On resolvable Golomb rulers,symmetric configurations and progressive dinner parties

Journal of Algebraic Combinatorics - We define a new type of Golomb ruler, which we term a resolvable Golomb ruler. These are Golomb rulers that satisfy an additional “resolvability”...     

Inner Derivations and Primal Ideals of C*-Algebras

Let A be a C*-algebra. For a A let D(a, A) denote the innerderivation induced by a, regarded as a bounded operator on A,and let d(a, Z(A)) denote the distance of a from Z(A), the centreof A. Let K(A) be the smallest number in [0, ] such that d(a,Z(A)) K(A)||D(a, A)|| for all a A. It is shown that if A isnon-commutative and has an identity then either K(A) = , or K(A) = 1 / 3, or K(A) 1. Necessaryand sufficient conditions for these three possibilities aregiven in terms of the primitive and primal ideals of A. If Ais a quotient of an AW*-algebra then K(A) . Helly's Theorem is used to show that if A is aweakly central C*-algebra then K(A) 1.     

Relaxed chromatic numbers of graphs

Given any family of graphsP, theP chromatic number _p (G) of a graphG is the smallest number of classes into whichV(G) can be partitioned such that each class induces a subgraph inP. We study this for hereditary familiesP of two broad types: the graphs containing no subgraph of a fixed graphH, and the graphs that are disjoint unions of subgraphs ofH. We generalize results on ordinary chromatic number and we computeP chromatic number for special choices ofP on special classes of graphs.Research supported in part by ONR Grant N00014-85K0570 and by a grant from the University of Illinois Research Board.     

Photochemical Reactions of Alkyl Phenylglyoxylates(1)

Three new photoproducts, ethyl O-benzoyl mandelate (5a), ethyl O-acetylmandelate (6a), and biphenyl triketone (7a) are isolated and identified in the reactions of ethyl phenylglyoxylate (1a) in benzene. Quantum yields and initial rate constants of product formation are shown to be concentration dependent. For the formation of carbonyl product 3 at lower starting material concentrations (<0.01 M), quantum yields greater than 1 are observed. Variations in the quantum yields as a function of reaction time are due to the accumulation of alpha-hydroxyphenyl ketene (D). The relative reactivities of triplet excited states of phenylglyoxylates 1 and phenyl ketones are compared. A mechanism involving both intramolecular gamma-H abstraction and intermolecular H abstraction, which leads to radical chain reaction, is proposed. Rate constants for intramolecular gamma-H abstraction (k(N)) and intermolecular H abstraction (k(I)) of methyl phenylglyoxylate (1d) are measured.     

Eclipsed Isopropyl Conformations of Two Dimeric Hydrazines

The X-ray structures of 4,10-di-tert-butyl-5,9-diisopropyl-4,5,9,10-tetraazatetracyclo[6.2.2.2(3,6)]tetradecane (s4iPr) and its 4,9-di-tert-butyl-5,10-diisopropyl isomer (a4iPr) are reported. Both compounds are in conformations having their in-N-alkyl groups (directed toward the central CH-CH bond of the molecule) anti to each other, as expected from previous work. The principal feature of interest is that one in-isopropyl group in each compound is in an eclipsed conformation, NN,C(alpha)Me twist angle -0.5(5) degrees for s4iPr and -6.4(4) degrees for a4iPr. Low energy (somewhat less) eclipsed in-isopropyl conformations are predicted by both molecular mechanics (MM2) and semiempirical quantum mechanical (AM1) calculations. The asymmetry of the potentially C(2) symmetric a4iPr because the two in-isopropyl groups are in different rotamers is apparently not a result of crystal packing forces, because a conformation with different isopropyl rotamers is the more stable one by at least 1.0 kcal/mol in solution, determined by (13)C-NMR spectroscopy. This result is not predicted by either calculation method. The "monomer", 2-tert-butyl-3-isopropyl-2,3-diazabicyclo[2.2.2]octane (3), proves to be a poor model for the conformations of 4iPr.     

Quantum-chaotic scattering effects in semiconductor microstructures

We show that classical chaotic scattering has experimentally measurable consequences for the quantum conductance of semiconductor microstructures. These include the existence of conductance fluctuations-a sensitivity of the conductance to either Fermi energy or magnetic field-and weak-localization-a change in the average conductance upon applying a magnetic field. We develop a semiclassical theory and present numerical results for these two effects in which we model the microstructures by billiards attached to leads. We find that the difference between chaotic and regular classical scattering produces a qualitative difference in the fluctuation spectrum and weak-localization lineshape of chaotic and nonchaotic structures. While the semiclassical theory within the diagonal approximation accounts well for the weak-localization lineshape and for the spectrum of the fluctuations, we uncover a surprising failure of the semiclassical diagonal-approximation theory in describing the magnitude of these quantum transport effects.