There are several formulations of the second law, and they may, in principle, have different domains of validity. Here a simple mathematical theorem is proven which serves as the most general basis for the second law, namely the Thomson formulation (“cyclic changes cost energy”), applied to...

In the marriage problem, a variant of the bi-parted matching problem, each member has a “wish-list” expressing his/her preference for all possible partners; this list consists of random, positive real numbers drawn from a certain distribution. One searches the lowest cost for the society, at...

This paper starts with a review of work on singularities in the spectral density of random harmonic chains. Detailed descriptions of singular behavior near the maximal frequency, near special frequencies and near island frequencies are discussed. Next the site-disordered Anderson model is...

Dynamical properties are studied of particles which perform random walks on square lattices, where a fraction c of the sites has been replaced by impurities. The latter may be either better conducting than the host, or worse conducting or even non-conducting. The quantities studied are the...

A generalized Thouless formula is proposed as an approximate criterion for Anderson localization in two- and three-dimensional systems. The criterion is exact for ordered systems and in the mean field limit of infinite dimensionality. For random systems it is expected that possible corrections...

For chains of harmonic oscillators with random masses a set of equations is derived, which determine the spatial Fourier components of the average one-particle Green's function. These equations are valid for complex values of the frequency. A relation between the spectral density and functions...