Group analysis of differential equations and generalized functions

We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows to compute symmetries of linear and nonlinear differential equations containing generalized function terms. Thereby, the group generators and group actions may be given by generalized functions themselves.Comment: 27 pages, LaTe

The geography of creative people in Germany

It has been argued that creativity is an important source of regional growth. This paper investigates the geography of people in creative occupation in Germany. The population share of the Creative Class as well as of bohemians and artists is relatively high in larger cities, but smaller places and rural regions may also have a considerable proportion of people with a creative job. While ethnical and cultural diversity and a high level of public supply in health care and education can explain the distribution of creative people, employment opportunities seem to play only a minor role. A high share of creative occupations seems to be conducive to regional growth; however, the exact nature of this relationship is still unclear

Asymmetric response of a jammed plastic bead raft

Fluctuation-dissipation relations have received significant attention as a potential method for defining an effective temperature in nonequilibrium systems. The successful development of an effective temperature would be an important step in the application of statistical mechanics principles to systems driven far from equilibrium. Many of the systems of interest are sufficiently dense that they are close to the jamming transition, a point at which interesting correlations develop. Here we study the response function in a driven system of plastic beads as a function of the density in order to elucidate the impact of the jamming transition on the use of fluctuation-dissipation relations. The focus is on measuring the response function for applied shear stress. We find that even when the amplitude of the applied stress leads to a linear response in the strain, the time scale of the response is dependent on the direction of the applied stress

Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion

We propose a robust autofocus method for reconstructing digital Fresnel holograms. The numerical reconstruction involves simulating the propagation of a complex wave front to the appropriate distance. Since the latter value is difficult to determine manually, it is desirable to rely on an automatic procedure for finding the optimal distance to achieve high-quality reconstructions. Our algorithm maximizes a sharpness metric related to the sparsity of the signal’s expansion in distance-dependent waveletlike Fresnelet bases. We show results from simulations and experimental situations that confirm its applicability

The State Of Play: A Notional Machine for Learning Programming

Comprehension of programming and programs is known to be a difficult task for many beginning students, with many computing courses showing significant drop out and failure rates. In this paper, we present a new notional machine de- sign and implementation to help with understanding of pro- gramming and its dynamics for beginning learners. The no- tional machine offers an abstraction of the physical machine designed for comprehension and learning purposes. We in- troduce the notional machine and a graphical notation for its representation. We also present Novis, an implementation of a dynamic real-time visualiser of this notional machine, integrated into BlueJ