Formally self adjointness for the Dirac operator on homogeneous spaces

Introduction. In [5], Wolf proved that the Dirac operator is essentially self adojoint over a Riemannian spin manifold M and he used it to give explicit realization of unitary representations of Lie groups. Let K be a Lie group and a a Lie group homomorphism of K into SO(n) which factors through Spin (ή). He defined the Dirac operator on spinors wit

Instanton-noninstanton transition in nonintegrable tunneling processes: A renormalized perturbation approach

The instanton-noninstanton (I-NI) transition in the tunneling process, which has been numerically observed in classically nonintegrable quantum maps, can be described by a perturbation theory based on an integrable Hamiltonian renormalized so as to incorporate the integrable part of the map. The renormalized perturbation theory is successfully applied to the two quantum maps, the H\'enon and standard maps. In spite of different nature of tunneling in the two systems, the I-NI transition exhibits very common characteristics. In particular, the manifestation of I-NI transition is obviously explained by a remarkable quenching of the renormalized transition matrix element. The enhancement of tunneling probability after the transition can be understood as a sudden change of the tunneling mechanism from the instanton to quite a different mechanism supported by classical flows just outside of the stable-unstable manifolds of the saddle on the top of the potential barrier.Comment: 6 pages, 4 figure

An allometric smoothing function to describe the relation between otolith and somatic growth over the lifespan of walleye pollock (Theragra chalcogramma)

We propose a new equation to describe the relation between otolith length (OL) and somatic length (fork length [FL]) of fish for the entire lifespan of the fish. The equation was developed by applying a mathematical smoothing method based on an allometric equation with a constant term for walleye pollock (Theragra chalcogramma) —a species that shows an extended longevity (>20 years). The most appropriate equation for defining the relation between OL and FL was a four-phase allometric smoothing function with three inflection points. The inflection points correspond to the timing of settlement of walleye pollock, changes in sexual maturity, and direction of otolith growth. Allometric smoothing functions describing the relation between short otolith radius and FL, long otolith radius and FL, and FL and body weight were also developed. The proposed allometric smoothing functions cover the entire lifespan of walleye pollock. We term these equations “allometric smoothing functions for otolith and somatic growth over the lifespan of walleye pollock.

Petrological and geochemical study of the Yamato-74359 and Yamato-74360 achondrites

Two Antarctic achondrites, Yamato (Y)-74359 and Y-74360,are very similar in bulk chemical compositions to the silicate portions of H chondrites, but the siderophile elements are extremely depleted in comparison to those of H chondrites. They consist of olivine, pyroxene and cryptocrystalline feldspar with minor amounts of chromite, kamacite, and troilite, and the chemical compositions of the constituent minerals are similar to those in H chondrites. Taking into consideration that the two achondrites have oxygen isotopic compositions typical of H chondrites, they were produced from H chondrites or the precursors of H chondrites. Chondrules are not observed, and the texture of the two achondrites is similar to one another, although Y-74360 is coarser-grained than Y-74359. Olivine occurs as euhedral or subhedral grains, mostly smaller than 100 microns in diameter, and they show slight chemical zoning, from magnesian cores (Fo_) to ferroan rims (Fo_). Orthopyroxene occurs as euhedral grains, larger than olivine grains, and also shows slight chemical zoning, from Ca-poorer magnesian cores to Ca-richer ferroan rims. Clinopyroxene occurs as rims of orthopyroxenes and shows remarkable chemical zoning continuously from pigeonitic interiors to augitic rims. Cryptocrystalline feldspar occurs in interstitial spaces between olivine and/or pyroxene, and seems to have crystallized in interstitial liquids in a rapid cooling condition. The cryptocrystalline feldspars in Y-74359 are chemically heterogeneous, classified into three groups, albite (An_Ab_Or_), anorthoclase (An_Ab_Or_), and intermediate alkali feldspar (An_Ab_Or_), and occur in different portions of the thin section. The two achondrites seem to have formed by partial melting from heterogeneous silicate precursors of unequilibrated H chondrites