Related papers: K-Primitivity : A Literature Survey

We show that a formal power series in $2N$ non-commuting indeterminates is a positive non-commutative kernel if and only if the kernel on $N$-tuples of matrices of any size obtained from this series by matrix substitution is positive. We…

Given proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, an $m \times n$ matrix $A$ with real entries is said to be semipositive if there exists a $x \in K_1^{\circ}$ such that $Ax \in K_2^{\circ}$, where…

Let $k$ be an uncountable field. We show that a finitely generated prime Goldie $k$-algebra of quadratic growth is either primitive or satisfies a polynomial identity, answering a question of Small in the affirmative.

A set $\mathcal{A}\subset \mathbb{N}$ is called additively decomposable (resp. asymptotically additively decomposable) if there exist sets $\mathcal{B},\mathcal{C}\subset \mathbb{N}$ of cardinality at least two each such that…

A finite non-regular primitive permutation group $G$ is extremely primitive if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, finding various applications. The…

Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes…

A subset of $\{1,2,\ldots,2n\}$ is said to be primitive if it does not contain any pair of elements $(u,v)$ such that $u$ is a divisor of $v$. Let $D(n)$ denote the number of primitive subsets of $\{1,2,\ldots,2n\}$ with $n$ elements.…

The $girth$ of a primitive Boolean matrix is defined to be the $girth$ of its associated digraph. In this paper, among all primitive Boolean matrices of order $n$, the primitive exponents of those of girth $g$ are considered. For the…

Let $F_k$ be the free group on $k$ generators, and let $H\le J\le \F_k$ be subgroups of finite rank. We present a new elementary algorithm to determine whether $H$ is a free factor of $J$. In particular, this algorithm can determine whether…

We introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and $k$-positivity for maps between matrix algebras. Starting from a compact family of $\alpha$-admissible unit…

A number field $K$ is primitive if $K$ and $\mathbb{Q}$ are the only subextensions of $K$. Let $C$ be a curve defined over $\mathbb{Q}$. We call an algebraic point $P\in C(\overline{\mathbb{Q}})$ primitive if the number field…

We provide an extensive list of desirable properties for an O-notation --- as used in algorithm analysis --- and reduce them to 8 primitive properties. We prove that the primitive properties are equivalent to the definition of the…

The purpose of the paper is to derive formulas that describe the structure of the induced supermodule H^0_G(\la) for the general linear supergroup G=GL(m|n) over an algebraically closed field K of characteristic p\neq 2. Using these…

Hermitian positive definite, totally positive, and nonsingular M-matrices enjoy many common properties, in particular: (A) positivity of all principal minors, (B) weak sign symmetry, (C) eigenvalue monotonicity, (D) positive stability. The…

Let q be a nonzero complex number that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O_q(M_{m,n}) of quantum matrices. Next, we describe all height one primes of O_q(M_{m,n}); these two problems…

Let $A$ be a matrix with nonnegative real entries. A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we…

Let $q$ be an even prime power and $m\geq2$ an integer. By $\mathbb{F}_q$, we denote the finite field of order $q$ and by $\mathbb{F}_{q^m}$ its extension degree $m$. In this paper we investigate the existence of a primitive normal pair…

Suppose $ m,n\geq 2 $ are co prime integers. We prove certain new symmetries of the base $ n $ representation of $ 1/m $, and in particular characterize the subgroup generated by $ n $ inside $ (\mathbb{Z}/m\mathbb{Z})^\times $. As an…

Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in…

We study the class of Hankel matrices for which the $k\times k$-block-matrices are positive semi-definite. We prove that a $k\times k$-block-matrix has non zero determinant if and only if all $k\times k$-block matrices have non zero…