Two main approaches have been considered for modelling the dynamics of the SIS model on complex metapopulations, i.e, networks of populations connected by migratory flows whose configurations are described in terms of the connectivity distribution of nodes (patches) and the conditional probabilities of connections among classes of nodes sharing the same degree. In the first approach migration and transmission/recovery process alternate sequentially,...

$\mathrm{𝐒𝐩𝐅𝐢}$ is the category of spaces with filters: an object is a pair $(X,ℱ)$, $X$ a compact Hausdorff space and $ℱ$ a filter of dense open subsets of $X$. A morphism $f(Y,𝒢)\to (X,ℱ)$ is a continuous function $fY\to X$ for which $f^{-1}\left(F\right)\in 𝒢$ whenever $F\in ℱ$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these...

The interval eigenproblem in max-min algebra is studied. A classification of interval eigenvectors is introduced and six types of interval eigenvectors are described. Characterization of all six types is given for the case of strictly increasing eigenvectors and Hasse diagram of relations between the types is presented.

In this paper, we mainly use the properties of the minimum eigenvalue of the Fan product of $M$-matrices and Cauchy-Schwarz inequality, and propose some new bounds for the minimum eigenvalue of the Fan product of two $M$-matrices. These results involve the maximum absolute value of off-diagonal entries of each row. Hence, the lower bounds for the minimum eigenvalue are easily calculated in the practical examples. In theory, a comparison is given in this paper. Finally, to illustrate our results, a simple...

We prove that the principal eigenvector of a positive matrix represents the relative dominance of its rows or ranking of alternatives in a decision represented by the rows of a pairwise comparison matrix.

Soit $\beta$ un nombre de Pisot ; nous montrons que pour tout entier $n$ assez grand il existe une matrice carrée à coefficients positifs ou nuls dont l’ordre est égal au degré de $\beta$ et dont ${\beta }^n$ est valeur propre.Soit $\beta =a_1/\beta +a_2/{\beta }^2+\cdots +a_n/{\beta }^n+\cdots$ le $\beta$-développement de $\beta$ ; si $\beta$ est un nombre de Pisot, alors la suite ${\left(a_n\right)}_{n\ge 1}$ est périodique après un certain rang $n_0$ (pour $n\ge n_0$, $a_{n+k}=a_n$) et le polynôme$$X^{n_0+k}-(a_1{X}^{n_0+k-1}+\cdots +a_{n_0+k})-(X^{n_0}-(a_1{X}^{n_0}+\cdots +a_{n_0}))$$est appelé polynôme de Parry. Nous montrons qu’il existe un ensemble relativement dense d’entiers $n$ tels que le polynôme minimal de ${\beta }^n$ est égal à son polynôme...

Four applications are outlined of pseudospectra of highly nonnormal linear operators.

We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F\left(x\right)=ux{u}^{-1}$ or $F\left(x\right)=u{x}^t{u}^{-1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0...