On three-dimensional conformally flat quasi-Sasakian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O₁]     

Convex nonagons with five intervertex distances

The vertices of a convex planar nonagon determine exactly five distances if and only if they are nine vertices of a regular 10-gon or a regular 11-gon. This result has important ties to related concerns, including the maximum number of points in the plane that determine exactly five distances and, for each n7, the samllest t for which there exists a convex n-gon whose vertices determine t distances and are not all on one circle.     

Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.     

Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum A. By weak we denote the relative weak topology induced on A by (A ^*,A ^**). In this note we study some properties of the topological space (A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.     

Inhomogeneous quantum groups and their quantized universal enveloping algebras

The quantum group IGL _q (N), the inhomogenization of GL _q (N), is formulated with -matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space.     

Long-range order in a simple model of interacting fermions

We consider a model of spinless fermions on a lattice, interacting through a nearest neighbor repulsion. In the half-filled band case and for dimensionsd 2, we rigorously prove that there is long-range order in some domain of the parameters=(k _B T)^–1 andt/U, wheret is the hopping amplitude of the particles,U the strength of their repulsion, and the inverse temperature. Our technique is based on the usual Peierls argument of classical statistical mechanics but fails for the groundstate. We discuss the specific difficulties introduced by the Fermi statistics.Work supported in part by U.S. NSF grant PHY 90-19433-A02.     

Time-asymptotic boson states from infinite mean field quantum systems coupled to the boson gas

By means of cocycle techniques in a recent paper, the global dynamics of mean field-boson couplings has been studied. Here, by restricting to the bosonic system the infinite time limit (t ) for very general initial states, one obtains time-asymptotic states on the bosonicC ^*-Weyl algebra, in which one partially rediscovers the collective ordering of the infinite mean field lattice.     

Theq-harmonic oscillator and the Al-Salam and Carlitz polynomials

One more model of aq-harmonic oscillator based on theq-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form ofq-creation andq-annihilation operators,q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.     

Crystallization temperatures of thin Co−Zr amorphous alloy films from electron diffraction

The transformation of amorphous thin alloy films of Co–Zr to the crystalline state was observed with an electron microscope in the diffraction mode. This investigation elucidates the crystallization temperatures,T _x , of thin films as opposed to theT _x presented for thicker samples; usually melt-quenched in the thickness range of 20–50 m. TheT _x of the thin films are compared also with theT _x for sputtered alloys of Co–Zr at 5 m.     

Biological production of ethanol from coal synthesis gas

The anaerobic bacteriaClostridium ljungdahlii produces ethanol and acetate from CO, CO₂, and H₂ in synthesis gas. Early studies with the bacterium showed that relatively high concentrations of ethanol could be produced by lowering the fermentation pH and eliminating yeast extract from the medium in favor of a defined medium. This article presents the results from a medium development study based on the aerobic bacteriumEscherichia coli. The results of continuous-reactor studies in a continuously stirred tank reactor (CSTR) with and without cell recycle are shown to demonstrate the utility of this improved medium.