Related papers: A short proof for the characterisation of tight fr…

This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite…

Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete…

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of…

The Titchmarsh convolution theorem is a celebrated result about the support of the convolution of two functions. We present a simple proof based on the canonical factorization theorem for bounded holomorphic functions on the unit disk.

The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction…

A common criterion in the design of finite Hilbert space frames is minimal coherence, as this leads to error reduction in various signal processing applications. Frames that achieve minimal coherence relative to all unit-norm frames are…

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…

This paper gives two different proofs to a structural theorem of decreasing minimization (lexicographic optimization) on integrally convex sets. The theorem states that the set of decreasingly minimal elements of an integrally convex set…

Thin sums matroids were introduced to extend the notion of representability to non-finitary matroids. We give a new criterion for testing when the thin sums construction gives a matroid. We show that thin sums matroids over thin families…

Probabilistic algorithms are applied to prove theorems about the finite general linear and unitary groups which are typically proved by techniques such as character theory and Moebius inversion. Among the theorems studied are Steinberg's…

We present coincidence and common fixed point results of selfmappings satisfying a contraction type in partially ordered metric spaces. As an application, we give an existence theorem for a common solution of integral equations.

We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk$-linear monoidal functor can be lifted…

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…

We give a direct and elementary proof of the theorem on formal functions by studying the behaviour of the Godement resolution of a sheaf of modules under completion.

The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated…

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to…

We provide a new method for constructing equiangular tight frames (ETFs). The construction is valid in both the real and complex settings, and shows that many of the few previously-known examples of ETFs are but the first representatives of…

The tensor product of two ordered vector spaces can be ordered in more than one way, just as the tensor product of normed spaces can be normed in multiple ways. Two natural orderings have received considerable attention in the past, namely…

We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, We can give some control on multiplicities (a result that needs the Classification of Finite Simple…

We show that much of the theory of finite tight frames can be generalised to vector spaces over the quaternions. This includes the variational characterisation, group frames, and the characterisations of projective and unitary equivalence.…