The Pearcey Process

The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fixed time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel.     

Finite Dimensional Unitary Representations of Quantum Anti–de Sitter Groups at Roots of Unity

We study irreducible unitary representations of U _q (SO(2,1)) and U _q (SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary many-particle representations is defined for . Received: 27 November 1996 / Accepted: 28 July 1997     

Comparative study of the cope rearrangement of hexa-1,5-diene and barbaralane

The classical rules for Cope rearrangements predict a transition state with chair form to be favored over the boat form. On the other hand, bridged homotropylidenes, which allow only a boat-form transition state by steric reasons, have extremely low barriers. A controversy about the correct pathway and the different possible intermediates and transition states of the reaction has gone on for years. In this work, the hypersurfaces of barbaralane, in comparison with the boat- and chair-form of hexa-1,5-diene, are computed by the ab inito CASSCF (6,6)/6-31G** method starting with UMP2/6-31G** natural orbitals (NO's). All three hypersurfaces show characteristic features, and, moreover differ from each other. A hitherto undiscussed intermediate, bicyclo[2.2.0]hexane, was localized on the boat-hexa-1,5-diene pathway. So it is noteworthy that our transition state for the boat-hexa-1,5-diene does not correspond to the transition states found by other authors for this conformation. The computed enthalpies of activation of boat- and chair-hexa-1,5-diene, and barbaralane are in good agreement with the experimental data.     

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.     

Some approaches to the multiple-minima problem in the calculation of polypeptide and protein structures

Various methods are used to surmount the multiple-minima problem that is encountered in the multidimensional conformational energy surface of a polypeptide. A summary is given here of two of these methods: (i) The build-up procedure that is modified to include statistical data on the positional frequencies of occurrence of amino acid residues along the chain, and (ii) the diffusion equation method that smoothes out the potential surface, leaving only the potential well containing the global minimum.     

An algorithm for packing regular multistrand polypeptide structures by energy minimization

An algorithm has been developed for packing polypeptide chains by energy minimization subject to regularity conditions, in which regularity is maintained without the addition of pseudoenergy terms by defining the energy as a function of appropriately chosen independent variables. The gradient of the energy with respect to the independent variables is calculated analytically. The speed and efficiency of convergence of the algorithm to a local energy minimum are comparable to those of existing algorithms for minimizing the energy of a single polypeptide chain. The algorithm has been used to reinvestigate the minimum-energy regular structures of three-stranded (L -Ala)₈, three-stranded (L -Val)₆, five-stranded (L -Ile)₆, and the regular and truncated three-stranded (Gly-L -Pro-L -Pro)₄ triple helices. Local minima with improved packing energies, but with essentially unchanged geometrical properties, were obtained in all cases. The algorithm was also used to reinvestigate the structures proposed previously for the I and II forms of crystalline silk fibroin. The silk II structure was reproduced with slightly improved packing and little other change. The orthorhombic silk I structure showed more change and considerably improved packing energy, but the new regular monoclinic silk I structure had considerably higher energy. The results support the structure proposed previously for silk II and the orthorhombic structure, but not the monoclinic structure proposed for silk I. © 1994 by John Wiley & Sons, Inc.     

Equivalence of regularity for the Bergman projection and the\bar \partial-neumann operator

A necessary and sufficient condition for regularity of the -Neumann operator on (0,q)-forms in a smooth bounded pseudoconvex domain in Cⁿ is that the orthogonal projections onto -closed forms of degrees (0,q−1), (0,q), and (0,q+1) all be regular. The first author partially supported by NSF Grant DMS-8701038