Formulation of N- and ν-representable density functional theory. V. Exchange-only self-consistent field

The concept of a self-consistent field is developed within the version of density functional theory based on local-scaling transformations. It is shown that in this context there arise two types of consistency: one relating to the charge-consistency within an orbit and another to “orbit jumping.” The latter is analyzed in terms of one-particle equations. The connection with other methods is discussed. © 1992 John Wiley & Sons, Inc.     

Regularity and other properties of absolutely continuous invariant measures for the quadratic family

In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.This paper was written during the author's stay at the IAS while supported by NSF grant DMS-860 1978     

Growth and characterisation of Eu doped GaN thin films

We have studied the optical properties of Eu doped GaN thin films. We have grown high quality Eu doped GaN thin films by using Gas Source Molecular Beam Epitaxy (GSMBE), with 1.4% Eu concentration. The Full Width at Half Maximum (FWHM) of the X-ray diffraction in an omega scan was found to be 288 arcsecs. Low Eu concentration (0.08%) doped GaN thin films were grown, where Eu-related photoluminescence at 622 and 613 nm was detected using above band-gap excitation at 2 K. For high Eu concentration of 30% GaN:Eu crystal photoluminescence (PL) and cathodoluminescence (CL) spectra show strong and intense transitions at 622 and 664 nm, but also at 593 nm for CL spectra, with a similar transition observed from the low Eu concentration sample.     

Synthesis of all diastereomeric methyl 2,5-anhydro-3-deoxy-hexonates: precursors to C-2-deoxynucleosides and THF-templated γ- and δ-amino acids

Efficient syntheses of all diastereomers of methyl 2,5-anhydro-3-deoxy-hexonate from mannono- or gulono-lactones provide precursors for C-nucleosides of 2-deoxyribose and for THF-templated γ- and δ-amino acids.     

A chemoenzymatic synthesis of the linear triquinane (−)-hirsutene and identification of possible precursors to the naturally occurring (+)-enantiomer

An enantiomerically pure cis-1,2-dihydrocatechol, which is readily obtained via a toluene dioxygenase-mediated dihydroxylation of toluene in a whole-cell biotransformation process, has been converted over 17 steps into the linear triquinane (−)-hirsutene. Since the enantiomer of the starting material is also available this work constitutes a formal total synthesis of the naturally occurring (+)-form of hirsutene. Furthermore, minor modifications of the route used here offer the possibility of accessing (+)-hirsutene from the original starting material.     

On the Neighbour-Distinguishing Index of a Graph

A proper edge colouring of a graph G is neighbour-distinguishing provided that it distinguishes adjacent vertices by sets of colours of their incident edges. It is proved that for any planar bipartite graph G with Δ(G)≥12 there is a neighbour-distinguishing edge colouring of G using at most Δ(G)+1 colours. Colourings distinguishing pairs of vertices that satisfy other requirements are also considered.     

2,12‐Dichloro‐10,20‐diphenyl‐5,7,15,17‐tetrahydro‐6H,16H‐dibenzo[d,l][1,9,2,6,10,14]dioxotetraazacyclohexadecine‐6,16‐dione dioxane solvate as a potential macrocyclic hexadentate ligand

The macrocyclic title compound crystallizes as a dioxane solvate, C₃₀H₂₂Cl₂N₄O₄·C₄H₈O₂, with two independent formula units in the unit cell. The observed syn conformation is controlled by both intramolecular N—H...O hydrogen bonds and intermolecular C—H...π interactions. The relative macrocyclic inner bore is estimated to be 4.19 Å. In the crystal structure, molecules form dimers via intermolecular C—H...π interactions, and these dimers are, in turn, linked to form columns along the a axis by intermolecular C—H...O hydrogen bonds. Both X‐ray diffraction analysis and density functional theory (DFT) calculations reveal that the macrocycle possesses very high flexibility. This property, as well as the presence of six donor atoms accessible for coordination, makes the title macrocycle a very promising ligand for complexation with the majority of transition metals.     

Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diagS(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand pair (S(n)×S(n−1),diagS(n−1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n−1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).     

Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory

In this paper the nonlinear wave equation $$u_u - u_{xx} + v(x)u(x,t) + \varepsilon u^3 (x,t) = 0$$ is studied. It is shown that for a large class of potentials,v(x), one can use KAM methods to construct periodic and quasi-periodic solutions (in time) for this equation.