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.     

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.     

Arc erosion reduction on electrical contacts using transversecurrent injection

Electrical contact lifetime is often directly determined by arc erosion. A method for reducing are erosion was developed consisting of injecting from an external current source an additional electrical current flowing parallel to the contact surface. This method was examined in three are environments using the additional transverse electrical current with a density less than 1 A/mm²: 1) automobile ignition contacts; 2) pulsed air arc; and 3) low pressure (P=100 mTorr) arc in nitrogen, SEM and X-ray examination showed that application of a transverse current in a contact during arcing changed the phase composition and microstructure of the contact surface. Under optimal conditions, the microstructure which is formed is significantly more erosion resistant than without the transverse current injection     

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.     

The unitary irreducible representations of the quantum heisenberg algebra

The goal of this work is to describe the irreducible representations of the quantum Heisenberg algebra and the unitary irreducible representation of one of its real forms. The solution of this problem is obtained through the investigation of theleft spectrum of the quantum Heisenberg algebra using the result about spectra of generic algebras of skew differential operators (cf. [R]).