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Let $\mathcal A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $\mathcal A''$ of $\mathcal A$ to any hyperplane endowed with the natural multiplicity $\kappa$ is then a free multiarrangement. Recently, in [Hoge-R\"ohrle2022], an analogue of Ziegler's theorem for the stronger notion of inductive freeness was proved: if $\mathcal A$ is inductively free, then so is the free multiarrangement $(\mathcal A'',\kappa)$. In [Hoge-R\"ohrle2018], all reflection arrangements which admit inductively free Ziegler restrictions are classified. The aim of this paper is an extension of this classification to all restrictions of reflection arrangements utilizing the aforementioned fundamental result from [Hoge-R\"ohrle2022]., Comment: v1 21 pages. arXiv admin note: text overlap with arXiv:1705.02767, arXiv:2204.09540

We provide a framework consisting of tools and metatheorems for the end-to-end verification of security protocols, which bridges the gap between automated protocol verification and code-level proofs. We automatically translate a Tamarin protocol model into a set of I/O specifications expressed in separation logic. Each such specification describes a protocol role's intended I/O behavior against which the role's implementation is then verified. Our soundness result guarantees that the verified implementation inherits all security (trace) properties proved for the Tamarin model. Our framework thus enables us to leverage the substantial body of prior verification work in Tamarin to verify new and existing implementations. The possibility to use any separation logic code verifier provides flexibility regarding the target language. To validate our approach and show that it scales to real-world protocols, we verify a substantial part of the official Go implementation of the WireGuard VPN key exchange protocol.

Amorphous gallium oxide thin films were grown by plasma-enhanced atomic layer deposition on (100) silicon substrates from trimethylgallium Ga(CH3)3 precursor and oxygen plasma. At 200 ��C, the growth per cycle is in the range of 0.65���0.70 �� for O2 plasma exposure times ranging from 3 up to 30 s during each cycle. The effect of O2 plasma exposure times on the interfacial SiOx regrowth and the electrical properties was investigated. In situ spectroscopic ellipsometry shows that the SiOx regrowth occurs during the first three cycles and is limited to 0.27 nm for plasma times as long as 30 s. Increasing the O2 plasma exposure during each ALD cycle leads to a drastic decrease in the leakage current density (more than 5 orders of magnitude for 30 nm films), which is linked to the suppression of oxygen vacancy states as evidenced by spectroscopic ellipsometry. Interestingly, an increase in the dielectric constant with increasing O2 plasma exposure time is observed, reaching a value of �����������14.2, larger than that of single crystalline ��-Ga2O3. This study highlights the crucial role of oxygen plasma exposure time in the control and tuning of the electrical properties of amorphous gallium oxide films.