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.     

Planar graphs,Hamilton cycles and extreme independence number

If a graph has too many edges, it must be Hamiltonian. We show how to construct graphs near the threshold: they have as many edges as possible without sacrificing planarity but are not Hamiltonian. We base our construction on the concepts of independent sets and 1-toughness.The work of the author was partially supported by CNPq-Brasil.     

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.     

R andRτ ratios at the five-loop level of perturbative QCD

We study perturbative QCD at the five-loop level. In particular we considerR = _tot(e ⁺ e ^– hadrons)/(e ⁺ e ^{– + –}) andR = ( v+hadrons)/( ev). We use our method to estimate the five-loop coefficients. As a result, we obtain _s (M _z ) = 0.1186(11) and _s (34 GeV) = 0.1396(16), which are accurate at the 1% level. We also findR = 3.8350(18), which is consistent withR and is accurate to 0.05%.     

Cutting planes,connectivity, and threshold logic

Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP _q ofCP, forq2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP ₂-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE ⁺, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem.Supported in part by NSF grant DMS-9205181 and by US-Czech Science and Technology Grant 93-205Partially supported by NSF grant CCR-9102896 and by US-Czech Science and Technology Grant 93-205     

Aurifeuillian factorizations and the period of the Bell numbers modulo a prime

We show that the minimum period modulo of the Bell exponential integers is for all primes and several larger . Our proof of this result requires the prime factorization of these periods. For some primes the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.

     

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.