Related papers: Cauchy Pairs and Cauchy Matrices

Following Lawvere's description of metric spaces using enriched category theory, we introduce a change in the base of enrichment that allows description of some aspects of (relativistic) causal spaces. All such spaces are Cauchy complete,…

This paper continues the authors' work on the question of unitary equivalence of matrices with entries in the complex-valued functions of a topological space (matrices over spaces). Specifically, we here consider the question of unitary…

We characterize those pairs of weights $ \sigma $ on $ \mathbb{R}$ and $ \tau $ on $ \mathbb{C}_+$ for which the Cauchy transform $\mathsf{C}_{\sigma} f (z) \equiv \int_{\mathbb{R}} \frac {f(x)} {x-z} \; \sigma (dx)$, $ z\in \mathbb{C}_+$,…

Lax pairs are a useful tool in finding conserved quantities of some dynamical systems. In this expository article, we give a motivated introduction to the idea of a Lax pair of matrices $(L,A)$, first for mechanical systems such as the…

In this paper, we discuss the representability almost everywhere (a.e.) in the plane of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite…

For every Banach space $X$ with a Schauder basis consider the Banach algebra $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ of all diagonal operators that are of the form $\lambda I + K$. We prove that $\mathbb{R}…

An algebraic isopair is a commuting pair of pure isometries that is annihilated by a polynomial defining a distinguished variety $\mathcal{V}$. The notion of the rank of a pure algebraic isopair with finite bimultiplicity is introduced. For…

In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair…

The current article continues our project on representation theory, Euler elements, causal homogeneous spaces and Algebraic Quantum Field Theory (AQFT). We call a pair (h,k) of Euler elements orthogonal if $e^{\pi i \ad h} k = -k$. We show…

Let $1 \leq k < n$ be integers. Two $n \times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + \mu B) = w_k(A) + w_k(B)$ for some scalar $\mu$ with $|\mu| = 1$; they form a TEA…

Suppose $G$ is a $p$-solvable group, where $p$ is odd. We explore the connection between lifts of Brauer characters of $G$ and certain local objects in $G$, called vertex pairs. We show that if $\chi$ is a lift, then the vertex pairs of…

The classical Cauchy completion of a metric space (by means of Cauchy sequences) as well as the completion of a uniform space (by means of Cauchy filters) are well-known to rely on the symmetry of the metric space or uniform space in…

A real square matrix $A$ is called a $Q$-matrix if the linear complementarity problem $LCP(A,q)$ has a solution for all $q \in \mathbb{R}^n$. This means that for every vector $q$ there exists a vector $x$ such that $x \geq 0, y=Ax+q\geq 0$…

Let $V$ be a nonempty finite set and $A=(a_{ij})_{i,j\in V}$ be a matrix with entries in a field $\mathbb{K}$. For a subset $X$ of $V$, we denote by $A[X]$ the submatrix of $A$ having row and column indices in $X$. We study the following…

Let $N$ be a natural number. We consider weighted Cauchy matrices of the form \[ \mathcal{C}_{a,A}=\left\{\frac{\sqrt{A_j A_k}}{a_k+a_j}\right\}_{j,k=1}^N, \] where $A_1,\dots,A_N$ are positive real numbers and $a_1,\dots,a_N$ are distinct…

Let \K denote an algebraically closed field and let V denote a vector space over \K with finite positive dimension. We consider an ordered pair of linear transformations A:V\to V,A*:V \to V that satisfy the following conditions:(i)Each of…

It is shown that a family of distributions on the complex space is characterized as the only family such that the orbit of one distribution under a certain group of transformations on the complex space is the same as that under the group of…

We study non-commutative real algebraic geometry for a unital associative *-algebra A viewing the points as pairs ({\pi},v) where {\pi} is an unbounded *-representation of A on an inner product space which contains the vector v. We first…

In this letter, a definition of the higher dimensional Lax pair for a lower dimensional system which may be a chaotic system is given. A special concrete (2+1)-dimensional Lax pair for a general (1+1)-dimensional three order autonomous…

We show that the K_1 group of a C*-algebra $A$ can be defined as homotopy classes of pairs, called balanced, of not necessarily unitary matrices over $A$ that have equal defects from being unitary. We also consider pairs of order zero…