Four types of special functions of G_2 and their discretization

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M \subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.Comment: 18 pages, 4 figures, 4 table

Stereoselective synthesis of highly substituted tetrahydrofurans by diverted carbene O–H insertion reaction

Copper or rhodium catalyzed reaction of diazocarbonyl compounds with β-hydroxyketones gives highly substituted tetrahydrofurans with excellent diastereoselectivity, under mild conditions, in a single step process that starts as a carbene O–H insertion reaction but is diverted by an intramolecular aldol reaction

Stereoselective synthesis of highly substituted tetrahydrofurans by diverted carbene O–H insertion reaction

Copper or rhodium catalyzed reaction of diazocarbonyl compounds with β-hydroxyketones gives highly substituted tetrahydrofurans with excellent diastereoselectivity, under mild conditions, in a single step process that starts as a carbene O–H insertion reaction but is diverted by an intramolecular aldol reaction

Quantum Dot Version of Berry's Phase: Half-Integer Orbital Angular Momenta

We show that Berry's geometrical (topological) phase for circular quantum dots with an odd number of electrons is equal to \pi and that eigenvalues of the orbital angular momentum run over half-integer values. The non-zero value of the Berry's phase is provided by axial symmetry and two-dimensionality of the system. Its particular value (\pi) is fixed by the Pauli exclusion principle. Our conclusions agree with the experimental results of T. Schmidt {\it at el}, \PR B {\bf 51}, 5570 (1995), which can be considered as the first experimental evidence for the existence of a new realization of Berry's phase and half-integer values of the orbital angular momentum in a system of an odd number of electrons in circular quantum dots.Comment: 4 pages, 2 figure

A dimensionally continued Poisson summation formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.Comment: 12 pages, version accepted by JFAA, with various additions and improvement

Berry's Phase in the Presence of a Stochastically Evolving Environment: A Geometric Mechanism for Energy-Level Broadening

The generic Berry phase scenario in which a two-level system is coupled to a second system whose dynamical coordinate is slowly-varying is generalized to allow for stochastic evolution of the slow system. The stochastic behavior is produced by coupling the slow system to a heat resevoir which is modeled by a bath of harmonic oscillators initially in equilibrium at temperature T, and whose spectral density has a bandwidth which is small compared to the energy-level spacing of the fast system. The well-known energy-level shifts produced by Berry's phase in the fast system, in conjunction with the stochastic motion of the slow system, leads to a broadening of the fast system energy-levels. In the limit of strong damping and sufficiently low temperature, we determine the degree of level-broadening analytically, and show that the slow system dynamics satisfies a Langevin equation in which Lorentz-like and electric-like forces appear as a consequence of geometrical effects. We also determine the average energy-level shift produced in the fast system by this mechanism.Comment: 29 pages, RevTex, submitted to Phys. Rev.

Schwinger Terms and Cohomology of Pseudodifferential Operators

We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component.Comment: 19 pages, plain te

Benzoate dioxygenase from<em> Ralstonia eutropha</em> B9 – unusual regiochemistry of dihydroxylation permits rapid access to novel chirons

Oxidation of benzoic acid by a microorganism expressing benzoate dioxygenase leads to the formation of an unusualipso,orthoarenecis-diol in sufficient quantities to be useful for synthesis.</p

Optical Holonomic Quantum Computer

In this paper the idea of holonomic quantum computation is realized within quantum optics. In a non-linear Kerr medium the degenerate states of laser beams are interpreted as qubits. Displacing devices, squeezing devices and interferometers provide the classical control parameter space where the adiabatic loops are performed. This results into logical gates acting on the states of the combined degenerate subspaces of the lasers, producing any one qubit rotations and interactions between any two qubits. Issues such as universality, complexity and scalability are addressed and several steps are taken towards the physical implementation of this model.Comment: 16 pages, 3 figures, REVTE