Paolo Zanardo

ALGEBRAIC ENTROPY FOR ABELIAN GROUPS

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism φ of a torsion group as the sum of the algebraic entropies of the restriction to a φ-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.

A general notion of algebraic entropy and the rank-entropy

Abstract We give a general definition of a subadditive invariant i of Mod(R), where R is any ring, and the related notion of algebraic entropy of endomorphisms of R-modules, with respect to i. We examine the properties of the various entropies that arise in different circumstances. Then we focus on the rank-entropy, namely the entropy arising from the invariant ‘rank’ for Abelian groups. We show that the rank-entropy satisfies the Addition Theorem. We also provide a uniqueness theorem for the rank-entropy.

Completely positive matrices and positivity of least squares solutions

Abstract A sufficient condition for a doubly nonnegative matrix to be completely positive is given, in terms of the positivity of the least squares solution of a linear system associated to the matrix. Some known results on completely positive matrices are derived by this condition.

Fully inert subgroups of free Abelian groups

A subgroup

Kurosch invariants for torsion-free modules over Nagata valuation domains

H

Idempotent Pairs and PRINC Domains

H of an Abelian group

Generalised E-Algebras over valuation domains

G