenmscand@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 $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ 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/121092Stable homotopy, $1$-dimensional NCCW complexes, and Property (H)2021-01-15T02:36:56+01:00Qingnan Anmscand@math.au.dkGeorge A. Elliottmscand@math.au.dkZhichao Liumscand@math.au.dkYuanhang Zhangmscand@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 (H).</p>2020-09-03T14:34:41+02:00##submission.copyrightStatement##https://www.mscand.dk/article/view/121826Issue covers2020-09-03T14:39:13+02:00Mathematica Scandinavicamscand@math.au.dk<p>Issue covers</p>2020-09-03T00:00:00+02:00##submission.copyrightStatement##https://www.mscand.dk/article/view/121827Volume index2020-09-03T14:40:55+02:00Mathematica Scandinavicamscand@math.au.dk<p>Volume index</p>2020-09-03T00:00:00+02:00##submission.copyrightStatement##