MATHEMATICA SCANDINAVICA https://www.mscand.dk/ Mathematica Scandinavica en-US MATHEMATICA SCANDINAVICA 0025-5521 Remarks on Diophantine approximation in function fields https://www.mscand.dk/article/view/109985 <p>We study some problems in metric Diophantine approximation over local fields of positive characteristic.</p> Arijit Ganguly Anish Ghosh ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 5 14 10.7146/math.scand.a-109985 Minimal complexes of cotorsion flat modules https://www.mscand.dk/article/view/110787 <p>Let $R$ be a commutative noetherian ring. We give criteria for a complex of cotorsion flat $R$-modules to be minimal, in the sense that every self homotopy equivalence is an isomorphism. To do this, we exploit Enochs' description of the structure of cotorsion flat $R$-modules. More generally, we show that any complex built from covers in every degree (or envelopes in every degree) is minimal, as well as give a partial converse to this in the context of cotorsion pairs. As an application, we show that every $R$-module is isomorphic in the derived category over $R$ to a minimal semi-flat complex of cotorsion flat $R$-modules.</p> Peder Thompson ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 15 33 10.7146/math.scand.a-110787 Blow-ups and Fano manifolds of large pseudoindex https://www.mscand.dk/article/view/109996 <p>We describe the Kleiman-Mori cones of Fano manifolds of large pseudoindex that admit a structure of smooth blow-up.</p> Carla Novelli ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 34 50 10.7146/math.scand.a-109996 On a question of Sadullaev concerning boundary relative extremal functions https://www.mscand.dk/article/view/105141 <p>We study the relation between certain alternative definitions of the boundary relative extremal function. For various domains we give an affirmative answer to the question of Sadullaev whether these extremal functions are equal.</p> Ibrahim K. Djire Jan Wiegerinck ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 51 60 10.7146/math.scand.a-105141 Duality of ODE-determined norms https://www.mscand.dk/article/view/109390 <p>Recently the author initiated a novel approach to varying exponent Lebesgue space $L^{p(\cdot)}$ norms. In this approach the norm is defined by means of weak solutions to suitable first order ordinary differential equations (ODE). The resulting norm is equivalent with constant $2$ to a corresponding Nakano norm but the norms do not coincide in general and thus their isometric properties are different. In this paper the duality of these ODE-determined $L^{p(\cdot)}$ spaces is investigated. It turns out that the duality of the classical $L^p$ spaces generalizes nicely to this class of spaces. Here the duality pairing and Hölder's inequality work in the isometric sense which is a notable feature of these spaces. The uniform convexity and smoothness of these spaces are characterized under the anticipated conditions. A kind of universal space construction is also given for these spaces.</p> Jarno Talponen ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 61 80 10.7146/math.scand.a-109390 The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian https://www.mscand.dk/article/view/109674 <p>In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.</p> Manfred Stoll ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 81 101 10.7146/math.scand.a-109674 Galoisian approach to complex oscillation theory of some Hill equations https://www.mscand.dk/article/view/110889 <p>We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponent of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. These are solutions obtainable by suitable differential field extension constructions. In particular, we establish a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential is a combination of four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of a biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality, and satisfy Fredholm integral equations over suitable integration regions in $\mathbb{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.</p> Yik-Man Chiang Guo-Fu Yu ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 102 131 10.7146/math.scand.a-110889 Bounded point derivations on $R^p(X)$ and approximate derivatives https://www.mscand.dk/article/view/109998 <p>It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p &gt;2$, then there is an approximate derivative at $x_0$. A similar result is proven for higher-order bounded point derivations. This extends a result of Wang which was proven for $R(X)$, the uniform closure of rational functions with poles off $X$.</p> Stephen Deterding ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 132 148 10.7146/math.scand.a-109998 Multilinear square functions and multiple weights https://www.mscand.dk/article/view/105504 <p>In this paper we prove weighted estimates for a class of smooth multilinear square functions with respect to multilinear $A_{\vec P}$ weights. In particular, we establish weighted estimates for the smooth multilinear square functions associated with disjoint cubes of equivalent side-lengths. As a consequence, for this particular class of multilinear square functions, we provide an affirmative answer to a question raised by Benea and Bernicot (Forum Math. Sigma 4, 2016, e26) about unweighted estimates for smooth bilinear square functions.</p> Loukas Grafakos Parasar Mohanty Saurabh Shrivastava ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 149 160 10.7146/math.scand.a-105504 Issue covers https://www.mscand.dk/article/view/111949 <p>Issue covers</p> Mathematica Scandinavica ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 Volume front pages https://www.mscand.dk/article/view/111950 <p>Volume front pages</p> Mathematica Scandinavica ##submission.copyrightStatement## 2019-01-13 2019-01-13 124 1 1 4