We prove a new irreducibility result for polynomials over ${\mathbb Q}$ and we use it to construct new infinite families of reciprocal monogenic quintinomials in ${\mathbb Z}[x]$ of degree $2^n$ .

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$ . We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.

We introduce the $\textbf{h}$ -minimum spanning length of a family ${\mathcal A}$ of $n\times n$ matrices over a field $\mathbb F$ , where $\textbf{h}$ is a p-tuple of positive integers, each no more than n. For an algebraically closed field $\mathbb F$ , Burnside’s theorem on irreducibility is essentially that the $(n,n,\ldots ,n)$ -minimum spanning length of ${\mathcal A}$ exists if ${\mathcal A}$ is irreducible. We show that the $\textbf{h}$ -minimum spanning length of ${\mathcal A}$ exists for every $\textbf{h}=(h_1,h_2,\ldots , h_p)$ if ${\mathcal A}$ is an irreducible family of invertible matrices with at least three elements. The $(1,1, \ldots ,1)$ -minimum spanning length is at most $4n\log _{2} 2n+8n-3$ . Several examples are given, including one giving a complete calculation of the $(p,q)$ -minimum spanning length of the ordered pair $(J^*,J)$ , where J is the Jordan matrix.

We introduce the notion of the slot length of a family of matrices over an arbitrary field ${\mathbb {F}}$ . Using this definition it is shown that, if $n\ge 5$ and A and B are $n\times n$ complex matrices with A unicellular and the pair $\{A,B\}$ irreducible, the slot length s of $\{A,B\}$ satisfies $2\le s\le n-1$ , where both inequalities are sharp, for every n. It is conjectured that the slot length of any irreducible pair of $n\times n$ matrices, where $n\ge 5$ , is at most $n-1$ . The slot length of a family of rank-one complex matrices can be equal to n.

We show that an irreducible family ${\mathcal{S}}$ of complex $n\times n$ matrices satisfies Paz’s conjecture if it contains a rank-one matrix. We next investigate properties of families of rank-one matrices. If ${\mathcal{R}}$ is a linearly independent, irreducible family of rank-one matrices then (i) ${\mathcal{R}}$ has length at most $n$, (ii) if all pairwise products are nonzero, ${\mathcal{R}}$ has length 1 or 2, (iii) if ${\mathcal{R}}$ consists of elementary matrices, its minimum spanning length $M$ is the smallest integer $M$ such that every elementary matrix belongs to the set of words in ${\mathcal{R}}$ of length at most $M$. Finally, for any integer $k$ dividing $n-1$, there is an irreducible family of elementary matrices with length $k+1$.

Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$, with $n\geq 3$ and $1\leq m<n$, is irreducible over $\mathbb{Q}$. By explicitly calculating the discriminant of $f(x)$, we prove that, when $\gcd (n,mB)=C=1$, there exist infinitely many values of $A$ such that the set $\{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$ is an integral basis for the ring of integers of $\mathbb{Q}(\unicode[STIX]{x1D703})$, where $f(\unicode[STIX]{x1D703})=0$.

Let $K$ be a number field with ring of integers $\mathbb{Z}_{K}$. We prove two asymptotic formulas connected with the distribution of irreducible elements in $\mathbb{Z}_{K}$. First, we estimate the maximum number of nonassociated irreducibles dividing a nonzero element of $\mathbb{Z}_{K}$ of norm not exceeding $x$ (in absolute value), as $x\rightarrow \infty$. Second, we count the number of irreducible elements of $\mathbb{Z}_{K}$ of norm not exceeding $x$ lying in a given arithmetic progression (again, as $x\rightarrow \infty$). When $K=\mathbb{Q}$, both results are classical; a new feature in the general case is the influence of combinatorial properties of the class group of $K$.

Node importance or centrality evaluation is an important methodology for network analysis. In this paper, we are interested in the study of objects appearing in several networks. Such common objects are important in network-network interactions via object-object interactions. The main contribution of this paper is to model multiple networks where there are some common objects in a multivariate Markov chain framework, and to develop a method for solving common and non-common objects’ stationary probability distributions in the networks. The stationary probability distributions can be used to evaluate the importance of common and non-common objects via network-network interactions. Our experimental results based on examples of co-authorship of researchers in different conferences and paper citations in different categories have shown that the proposed model can provide useful information for researcher-researcher interactions in networks of different conferences and for paper-paper interactions in networks of different categories.

We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The main new tool we use is the equivariant assignment of a point $Q(X)$ to any end $X\in\partial F_n$, given an action of $F_n$ on an $\bm{R}$-tree $T$ with trivial arc stabilizers; this point $Q(X)$ may be in $T$, or in its metric completion, or in its boundary.

AMS 2000 Mathematics subject classification: Primary 20F65; 20E05; 20E08

In the study of null-persistent Markov chains the sequence plays an important role. (See for example, Orey (1971), Athreya and Ney (1980).) In this paper we consider the rate of growth of UN to ∞ as N → ∞. This rate of growth is related to the rate of growth of as N → ∞ and also to the rate at which as n → ∞.

Let t ≥ 0 be a Markov transition probability semigroup on a general space satisfying a suitable φ-irreducibility condition. We show the existence of (i) a decay parameter λ ≥ 0 which is a common abscissa of convergence of the integrals ƒ estPt (x, A) dt for almost all x and all suitable A, (ii) a natural classification into λ-positive, λ-null and λ-transient cases. Moreover this classification is completely determined by any one of the h-skeleton chains of (Pt). We study the convergence of eλtPt(x, A) in the λ-positive case, and show that the limit f(X)π(A) (where f and π are the unique λ-invariant function and measure, normalized so that π(f) = 1) is reached at a uniform exponential rate of convergence, i.e. ||eλtPt(x, ·)-f(x)π(·)||f = O (e−αt) for some α > 0 and almost all x if there is a π-positive set such that the convergence is exponentially fast on this set. These results are used to deduce conditions for (Pt) to have quasi-stationary distributions.

If an n × n stochastic matrix has a column with no zeros, one can immediately conclude that the chain is ergodic and the state corresponding to that column is persistent and aperiodic. In this paper it is shown that it is decidable whether or not some power of a finite stochastic matrix has a positive column. Some problems regarding positive columns in infinite stochastic matrices are also considered.