Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Let denote the field of algebraic numbers in A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ M_d((G, σ)), regarded as an operator on l²(G)^d, the eigenvalues of A are algebraic numbers, where σ ∈ Z²(G, ) is an algebraic multiplier, and denotes the unitary elements of . Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σⁿ=1 for some positive integer n) this property holds for a larger class of groups containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators. The second and third authors acknowledge support from the Australian Research Council.     

On the Nuclear Energy Generation Rate in a Simple Analytic Stellar Model

This paper is concerned with the hydrostatic equilibrium of a purely gaseous star generating energy at a rate . By the common technique of integration theory of special functions the total net rate of nuclear energy generation is evaluated analytically for a linear density distribution function assumed for a simple stellar model.     

The distribution and the exact percentage points for Wilks'L mvc criterion

Summary Wilks'L _mvc is the likelihood ratio criterion for testing the hypothesis that the mean values are equal, the variances are equal and the covariances are equal, in ap-variate normal population. In this article the exact null distribution as well as the exact percentage points are given for the first time. The distribution is obtained for the most general cases and the inverse tables, namely, the values ofu for given values ofF(u) are computed for the values ofF(u)=0.01, 0.02, 0.05 and for the various values ofn andp whereF(u) is the exact distribution function of the test statistic,n=N−1 andN is the sample size. The exact tables are given forp=2, 3, 4, 5, 6, 7, 8, 9.     

Operator Algebra Quantum Homogeneous Spaces of Universal Gauge Groups

In this paper, we quantize universal gauge groups such as SU(∞), as well as their homogeneous spaces, in the σ-C*-algebra setting. More precisely, we propose concise definitions of σ-C*-quantum groups and σ-C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute these groups for the quantum homogeneous spaces associated to the quantum version of the universal gauge group SU(∞).     

Organocatalyzed cycloaddition reactions of azomethine imines—A lookback

The versatile and convergent nature of 1,3-dipolar cycloaddition reactions has made them an indispensable tool in organic chemistry for synthesizing five-membered heterocycles. Among the various dipoles, azomethine imines (AMI) constitute a versatile class, increasingly used to synthesize biologically relevant heterocycles. The organocatalytic cycloaddition reactions of azomethine imines are relatively unexplored compared to the corresponding transition metal-catalyzed reactions. This review highlights the unique organocatalytic cycloaddition reactions of AMI with different dipolarophiles. The cycloaddition of azomethine imines catalyzed by organic bases such as prolinol, proline, alkaloids, 1,4-diazabicyclo[2.2.2]octane, 1,8-diazabicyclo[5.4.0]undec-7-ene (DBU), primary amines, tertiary amines, N-heterocyclic carbene, and phosphine are discussed here. The mechanistic and electronic aspects, including broad functional group tolerance, catalyst loading, and reaction conditions, are evaluated. This review also demonstrates the scope and potential reactivity of azomethine imines in organocatalytic cycloadditions. We are positive that the reactions of azomethine imines discussed here will aid in discovering new and efficient reaction pathways in synthesizing biologically active and industrially relevant molecules.     

Quantum Hall Effect on the Hyperbolic Plane

In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case. Received: 17 March 1997 / Accepted: 24 April 1997     

A stereoselective synthesis of spiro-dioxolanes via the multicomponent reaction of dicarbomethoxycarbene, aldehydes and 1,2- or 1,4-diones

The three component reaction of acyclic carbonyl ylides generated from dicarbomethoxycarbene and aldehydes with 1,2- and 1,4-diones is described. The reaction afforded the corresponding spiro-dioxolanes in good yields.     

Protonated nitro group as a gas-phase electrophile: Experimental and theoretical study of the cyclization of <Emphasis Type="Italic">o</Emphasis>-nitrodiphenyl ethers,amines, and sulfides

A novel gas-phase electrophilic cyclization, initiated by the protonation of a nitro group, occurs for 2-nitrophenyl phenyl ether and for the analogous sulfide and amine, leading to heterocyclic intermediates in each case. Subsequently, the cyclic intermediates dissociate via two pathways: (1) unusual step-wise eliminations of two OH radicals to afford heterocyclic cations, [phenoxazine - H](+), [phenothiazine - H](+), and [phenazine + H](+), and (2) expulsion of H(2)O, to yield a heterocyclic ketone, followed by loss of CO. The proposed structures of the gas-phase product ions and reaction mechanisms are supported by chemical substitution, deuterium labeling, accurate mass measurements at high mass resolving power, product-ion mass spectra obtained by tandem mass spectrometry, mass spectra of reference compounds, and molecular orbital calculations. Using a mass spectrometer as a reaction vessel, we demonstrate that, upon protonation, a nitro group becomes an electrophile and participates in cyclization reactions in the gas phase.     

On bilinear forms in normal variables

Bilinear forms in normal variables when the matrices of the forms are rectangular are considered. Explicit expressions for the cumulants, joint cumulants and joint cumulants of bilinear and quadratic forms are given. Necessary and sufficient conditions are established for the independence of two bilinear forms as well as a bilinear and a quadratic form. Special cases are shown to agree with known results.