Quantum invariants at the sixth root of unity

A general topological formula is given for theSU (2) quantum invariant of a 3-manifoldM at the sixth root of unity. It is expressed in terms of the homology, Witt invariants and signature defects of the various 2-fold covers ofM, and thus ties in with basic 4-dimensional invariants. A discussion of the range of values of these quantum invariants is included, and explicit evaluations are made for lens spaces.     

Extreme point programming with nonlinear constraints

This paper summarizes previous results obtained by the authors on methods of solving extreme point mathematical programming problems with linear constraints. It is also shown how these results can be extended to yield an algorithm for solving extreme point mathematical programming problems with nonlinear constraints. Numerical examples to illustrate the algorithms are included.     

The centralizer of a semi-linear transformation

Let V be a finite-dimensional vector space over a division ring D, where D is finite-dimensional over its center F. Suppose T is a semi-linear transformation on V with associated automorphism σ of D. The centralizer of T is the ring C(T) of all linear transformations on V which commute with T. If σ^r is the identity on D for some r ? 1 and no smaller positive power of σ is an inner automorphism, then the center of C(T) is computed to be polynomials in T^r with coefficients from F₀, where F₀ is the subfield of F left elementwise fixed by σ. A matrix version of this theorem is also given.     

The interference of an electron beam with the surface reaction between oxygen and germanium

Electron beam assisted adsorption and desorption of oxygen was studied by Auger electron spectroscopy (AES). Beam assisted adsorption was observed on clean as well as on oxidized surfaces. After an oxygen exposure of 1000 × 10^?7 Torr min and continuous irradiation with beam voltage of 1.5 kV and beam current density 2 microA mm^?2, the oxygen 510 eV signal amplitude from the point of beam impact was 2.5 times greater than the signal from the non-irradiated region. The Ge 89 eV signal showed a corresponding decrease. Enhanced adsorption occurred at beam energies as low as 16.5 eV. After irradiation, the oxidized surface was not carbon contaminated. Following an oxygen exposure of 30 min at 0.1 Torr and 550°C and subsequent additional beam assisted exposure of 1000 × 10^?7 Torr min, the maximum oxide thickness was about 18 Å. Beam assisted desorption did not occur from thin oxygen layers (0–510 eV signal strength less than 5 units, calculated oxide thickness about 6 Å), but occurred from thick oxides and stopped after the signal amplitude had decreased to 5 units. Based on these results, a model for the structure of the oxygen layer covering the Ge(111) surface is proposed. Mechanisms for adsorption and desorption are discussed. The implications of beam assisted adsorption and desorption on electron beam operated surface measurements (LEED, AES, ELS, APS etc.) are stressed.     

The centralizer of a matrix with real quaternion elements

Let T be a Q-endomorphism on a finite dimensional left vector space V over the division ring of real quaternions Q. In this paper we show that the centralizer of T can be regarded as an algebra over the complex numbers C and the dimension of this algebra is computed in terms of the invariant factors of T. Thus the number of C-linearly independent matrices which commute with a given matrix representing T is determined.     

Thiocyanogen (SCN)2− preparation,ultraviolet photoelectron spectrum and structure

The unstable thiocyanogen molecule, (SCN)₂, has been investigated in the gas phase by ultraviolet photoelection spectroscopy. Semi-empincal and ab intio calculations are used to assist with the orbital assignments and to provide an assessment of the structure A disulphide structure, with a dihedral angle of 87.8° is indicated, similar to those of other well known S₂X₂ molecules.