The history of differential forms from Clairaut to Poincaré

The history of differential forms is examined from its origins in the work of Clairaut on the theory of differential equations through the end of the 19th century. In particular, we explore the developments leading to the concept of the exterior derivative, the Poincaré lemma and its converse, and the notion of the period of an exact differential. We also note that whereas the original motivation for the idea of the integral of an n-form lay in the theory of complex variables, much of its development was done by physicists.     

Preferred frames and oscillating universes

A metric theory of gravitation is presented. It is based on the existence of a preferred ‘cosmic’ time. It agrees with all present experimental facts regarding gravitation and leads to singularity-free oscillating universes.     

Quasi-phase-matching in chiral materials

The second-order nonlinear optical coefficients associated with chirality differ in sign for the two mirror-image forms (enantiomers) of a chiral material. Structures comprised of alternating stacks of the enantiomers can therefore be used for quasi-phase-matched frequency conversion, as we demonstrate here by second-harmonic generation from Langmuir-Blodgett films of a helicenebisquinone. Such structures could lead to new types of frequency converters in which both the second-order nonlinear response and quasi-phase-matching arise from the chirality of a material rather than its polar order.     

Grand unification signal from ultrahigh energy cosmic rays?

The spectrum of ultrahigh energy (above approximately 10(9) GeV) cosmic rays is consistent with the decay of grand unification scale particles. The predicted mass is m(X) = 10(b) GeV, where b = 14.6(+1.6)(-1.7).     

Polychromatic 4-coloring of cubic bipartite plane graphs

It is proved that the vertices of a cubic bipartite plane graph can be colored with four colors such that each face meets all four colors. This is tight, since any such graph contains at least six faces of size four.     

Self-regeneration of Pd-LaFeO3 catalysts: new insight from atomic-resolution electron microscopy

Aberration-corrected transmission electron microscopy was used to study atomic-scale processes in Pd-LaFeO(3) catalysts. Clear evidence for diffusion of Pd into LaFeO(3) and out of LaFe(0.95)Pd(0.05)O(3-δ) under high-temperature oxidizing and reducing conditions, respectively, was found, but the extent to which these processes occurred was quite limited. These observations cast doubt that such phenomena play a significant role in a postulated mechanism of self-regeneration of this system as an automotive exhaust-gas catalyst.