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.

Rationale: Asthma prevalence, onset, remission and relapse, and healthcare use have been intensively studied. However, asthma symptom progression through childhood and adolescence has not been well studied, in part due to the challenges in obtaining consistent and robust long-term follow-up data on a large series of subjects with asthma.Objectives: To use the asthma diary symptom data of the Childhood Asthma Management Program placebo group (5 yr, 418 subjects, and total 564,518 records) to establish sex-specific highresolution time courses of the natural progression of asthma symptoms through childhood and adolescence.Methods: We used the asthma diary symptom code as a measure of daily disease severity. Annual records of Tanner stage were used to determine the influence of puberty on severity. A data alignment technique was used to derive 13-year time courses of mean symptoms and mean Tanner stage.Measurements and Main Results: Data analyses showed three age-and sex-related phases of asthma symptom progression: Phase 1 (ages 5 and 6 yr)-greater severity in boys; Phase 2 (ages 7 to 9 yr)-no sex difference in severity; and Phase 3 (age 10-17 yr)-greater severity in girls. The continuous decline of symptoms in both sexes stops abruptly at the onset of puberty. Conclusions:The severity of asthma symptoms varies through childhood and adolescence, and patterns differ by sex. Puberty has a strong influence on symptom progression in both sexes. Progression of symptoms is a distinct aspect of asthma epidemiology.

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