Random banded and sparse matrices

This article discusses some mathematical results and conjectures about random band matrix ensembles (RBM) and sparse matrix ensembles. Spectral problems of RBM and sparse matrices can be expressed in terms of supersymmetric (SUSY) statistical mechanics that provides a dual representation for disordered quantum systems. This representation offers important insights into nonperturbative aspects of the spectrum and eigenfunctions of RBM. The article first presents the definition of RBM ensembles before considering the density of states, the behaviour of eigenvectors, and eigenvalue statistics for RBM and sparse random matrices. In particular, it highlights the relations with random Schrödinger (RS) and the role of the dimension of the lattice. It also describes the connection between RBM and statistical mechanics, the spectral theory of large random sparse matrices, conjectures and theorems about eigenvectors and local spacing statistics, and the RS operator on the Cayley tree or Bethe lattice.

Vortex Shedding in a Tandem Circular Cylinder System With a Yawed Downstream Cylinder

This investigation examines the flow produced by a tandem cylinder system with the downstream cylinder yawed to the mean flow direction. The yaw angle was varied from α=90deg (two parallel tandem cylinders) to α=60deg; this has the effect of varying the local spacing ratio between the cylinders. Fluctuating pressure and hot-wire measurements were used to determine the vortex-shedding frequencies and flow regimes produced by this previously uninvestigated flow. The results showed that the frequency and magnitude of the vortex shedding varies along the cylinder span depending on the local spacing ratio between the cylinders. In all cases the vortex-shedding frequency observed on the front cylinder had the same shedding frequency as the rear cylinder. In general, at small local spacing ratios the cylinders behaved as a single large body with the shear layers separating from the upstream cylinder and attaching on the downstream cylinder, this caused a correspondingly large, low frequency wake. At other positions where the local span of the tandem cylinder system was larger, small-scale vortices began to form in the gap between the cylinders, which in turn increased the vortex-shedding frequency. At the largest spacings, classical vortex shedding persisted in the gap formed between the cylinders, and both cylinders shed vortices as separate bodies with shedding frequencies typical of single cylinders. At certain local spacing ratios two distinct vortex-shedding frequencies occurred indicating that there was some overlap in these flow regimes.

Vortex Shedding in a Yawed-Tandem Circular Cylinder Arrangement

This investigation examines the flow produced by a tandem cylinder system with the downstream cylinder yawed to the mean flow direction. The yaw angle was varied from α = 90° (two parallel tandem cylinders) to α = 60°; this has the effect of varying the local spacing ratio between the cylinders. Fluctuating pressure and hot-wire measurements were used to determine the vortex-shedding frequencies and flow regimes produced by this previously uninvestigated flow. The results showed that the frequency and magnitude of the vortex-shedding varies along the cylinder span depending on the local spacing ratio between the cylinders. In all cases the vortex-shedding frequency observed on the front cylinder had the same shedding frequency as the rear cylinder. In general, at small local spacing ratios the cylinders behaved as a single large body with the shear layers separating from the upstream cylinder and attaching on the downstream cylinder, this caused a correspondingly large, low frequency wake. At other positions where the local span of the tandem cylinder system was larger, small scale vortices began to form in the gap between the cylinders which in turn increased the vortex-shedding frequency. At the largest spacings, classical vortex-shedding persisted in the gap formed between the cylinders and both cylinders shed vortices as separate bodies with shedding frequencies typical of single cylinders. At certain local spacing ratios two distinct vortex-shedding frequencies occurred indicating that there was some overlap in these flow regimes.