Related papers: Integer factorization of a positive-definite matri…

For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the…

A dimension group is a partially ordered countable group such that (1) every finite subset is contained in an ordered subgroup which is a finite direct power of Z and (2) the group has an order unit i.e. a positive element u such that every…

Matrix Code gives imperative programming a mathematical semantics and heuristic power comparable in quality to functional and logic programming. A program in Matrix Code is developed incrementally from a specification in pre/post-condition…

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…

It is known that any symmetric matrix $M$ with entries in $\R[x]$ and which is positive semi-definite for any substitution of $x\in\R$, has a Smith normal form whose diagonal coefficients are constant sign polynomials in $\R[x]$. We…

We characterize injectivity of von Neumann algebras in terms of factoring bilinear maps as products of linear maps.

For every prime integer $p$, an explicit factorization of the principal ideal $p\z_K$ into prime ideals of $\z_K$ is given, where $K$ is a quartic number field defined by an irreducible polynomial $X^4+aX+b\in\z[X]$.

In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…

We present a simple proof of the factorization of (complex) symmetric matrices into a product of a square matrix and its transpose, and discuss its application in establishing a uniqueness property of certain antilinear operators.

In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral…

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…

We prove that the variety defined by the determinant of the matrix of diagonals is $F$-pure for matrices of all sizes and in all positive prime characteristics. Moreover, we find a system of parameters for it.

Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest…

We present a convenient notation for positive/negative-conditional equations. The idea is to merge rules specifying the same function by using case-, if-, match-, and let-expressions. Based on the presented macro-rule-construct,…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…

We assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) set having the group property. The…

A well-known fact in linear algebra is that $A^T A$ is always positive semi-definite for any real matrix $A$. We consider a generalization of this fact via the following decision problem. Given a symbolic product of length $k$, consisting…

The construction of a generic representation of $g\ell(n+1)$ or of the trigonomentric deformation of its enveloping algebra known as algebraic induction is conveniently formulated in term of Lax matrices. The Lax matrix of the constructed…

By the theory of elliptic curves, we study the integers representable as the product of the sum of four integers with the sum of their reciprocals and give a sufficient condition for the integers with a positive representation.

Some real functions f induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define linear mapping beta on matrices (which is basic in the…