Related papers: A Model Theoretic Perspective on Matrix Rings

We give a sufficient condition for a model theoretic structure $B$ to 'inherit' quantifier elimination from another structure $A$. This yields an alternative proof of one of the main result from \cite{kle}, namely quantifier elimination for…

We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra $\mathcal{N}$ is never model complete if its…

This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next…

Let $R$ be an o-minimal expansion of a group in a language in which $\textrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate for a valuational cut in $R$. We identify a condition that implies quantifier elimination for…

Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of…

The issue of non-perturbative background independent quantization of matrix models is addressed. The analysis is carried out by considering a simple matrix model which is a matrix extension of ordinary mechanics reduced to 0 dimension. It…

The matrix model formulation of M theory can be generalized to compact transverse backgrounds such as tori. If the number of compact directions is K then the matrix model must be generalized to K+1 dimensional super Yang Mills theory on a…

Let $\mathfrak{R}$ and $\mathfrak{R}'$ be two associative rings (not necessarily with the identity elements). A bijective map $\varphi$ of $\mathfrak{R}$ onto $\mathfrak{R}'$ is called a \textit{$m$-multiplicative isomorphism} if {$\varphi…

The general linear group acts on $m$-tuples of $N\times N$ matrices by simultaneous conjugation. Quantum deformations of the corresponding rings of invariants and the so-called trace rings are investigated.

We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.

The decomposition of arbitrary unitary transformations into sequences of simpler, physically realizable operations is a foundational problem in quantum information science, quantum control, and linear optics. We establish a 1D Quantum Field…

In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…

A proposal for the matrix model formulation of the M-theory on a space with a boundary is given. A general machinery for modding out a symmetry in M(atrix) theory is used for a Z_2 symmetry changing the sign of the X_1 coordinate. The…

Let K be an infinite field and let R be a K-algebra endowed with a homogeneous polynomial norm N of degree n. If N satisfies a formal analogue of the Cayley-Hamilton Theorem the we will show that R is a quotient of the ring of the…

We extend Latimer and MacDuffee's theorem to a general commutative domain and apply this result to study similarity of matrices over integral rings of number fields. We also conjecture similarity over discrete valuation rings can be descent…

This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite…

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a…

We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first…

In this article, we explore the problem of constructing high-dimensional expanders through the study of relations between expansion constants over different rings. We investigate expansion constants of integer matrices regarded as morphisms…

The principle of equivalence is translated into the language of the world-volume field theories that define matrix and string theories. This idea leads to explore possible matrix descriptions of M-theory compactifications. An interesting…