Related papers: A Model Theoretic Perspective on Matrix Rings

Recent improvement on Tarski's procedure for quantifier elimination in the first order theory of real numbers makes it feasible to solve small instances of the following problems completely automatically: 1. listing all equality and…

Let $n\geq2$ be a natural number. Let $M_n(\mathbb{K})$ be the ring of all $n \times n$ matrices over a field $\mathbb{K}$. Fix natural number $k$ satisfying $1<k\leq n$. Under a mild technical assumption over $\mathbb{K}$ we will show that…

We compute the algebraic K-theory of the non-commutative ring k<x_1,...,x_n>/(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in…

Let $M_n(\mathbb{Z})$ the ring of $n$-by-$n$ matrices with integral entries, and $n \geq 2$. This paper studies the set $G_n(\mathbb{Z})$ of pairs $(A,B) \in M_n(\mathbb{Z})^2$ generating $M_n(\mathbb{Z})$ as a ring. We use several…

We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite…

We give a number of constructions where inverse limits seriously degrade properties of regular rings, such as unit-regularity, diagonalisation of matrices, and finite stable rank. This raises the possibility of using inverse limits to…

Let $K$ be a 2-torsion free ring with identity. We give a description of the Lie derivations of $R=R_{n}(K,J)=NT_{n}(K)+M_{n}(J)$, the ring of all $n\times n$ matrices over $K$ such that the entries on and above the main diagonal are…

It is well known that the full matrix ring over a skew-field is a simple ring. We generalize this theorem to the case of semirings. We characterize the case when the matrix semiring $\mathbf{M}_n(S)$, of all $n\times n$ matrices over a…

For certain rings $\mathcal{R}$, we construct explicit matrices representing nonzero classes in the algebraic $K$ theory group $NK_{1}(\mathcal{R})$.

We study two-dimensional classically integrable field theory with independent boundary condition on each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically…

Under suitable hypotheses on the ground field and on the matrix $M$, we discuss existence, uniqueness and properties of some additive decompositions of $M$ and of its image through a convergent series.

Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field,…

Matrix regularity is a key to various problems in applied mathematics. The sufficient conditions, used for checking regularity of interval parametric matrices, usually fail in case of large parameter intervals. We present necessary and…

We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers.

Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form -- usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently…

We consider the scattering matrices of massive quantum field theories with no bound states and a global $O(N)$ symmetry in two spacetime dimensions. In particular we explore the space of two-to-two S-matrices of particles of mass $m$…

Let $n$ and $s$ be fixed integers such that $n\geq 2$ and $1\leq s\leq \frac{n}{2}$. Let $M_n(\mathbb{K})$ be the ring of all $n\times n$ matrices over a field $\mathbb{K}$. If a map $\delta:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$…

Question 3 of [3] asks whether the matrix ring Mn(R) is nil clean, for any nil clean ring R. It is shown that positive answer to this question is equivalent to positive solution for Kothe's problem in the class of algebras over the field…

We formulate a notion of "geometric reductivity" in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result…

The issue of justifying the matrix-theory proposal is revisited. We first discuss how the matrix-string theory is derived directly starting from the eleven dimensional supermembrane wrapped around a circle of radius $R=g_s\ell_s$, without…