Non-degenerate Surfaces Research Articles

Symmetries of cosmological Cauchy horizons

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions).

ON LOCAL AUTOMORPHISMS OF REAL ANALYTIC HYPERSURFACES

In this paper the author studies biholomorphic transformations of carrying a nondegenerate real analytic surface into itself and leaving a particular point fixed. The first estimates of the dimensions of such groups of transformations were obtained by V. K. Belosapka (see MR 80h: 32039). In the present paper it is proved that the dimension of such groups for a nonspherical surface does not exceed . Bibliography: 2 titles.

On the calculation of surface states for (111) faces of diamond lattices

Within the framework of the Heine-Jones model, the matching at (111) faces of diamond lattices is carefully analysed and shown to be feasible at every point of the boundary plane. The resulting secular equation for surface states allows one to generalize the existence theorems of Shockley and Forstmann and gives two nondegenerate surface states at the point J, in sharp contrast with previous calculations.

Nondegenerate surfaces and fine-cyclic surfaces

Nondegenerate surfaces and fine-cyclic surfaces

Electron Self-Energy and Temperature-Dependent Effective Masses in Semiconductors:n-Type Ge and Si

The electronic self-energy due to interaction with acoustic phonons is evaluated as a function of the electron propagation vector k, and a relation is established connecting the Sommerfeld-Bethe interaction constant with the energy band separation and effective masses. For nondegenerate prolate ellipsoidal energy surfaces of revolution, the self-energy depends linearly on the temperature $T$ at high temperatures and quadratically on $T$ at low temperatures, this behavior being substantiated by the experimental results of Macfarlane and Roberts. The temperature dependence of the principal effective masses ${m}_{i}(T)$ at high temperatures is given by ($i=l or t$) $\frac{{m}_{i}(0)}{{m}_{i}(T)}=1+(\frac{128\ensuremath{\pi}}{9\ensuremath{\rho}{h}^{3}s{\ensuremath{\Theta}}_{D}}){{m}_{i}}^{2}(0){\ensuremath{\alpha}}_{i}(E){〈{C}^{2}〉}_{\mathrm{Av}}T$ thus indicating a decrease in effective mass with rising temperatures. The result does not explain the deviation from the ${T}^{\ensuremath{-}\frac{3}{2}}$ law for the lattice mobility as observed by Morin and Maita. The percentage decrease at room temperature for each of the electron effective masses amounts to less than 1%. These results do not account fully for the possible change determined by Lax and Mavroides.

Electron Self-Energy and Temperature-Dependent Effective Masses in Semiconductors:n-Type Ge and Si

The electronic self-energy due to interaction with acoustic phonons is evaluated as a function of the electron propagation vector k, and a relation is established connecting the Sommerfeld-Bethe interaction constant with the energy band separation and effective masses. For nondegenerate prolate ellipsoidal energy surfaces of revolution, the self-energy depends linearly on the temperature $T$ at high temperatures and quadratically on $T$ at low temperatures, this behavior being substantiated by the experimental results of Macfarlane and Roberts. The temperature dependence of the principal effective masses ${m}_{i}(T)$ at high temperatures is given by ($i=l or t$) $\frac{{m}_{i}(0)}{{m}_{i}(T)}=1+(\frac{128\ensuremath{\pi}}{9\ensuremath{\rho}{h}^{3}s{\ensuremath{\Theta}}_{D}}){{m}_{i}}^{2}(0){\ensuremath{\alpha}}_{i}(E){〈{C}^{2}〉}_{\mathrm{Av}}T$ thus indicating a decrease in effective mass with rising temperatures. The result does not explain the deviation from the ${T}^{\ensuremath{-}\frac{3}{2}}$ law for the lattice mobility as observed by Morin and Maita. The percentage decrease at room temperature for each of the electron effective masses amounts to less than 1%. These results do not account fully for the possible change determined by Lax and Mavroides.

The tangent planes to a degenerate surface

1. In the Lehrbuch der abzählenden Methoden Zeuthen considers the problem of determining the nature and multiplicity of the tangent lines and planes to a degenerate surface in ordinary space. The surface φ in question consists of n arbitrary planes; these meet in pairs, forming double lines, and in threes, forming triple points. The tangent cone to φ from an arbitrary point consists of plane pencils, each counted twice, as follows from the known theory of degenerate plane curves. According to Zeuthen the tangent planes are of two types:(i) those passing through one of the triple points; there are thus such systems, and each is to be counted six times;(ii) those passing through certain points S lying on the double lines; these depend upon the passage to the limit, it being assumed that φ is obtained as the limiting form of some nondegenerate surface. Two such points lie on each double line, so that there are n(n−1) such systems of planes.

A set of properties characteristic of a class of congruences connected with the theory of functions

be a functional relation between the complex variables z and w. Let us first plot corresponding values of z and w in one and the same plane, and let us then project these points upon the unit sphere by stereographic projection. The lines, which join all pairs of points thus obtained upon the sphere, for a given relation w = F (z), form the congruence in question. If w is not a linear function. of z, every congruence of this class has the following properties. Ia. It is a W-congruence whose focal sheets are distinct, non-degenerate, and non-ruled surfaces. Ib. The focal surfaces are real and have a positive measure of curvature. II. The developables of the congruence determtne isothermally conjugate systems of curves on both sheets of the focal surface. III. The asymptotic curves of both sheets of the focal surface belong to linear complexes. IV. The directrix of the first kind, for every point of either focal sheet, coincides with the directrix of the second kind for the corresponding point of the other focal sheet. Va. The directrix quadrics of both-focal sheets are non-degenerate, and coincide with each other. Vb. Both of these directrix quadrics coincide with the Riemann'sphere. If we prefer, we may replace property II by