Rigidity and flexibility of virtual polytopes

All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.     

Activation of peroxyl and molecular oxygen using bis -benzimidazole diamide copper (II) compounds

New tetradentatebis-benzimidazole ligands have been synthesized and utilized to prepare copper (II) complexes. Some of these copper (II) complexes have been characterized structurally. The copper (II) in these complexes is found to possess varying geometries. A distorted octahedral geometry is found with a highly unsymmetrical bidentate nitrate group. An unusual polymeric one-dimensional structure is observed where copper (II) is in a distorted square pyramidal geometry with a monodentate nitrate ion, having long Cu-O bond, while a distorted triagonal bipyramidal geometry is found with two carbonyl O atoms and a Cl atom in the equatorial plane, and two benzimidazole imine N atoms occupy the axial position. These compounds are found to activate the cumylperoxyl group, and this has been utilized in the facile oxidation of aromatic alcohols to aldehydes, where they act as catalysts with large turnovers. The yields of the respective products vary from 32 to 65%. The role of molecular oxygen has been studied and an attempt has been made to identify the “active copper species”. Activation of molecular oxygen has also been observed and has been used for oxidative dealkylation of a hindered phenol, producing di-butyl quinones with yields of 20–25% and 10–12 fold catalytic turnover. Dihydroxybenzenes and substituted catechols are also readily oxidized to the corresponding quinones, in oxygen-saturated solvents. Yields of 84% have been observed with 34-fold catalyst turnover, with di-t-butylcatechol. The activity of these copper (II) —bis-benzmidazolediamide compounds is reminiscent of the functioning of copper centres in galactose oxidase, tyrosinase and catechol oxidase.     

A short note on the entrainment and exit boundary conditions

An attempt is made to find out the suitable entrainment and exit boundary conditions in laminar flow situations. Streamfunction vorticity formulation of the Navier–Stokes equations are solved by ADI method. Two‐dimensional laminar plane wall jet flow is used to test different forms of the boundary conditions. Results are compared with the experimental and similarity solution and the proper boundary condition is suggested. The Kind 1 boundary condition is recommended. It consists of zero first derivative condition for velocity variable and for streamfunction equation, mixed derivative at the entrainment and exit boundaries. Copyright © 2005 John Wiley & Sons, Ltd.     

Investigation of hypersonic nozzle flow uniformity using NO fluorescence

Planar laser-induced fluorescence visualisation is used to investigate nonuniformities in the flow of a hypersonic conical nozzle. Possible causes for the nonuniformity are outlined and investigated, and the problem is shown to be due to a small step at the nozzle throat. Entrainment of cold boundary layer gas is postulated as the cause of the signal nonuniformity. PACS 47.80.Jk, 47.40.Ki, 47.60.+i     

电路过渡过程的计算机辅助分析

电路过渡过程所列方程是微分方程，本文中采用的是方框图模型分析法，即将微分方程的复杂示解分解成最基本的加（减）、乘（除）、积分（微分）、增益等运算，采用VB设计用户界面产进行计算，并给出了一算例。     

Viscoelastic properties of amorphous polymers 3: low molecular weight poly(methylphenylsiloxane)

By making creep and recoverable creep measurements of a nearly monodisperse low molecular weight poly(methyl phenyl siloxane) sample, we have found on decreasing temperature towardsT _g that there is continuously a change in the viscoelastic spectrum concomitant with a decrease of the steadystate recoverable compliance. This behavior is exactly the same as previously observed in low molecular weight poly(styrene), proving that this spectacular anomaly in the viscoelasticity of low molecular weight polymers is general and deserves an explanation. Photon correlation spectroscopic measurements performed on the same sample have extended the observation of the viscoelastic response to shorter times and the result corroborates the trend of variation established by the creep data.Dedicated to Prof.Dr. E. W. Fischer on his 65th Birthday. Prof.Dr. Fischer is known for his valuable contribution to fosterine, international collaboration of research in polymer science. This work is an example of his contribution because it would not be possible without him bringing us together. One of us (KLN) would like to take this opportunity to thank Prof. Dr. Fischer for his unwaiving support of the 1st (Crete) and the 2nd (Alicante) International Discussion Meeting on Relaxations in Complex Systems     

Inner derivations and norm equality

We characterize when the norm of the sum of two bounded operators on a Hilbert space is equal to the sum of their norms.

     

On universal subsets of Banach spaces

We prove that to most of the known hypercyclic operators A on separable Banach spaces there exist compact (compact convex, compact connected) subsets K of E such that each compact (compact convex, compact connected) subset of E can be approximated with respect to Hausdorff's distance by for suitable . Received July 8, 1997, in final form October 17, 1997