Interpretations of the Beurling–Lax–Halmos Theorem on invariant subspaces of the unilateral shift are explored using the language of Hilbert modules. Extensions and consequences are considered in both the one and multivariate cases with an emphasis on the classical Hardy, Bergman and Drury–Arveson spaces.

Kirszbraun’s Theorem states that every Lipschitz map S → Rn, where S ⊆ Rm, has an extension to a Lipschitz map Rm → Rn with the same Lipschitz constant. Its proof relies on Helly’s Theorem: every family of compact subsets of Rn, having the property that each of its subfamilies consisting of at most n + 1 sets share a common point, has a non-empty intersection. We prove versions of these theorem...

The noise level of a high-throughput screening (HTS) experiment depends on various factors such as the quality and robustness of the assay itself and the quality of the robotic platform. Screening of compound mixtures is noisier than screening single compounds per well. A classification model based on naïve Bayes (NB) may be used to enrich such data. The authors studied the ability of the NB cl...

We give a complete and careful proof of Quillen’s theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories.

In locally compact spaces, (Borel-)measurable sets can be approximated by compact sets. Ulam extended this result to complete separable metric spaces. We give a constructive proof of Ulam’s theorem. It is first proved intuitionistically and then, using a logical ‘trick’ due to Ishihara, a proof acceptable in Bishop-style mathematics is obtained. We feel this proof provides some insight into Ish...

Under very general assumptions we show that Vafa-Witten theorem on vector symmetries in vector-like theories can be extended to some physically relevant gauge theories with non-positive definite integration measure as QCD with a Θ-vacuum term.

We extend the ‘∨-premorphisms’ part of the Ehresmann-ScheinNambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the ‘morphisms’ part. However, it is so-called ‘∧-premorphisms’ which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type t...

We consider the action of the operator ℒg(z) = (1 - z)(-1)∫ z (1)‍f(ζ)dζ on a class of "mixed norm" spaces of analytic functions on the unit disk, X = H α,ν (p,q) , defined by the requirement g ∈ X ⇔ r ↦ (1 - r) (α) M p (r, g ((ν))) ∈ L (q) ([0,1], dr/(1 - r)), where 1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > 0, and ν is a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet...

We briefly discuss here the Balog-Szemerédi theorem, compare its different versions and prove, following Gowers, the strongest one — also due to Gowers. 1. Discussion For a finite set A of elements of an abelian group and a group element s, by νA(s) we denote the number of representations of s as a sum of two elements of A: νA(s) = #{(a′, a′′) ∈ A× A : s = a′ + a′′}. We write 2A = {a′ + a′′ : a...

This paper deals with syzygies of the ideals of the Veronese embeddings. By Green’s Theorem we know that OPn(d) satisfies Green-Lazarsfeld’s Property Np ∀d ≥ p, ∀n. By Ottaviani-Paoletti’s theorem if n ≥ 2, d ≥ 3 and 3d − 2 ≤ p then OPn(d) does not satisfy Property Np. The cases n ≥ 3, d ≥ 3, d < p < 3d− 2 are still open (except n = d = 3). Here we deal with one of these cases, namely we prove ...