Invariant Stein domains in Stein symmetric spaces and a non-linear complex convexity theorem

We prove a complex version of Kostant's non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of non-compact Riemannian symmetric G/K spaces are given.Comment: 9 page

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns \Xi\subeq G_\C /K_\C of non-compact Riemannian symmetric spaces $G/K$. The basic result is that affine symmetric spaces of $G$ can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.Comment: 29 page

Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions

Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures $\mu$ from some set $M$ to functions in a reproducing kernel Hilbert space (RKHS) with kernel $k$. The RKHS distance of two mapped measures is a semi-metric $d_k$ over $M$. We study three questions. (I) For a given kernel, what sets $M$ can be embedded? (II) When is the embedding injective over $M$ (in which case $d_k$ is a metric)? (III) How does the $d_k$-induced topology compare to other topologies on $M$? The existing machine learning literature has addressed these questions in cases where $M$ is (a subset of) the finite regular Borel measures. We unify, improve and generalise those results. Our approach naturally leads to continuous and possibly even injective embeddings of (Schwartz-) distributions, i.e., generalised measures, but the reader is free to focus on measures only. In particular, we systemise and extend various (partly known) equivalences between different notions of universal, characteristic and strictly positive definite kernels, and show that on an underlying locally compact Hausdorff space, $d_k$ metrises the weak convergence of probability measures if and only if $k$ is continuous and characteristic.Comment: Old and longer version of the JMLR paper with same title (published 2018). Please start with the JMLR version. 55 pages (33 pages main text, 22 pages appendix), 2 tables, 1 figure (in appendix

Holomorphic horospherical transform on non-compactly causal spaces

We develop integral geometry for non-compactly causal symmetric spaces. We define a complex horospherical transform and, for some cases, identify it with a Cauchy type integral.Comment: Revised, final version; to appear in IMRN, 38

AlScN: A III-V semiconductor based ferroelectric

Ferroelectric switching is unambigiously demonstrated for the first time in a III-V semiconductor based material: AlScN -- A discovery which could help to satisfy the urgent demand for thin film ferroelectrics with high performance and good technological compatibility with generic semiconductor technology which arises from a multitude of memory, micro/nano-actuator and emerging applications based on controlling electrical polarization. The appearance of ferroelectricity in AlScN can be related to the continuous distortion of the original wurtzite-type crystal structure towards a layered-hexagonal structure with increasing Sc content and tensile strain, which is expected to be extendable to other III-nitride based solid solutions. Coercive fields which are systematically adjustable by more than 3 MV/cm, high remnant polarizations in excess of 100 \mu C/cm$^2$ which constitute the first experimental estimate of the previously inaccessible spontaneous polarization in a III-nitride based material, an almost ideally square-like hysteresis resulting in excellent piezoelectric linearity over a wide strain interval from -0.3% to +0.4% as well as a paraelectric transition temperature in excess of 600{\deg}C are confirmed. This intriguing combination of properties is to our knowledge as of now unprecedented in the field of polycrystalline ferroelectric thin films and promises to significantly advance the commencing integration of ferroelectric functionality to micro- and nanotechnology, while at the same time providing substantial insight to one of the central open questions of the III-nitride semiconductors - that of their actual spontaneous polarization