Optical and transport properties of materials depend heavily upon features of electronic band structures in proximity of energy extrema in the Brillouin zone (BZ). Such features are generally described in terms of multi-dimensional quadratic expansions and corresponding definitions of effective masses. Multi-dimensional quadratic expansions, however, are permissible only under strict conditions that are typically violated when energy bands become degenerate at extrema in the BZ. Even for energy bands that are non-degenerate at critical points in the BZ there are instances in which multi-dimensional quadratic expansions cannot be correctly performed. Suggestive terms such as "band warping", "fluted energy surfaces", or "corrugated energy surfaces" have been used to refer to such situations and ad hoc methods have been developed to treat them. While numerical calculations may reflect such features, a complete theory of band warping has not hitherto been developed. We define band warping as referring to band structures that do not admit second-order differentiability at critical points in k-space and we develop a generally applicable theory, based on radial expansions, and a corresponding definition of angular effective mass. Our theory also accounts for effects of band non-parabolicity and anisotropy, which hitherto have not been precisely distinguished from, if not utterly confused with, band warping. Based on our theory, we develop precise procedures to evaluate band warping quantitatively. As a benchmark demonstration, we analyze the warping features of valence bands in silicon using first-principles calculations and we compare those with previous semi-empirical models. As an application of major significance to thermoelectricity, we use our theory and angular effective masses to generalize derivations of tensorial transport coefficients for cases of either single or multiple electronic bands, with either quadratically expansible or warped energy surfaces. From that theory we discover the formal existence at critical points of transport-equivalent ellipsoidal bands that yield identical results from the standpoint of any transport property. Additionally, we illustrate with some basic multi-band models the drastic effects that band warping and anisotropy can induce on thermoelectric properties such as electronic conductivity and thermopower tensors.

The modeling and investigation of the dynamics and configurations of animal groups is a subject of growing attention. In this paper, we present a continuum model of flocking and use it to investigate the reaction of a flock to an obstacle or an attacking predator. We show that the flock response is in the form of density disturbances that resemble Mach cones whose configuration is determined by the anisotropic propagation of waves through the flock. We analytically and numerically test relations that predict the Mach wedge angles, disturbance heights, and wake widths. We find that these expressions are insensitive to many of the parameters of the model. * Electronic address: nmech@umd.edu;

We investigate geodesic orbits and manifolds for metrics associated with Schwarzschild geometry, considering space and time curvatures separately. For 'a-temporal' space, we solve a central geodesic orbit equation in terms of elliptic integrals. The intrinsic geometry of a two-sided equatorial plane corresponds to that of a full Flamm's paraboloid. Two kinds of geodesics emerge. Both kinds may or may not encircle the hole region any number of times, crossing themselves correspondingly. Regular geodesics reach a periastron greater than the rS Schwarzschild radius, thus remaining confined to a half of Flamm's paraboloid. Singular or s-geodesics tangentially reach the rS circle. These s-geodesics must then be regarded as funneling through the 'belt' of the full Flamm's paraboloid. Infinitely many geodesics can possibly be drawn between any two points, but they must be of specific regular or singular types. A precise classification can be made in terms of impact parameters. Geodesic structure and completeness is conveyed by computer-generated figures depicting either Schwarzschild equatorial plane or Flamm's paraboloid. For the 'curved-time' metric, devoid of any spatial curvature, geodesic orbits have the same apsides as in Schwarzschild space-time. We focus on null geodesics in particular. For the limit of light grazing the sun, asymptotic 'spatial bending' and 'time bending' become essentially equal, adding up to the total light deflection of 1.75 arc-seconds predicted by general relativity. However, for a much closer approach of the periastron to rS, 'time bending' largely exceeds 'spatial bending' of light, while their sum remains substantially below that of Schwarzschild space-time.