Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

The vibrational energy flow transition in organic molecules: Theory meets experiment

Most large dynamical systems are thought to have ergodic dynamics, whereas small systems may not have free interchange of energy between degrees of freedom. This assumption is made in many areas of chemistry and physics, ranging from nuclei to reacting molecules and on to quantum dots. We examine the transition to facile vibrational energy flow in a large set of organic molecules as molecular size is increased. Both analytical and computational results based on local random matrix models describe the transition to unrestricted vibrational energy flow in these molecules. In particular, the models connect the number of states participating in intramolecular energy flow to simple molecular properties such as the molecular size and the distribution of vibrational frequencies. The transition itself is governed by a local anharmonic coupling strength and a local state density. The theoretical results for the transition characteristics compare well with those implied by experimental measurements using IR fluorescence spectroscopy of dilution factors reported by Stewart and McDonald [Stewart, G. M. & McDonald, J. D. (1983) J. Chem. Phys. 78, 3907–3915].

Role of the integument in insect defense: pro-phenol oxidase cascade in the cuticular matrix.

The cuticle of the silkworm Bombyx mori was demonstrated to contain pro-phenol oxidase [zymogen of phenol oxidase (monophenol, L-dopa:oxygen oxidoreductase, EC 1.14.18.1)] and its activating cascade. The activating cascade contained at least one serine proteinase zymogen (latent form of pro-phenol oxidase activating enzyme). When the extracted cascade components were incubated with Ca2+, the latent form of pro-phenol oxidase activating enzyme was itself activated and, in turn, converted through a limited proteolysis of pro-phenol oxidase to phenol oxidase. Immuno-gold localization of prophenol oxidase in the cuticle using a cross-reactive hemolymph anti-pro-phenol oxidase antibody revealed a random distribution of this enzyme in the nonlamellate endocuticle and a specific orderly arrayed pattern along the basal border of the laminae in the lamellate endocuticle of the body wall. Furthermore, prophenol oxidase was randomly distributed in the taenidial cushion of the tracheal cuticle. At the time of pro-phenol oxidase accumulation in the body wall cuticle, no pro-phenol oxidase mRNA could be detected in the epidermal tissue, whereas free-circulating hemocytes contained numerous transcripts of pro-phenol oxidase. Our results suggest that the pro-phenol oxidase is synthesized in the hemocytes and actively transported into the cuticle via the epidermis.