Linear Stochastic State Space Theory in the White Noise Space Setting

We study state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely its average behavior

A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II : Supmech and Quantum Systems

Supmech, which is noncommutative Hamiltonian mechanics \linebreak (NHM) (developed in paper I) with two extra ingredients : positive observable valued measures (PObVMs) [which serve to connect state-induced expectation values and classical probabilities] and the `CC condition' [which stipulates that the sets of observables and pure states be mutually separating] is proposed as a universal mechanics potentially covering all physical phenomena. It facilitates development of an autonomous formalism for quantum mechanics. Quantum systems, defined algebraically as supmech Hamiltonian systems with non-supercommutative system algebras, are shown to inevitably have Hilbert space based realizations (so as to accommodate rigged Hilbert space based Dirac bra-ket formalism), generally admitting commutative superselection rules. Traditional features of quantum mechanics of finite particle systems appear naturally. A treatment of localizability much simpler and more general than the traditional one is given. Treating massive particles as localizable elementary quantum systems, the Schr$\ddot{o}$dinger wave functions with traditional Born interpretation appear as natural objects for the description of their pure states and the Schr$\ddot{o}$dinger equation for them is obtained without ever using a classical Hamiltonian or Lagrangian. A provisional set of axioms for the supmech program is given.Comment: 55 pages; some modifications in text; improved treatment of topological aspects and of Noether invariants; results unchange

Homotopic and Geometric Galois Theory (online meeting)

In his "Letter to Faltings'', Grothendieck lays the foundation of what will become part of his multi-faceted legacy to arithmetic geometry. This includes the following three branches discussed in the workshop: the arithmetic of Galois covers, the theory of motives and the theory of anabelian Galois representations. Their geometrical paradigms endow similar but complementary arithmetic insights for the study of the absolute Galois group $\mathrm{G}_{\mathbb{Q}}$ of the field of rational numbers that initially crystallized into a functorially group-theoretic unifying approach. Recent years have seen some new enrichments based on modern geometrical constructions - e.g. simplicial homotopy, Tannaka perversity, automorphic forms - that endow some higher considerations and outline new geometric principles. This workshop brought together an international panel of young and senior experts of arithmetic geometry who sketched the future desire paths of homotopic and geometric Galois theory

Model Theory and Groups

The aim of the workshop was to discuss the connections between model theory and group theory. Main topics have been the interaction between geometric group theory and model theory, the study of the asymptotic behaviour of geometric properties on groups, and the model theoretic investigations of groups of finite Morley rank around the Cherlin-Zilber Conjecture

Second International Workshop on Harmonic Oscillators

The Second International Workshop on Harmonic Oscillators was held at the Hotel Hacienda Cocoyoc from March 23 to 25, 1994. The Workshop gathered 67 participants; there were 10 invited lecturers, 30 plenary oral presentations, 15 posters, and plenty of discussion divided into the five sessions of this volume. The Organizing Committee was asked by the chairman of several Mexican funding agencies what exactly was meant by harmonic oscillators, and for what purpose the new research could be useful. Harmonic oscillators - as we explained - is a code name for a family of mathematical models based on the theory of Lie algebras and groups, with applications in a growing range of physical theories and technologies: molecular, atomic, nuclear and particle physics; quantum optics and communication theory