On the ranks of single elements of reflexive operator algebras

For any completely distributive subspace lattice on a real or complex reflexive Banach space and a positive integer , necessary and sufficient (lattice-theoretic) conditions are given for the existence of a single element of of rank . Similar conditions are given for the existence of single elements of infinite rank. From this follows a relatively simple lattice-theoretic condition which characterises when every non-zero single element has rank one. Slightly stronger results are obtained for the case where is finite, including the fact that every single element must then be of finite rank.

     

On complementary subspaces of Hilbert space

Every pair of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form on a Hilbert space . Here is possibly , is a positive injective contraction and denotes the graph of . For such a pair the following are equivalent: (i) is similar to a pair in generic position; (ii) and have a common algebraic complement; (iii) is similar to for some operators on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.

     

Crystal structure of the trigonal form of isotactic polypropylene as an example of density-driven polymer structure

Propene-hexene copolymers crystallize in a new polymorphic form of isotactic polypropylene when the concentration of hexene is higher than nearly 10-15 mol %. The hexene units are included in the crystals, inducing an increase of density that allows crystallization of 3-fold helical chains in a trigonal unit cell according to the space group R3c or Rc, where the helical symmetry of the chains is maintained in the crystal lattice. The structure of this new form is similar to those of isotactic polybutene and polystyrene and does not crystallize in polypropylene homopolymer because it would have too low density. The crystal structure of isotactic polypropylene is therefore no longer an exception to the principles of polymer crystallography, but the new structure represents the fulfillment of these principles and indicates that the packing of polymer molecules is mainly driven by density.     

Bailout embeddings and neutrally buoyant particles in three-dimensional flows

We use the bailout embeddings of three-dimensional volume-preserving maps to study qualitatively the dynamics of small spherical neutrally buoyant impurities suspended in a time-periodic incompressible fluid flow. The accumulation of impurities in tubular vortical structures, the detachment of particles from fluid trajectories near hyperbolic invariant lines, and the formation of nontrivial three-dimensional structures in the distribution of particles are predicted.     

Algebraic isomorphisms and -subspace lattices

The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -subspace lattice, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

     

Control of particles in microelectrode devices

The impact of the convective fluid motion induced by the electric fields on the dielectrophoretic manipulation of particles is investigated theoretically and experimentally. By means of a simplified model a channel with a periodic array of microelectrodes we show that electroconvective flows induce the formation of traps for particles, providing a dynamical mechanism to control microparticles in such devices. We demonstrate experimentally the theoretically predicted dynamical phenomena.     

Single elements of finite CSL algebras

An element of an (abstract) algebra is a single element of if and imply that or . Let be a real or complex reflexive Banach space, and let be a finite atomic Boolean subspace lattice on , with the property that the vector sum is closed, for every . For any subspace lattice the single elements of Alg are characterised in terms of a coordinatisation of involving . (On separable complex Hilbert space the finite distributive subspace lattices which arise in this way are precisely those which are similar to finite commutative subspace lattices. Every distributive subspace lattice on complex, finite-dimensional Hilbert space is of this type.) The result uses a characterisation of the single elements of matrix incidence algebras, recently obtained by the authors.

     

Circular Dichroism of Some Isomeric Alkyl Pyridines

The UV. and CD. spectra of the optically active 2-, 3- and 4-(l-methylpropyl)-pyridines ( 1a , b , c ) and of the corresponding (2-methylbutyl)pyridines ( 2a , b , c ) have been investigated in hexane at RT. and at ?85°, in hexane/TFA and in aqueous HCl-solution. In 1c and 2c both the n → π* and the π → π* transitions are optically active, an n → π* transition being located above 265 nm. The position and the sign of the CD. bands are greatly influenced by the position and the nature of the alkyl substituent.     

Synthesis of some 3-substituted quino[3,2-c][1,8]naphtyridines. A new heterocyclic ring system

Isatoic acid reacts with 7-mcthyl-2,3-dihydro-1,8-naphthyridin-4(1H) one ( 8 ) to give 3-methyl-5,6-dihvdroquino[3,2-c][1,8]naphthyriclin-7-carboxylic acid ( 9a ), which was transformed into the 3-methylquino[3,2-c][1,8]naphthyridine ( 7a ) by refluxing with copper chromite in quinoline. The same product ( 7a ) was also obtained by aromatization of the 3-methyl-5,6-dihydroquino-[3,2-c][1.8]naphthvridine ( 10a ), prepared by condensation of the ketone ( 8 ) and o-aminobenzaldehyde. Other 3-substituted quino[3,2-c][1,8]naphthyridines ( 7b,c,d,e ), which contain a new heterocyclic, ring structure, have been prepared using o-aminobenzaldehyde and 7-sub-stituted-2,3-dihv dro-1,8-naphthyridin-4(1H) ones ( 12 and 13 ) as starting materials. Also, the preparation of the parent nucleus ( 7f ) is described.     

Asymmetric Aldol Condensation of 1-Naphtol and Pyruvates in the Presence of Lewis Acids Preliminary Communication

Lewis acids catalyze teh condensation of methyl pyruvate and l-naphtol to give menthyl 2-hydroxy-2-(l-hydroxy-2-naphtyl)propionate in a ratio of up 96:4 in favor of the levorotatory diastereoisomer. Epimerization, at 25°C for five days, occurs on benzylic carbon atom to give a 30:70 mixture enriched with the dextrorotatory isomer.