Reflexivity of the commutant and local commutants of an algebraic operator

We show that an algebraic operator on a complex Banach space has reflexive commutant if and only if each zero of the minimal polynomial of the operator is simple. Further, for any operator, the local commutant at an eigenvector is reflexive. On the other hand, for an algebraic operator whose minimal polynomial has at least one zero that is not simple, the local commutant of the operator at a given vector is reflexive precisely when the vector is an eigenvector.     

Verallgemeinerung eines Satzes von B. Huppert und J. G. Thompson

Ohne Zusammenfassung     

Endliche Gruppen mit lauter nilpotenten zweitmaximalen Untergruppen

Ohne ZusammenfassungHerrnReinhold Baer zum 60. Geburtstag am 22. Juli 1962 gewidmet     

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

     

Towards Massively Parallel Computations in Algebraic Geometry

Foundations of Computational Mathematics - Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical...     

Algebraic Torsion in Contact Manifolds

We extract an invariant taking values in \mathbbNè{￥}{\mathbb{N}\cup\{\infty\}} , which we call the order of algebraic torsion, from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order 0 if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order 1 (though the converse is not true). We also construct examples for each k ? \mathbbN{k \in \mathbb{N}} of contact 3-manifolds that have algebraic torsion of order k but not k − 1, and derive consequences for contact surgeries on such manifolds.     

Replication in one-dimensional cellular automata

In a cellular automaton (CA), replication is the ability to indefinitely generate copies of a finite collection of patterns, starting from finite seeds. A transparent feature of additive CA, replication mechanisms are less clear in the absence of additivity; this paper investigates such dynamics through several examples. For the 1 Or 2 rule and its generalizations, replication is inevitable and we investigate self-organization properties. In the Perturbed Exactly 1 rule we study frequency of replicators, and the new phenomenon is called quasireplication. The last CA is the Extended 1 Or 3 rule, which allows for replication on different backgrounds. We employ a mixture of rigorous and empirical techniques.