Penot’s Compactness Property in Ultrametric Spaces with an Application

In this work, we investigate the compactness property in the sense of Penot in ultrametric spaces. Then, we show that spherical completeness is exactly the Penot’s compactness property introduced for convexity structures. The spherical completeness property misled some mathematicians to it to hyperconvexity in metric spaces. As an application, we discuss some fixed point results in spherically complete ultrametric spaces.

The lattice properties of 𝔛-local formations of finite groups

Let [Formula: see text] be a class of simple groups with a completeness property [Formula: see text]. Förster introduced the concept of [Formula: see text]-local formation in order to obtain a common extension of well-known theorems of Gaschütz–Lubeseder–Schmid and Baer [Publ. Mat. UAB 29(2–3) (1985) 39–76]. In this paper, it is proved that the lattice of all [Formula: see text]-local formations of finite groups is modular.

Rhythmic Keccak: SCA Security and Low Latency in HW

Glitches entail a great issue when securing a cryptographic implementation in hardware. Several masking schemes have been proposed in the literature that provide security even in the presence of glitches. The key property that allows this protection was introduced in threshold implementations as non-completeness. We address crucial points to ensure the right compliance of this property especially for low-latency implementations. Specifically, we first discuss the existence of a flaw in DSD 2017 implementation of Keccak by Gross et al. in violation of the non-completeness property and propose a solution. We perform a side-channel evaluation on the first-order and second-order implementations of the proposed design where no leakage is detected with up to 55 million traces. Then, we present a method to ensure a non-complete scheme of an unrolled implementation applicable to any order of security or algebraic degree of the shared function. By using this method we design a two-rounds unrolled first-order Keccak-

More on Uniform Paracompactness in Pointfree Topology

We revisit uniformly paracompact uniform frames and show that, in analogy with their spatial counterparts, they have a characterisation in terms of a “completeness property”. Namely, they are precisely those in which every weakly Cauchy filter clusters. We also give another characterisation in terms of the Čech-Stone compactification of the underlying frame. By tweaking the definition of uniformly paracompact frames, we define uniformly para-Lindelöf frames (analogously to same-named uniform spaces) and characterise them in terms of the Lindelöf coreflection of the underlying frame. This latter characterisation has no spatial counterpart.

Closure of dilates of shift-invariant subspaces

Let V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a point of A*-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.