Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces

We consider the motion of hypersurfaces in Riemannian manifolds by their curvature vectors. We show that the Harnack quadratic is an affine second fundamental form of the space-time track of the hypersurface. Given a solution to the Ricci flow, we show that with respect to an appropriate metric on space-time, the space-slices evolve by mean curvature flow. This enables us to identify the Harnack quadratic for the mean curvature flow with the trace Harnack quadratic for the Ricci flow.     

On the representation of unity by binary cubic forms

If is a binary cubic form with integer coefficients such that has at least two distinct complex roots, then the equation possesses at most ten solutions in integers and , nine if has a nontrivial automorphism group. If, further, is reducible over , then this equation has at most solutions, unless is equivalent under -action to either or . The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations for cubic and irreducible of positive discriminant . As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form and to Mordell's equation , where and are nonzero integers.

     

Photoemission studies of the surfactant-aided growth of Ge on Te-terminated Si(100)

The interactions of Ge adatoms with a Si(100) surface terminated by an ordered layer of Te have been studied in detail using XPS, SXPS, STM and LEED. It has been demonstrated that the Te layer has a surfactant action on the growth mode of the Ge in that the two dimensional growth regime is extended to at least 200 Å and the Te is seen to segregate to the growing Ge surface. The surface reconstruction of the Ge layer changes from (1 × 1) in the initial stages to (2 × 2) as growth proceeds and the surface population of Te is reduced. SXPS line shape analysis has indicated that the initial stages of Ge incorporation are characterised by the formation of small islands above those surface Si sites not fully coordinated with Te. Continued growth of such islands is, however, restricted due to their high surface free energy with respect to the surrounding Te-terminated areas. Ge atoms therefore site-exchange with Te atoms in bridge sites, thus becoming incorporated onto the Si lattice and displacing the Te to bridge sites on the growing surface. In this manner islanding is prevented and two-dimensional growth continues beyond the critical thickness. No evidence is seen for any significant incorporation of the Te within the growing Ge layer.     

Corner effects in free shear layers in rotating flows

Correct treatment of the 0(E ^1/2)×0(E ^1/2) corner regions leads to better agreement between the theory and observations of profiles of axial velocity in free shear layers.

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Sommaire Le traitement correct des 0(E ^1/2)×0(E ^1/2) des coins, permet un meilleur accord entre la théorie et les observations des profils de vitesse axial les couches libres de cisaillement (couches de Stewartson).

     

The no-contact extensometer

A no-contact extensometer using visible light reflected from the test piece is described which in different forms can be used to test both rubbers and rigid plastics. Examples of suitable applications are briefly discussed and the precision of the extensometer given.     

Hilbert's “Grundlagen der Geometrie”

A careful analysis is made, in the light of subsequent developments, of the background, content, aim, and influence of the ten German editions of Hilbert's “Grundlagen der Geometrie”. Especial attention is given to the sources of Hilbert's ideas, the critical reactions of contemporary geometers to the first edition, the latter's connections with Hilbert's famous problems, and its role as a model for later axiomatic approaches tho the foundations of mathematics and mathematical physics.     

Fluorescence spectrum of chlorine dioxide induced by the 4765 Å argon-ion laser line

The fluorescence spectrum induced in ClO₂ by excitation with the 4765 Å argon-ion laser line has been studied under moderate resolution with the laser in single mode operation and under high resolution with the laser in multimode operation. The most intense features of the fluorescence have been assigned to excitation by the 23_{2, 22} ← 24_{2, 23} transition of the (0, 0, 0) ← (0, 0, 0) band of ³⁷ClO₂. Another set of features has been assigned to excitation by the 38_{0, 38} ← 37_{0, 37} transition of the (0, 0, 0) ← (0, 0, 0) band of ³⁷ClO₂.The vibrational structure and the ν₁ vibrational dependence of the rotational structure in the fluorescence have been least squares fitted giving a new set of vibrational constants and vibration-rotation interaction constants, $\text{α}{}_1{}^{\mathrm{B}}$ and $\text{α}{}_1{}^{\mathrm{C}}$.     

Laser induced microexplosions of a photosensitive polymer

Laser ablation of a photosensitive triazene polymer was studied in a micro region by means of a nanosecond imaging technique. The propagation of a blast wave, 100 ns after laser irradiation, sufficiently matched a planar blast wave model including the decomposed source mass which indicates characteristics of a microexplosion. The measured velocities of the fronts indicates two blast waves: an initially fast unsupported wave around the peak of the laser pulse, and a relatively slow supported wave involving the main component of the decomposition.