On the phase space approach to complexity

We elaborate in some detail on a new phase space approach to complexity, due to Y.-C. Zhang. We show in particular that the connection between maximal complexity and power law noise or correlations can be derived from a simple variational principle. For a 1D signal we find 1/f noise, in accordance with Zhang.     

Wrinkle-free solutions in the theory of annular elastic membranes

Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Föppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.     

Multicenter ligand transformations of thioureas on ruthenium clusters

The reaction of trirutheniumdodecacarbonyl with various thioureas was found to give, depending upon the substituents and the reaction conditions, a large variety of tri-, tetra-, penta-, and hexanuclear ruthenium clusters. The ligand systems observed in the products are derived from the thioureas employed; the fragmentation of the thiourea molecules involve N-H, C-S, C-H, and C-N activation processes.     

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].