Traces and idempotents in group algebras

Let G be a group and E an idempotent matrix with entries in the complex group algebra C G. In this paper, we study arithmetic properties of the coefficients r _E (g), gG, of the Hattori-Stallings rank r _E of E. Bass proved in [2] that the r _E (g)s are algebraic numbers. Following Zaleskii, we proceed by reduction to positive characteristic and give an alternative proof of that assertion, while obtaining at the same time an upper bound for the degree of the minimum polynomial of r _E (g) over Q.     

The twenty-fourth Fermat number is composite

We have shown by machine proof that is composite. The rigorous Pépin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors. Each program employed a floating-point, FFT-based discrete weighted transform (DWT) to effect multiplication modulo . The final, respective Pépin residues obtained by these two machines were in complete agreement. Using intermediate residues stored periodically during one of the floating-point runs, a separate algorithm for pure-integer negacyclic convolution verified the result in a ``wavefront' paradigm, by running simultaneously on numerous additional machines, to effect piecewise verification of a saturating set of deterministic links for the Pépin chain. We deposited a final Pépin residue for possible use by future investigators in the event that a proper factor of should be discovered; herein we report the more compact, traditional Selfridge-Hurwitz residues. For the sake of completeness, we also generated a Pépin residue for , and via the Suyama test determined that the known cofactor of this number is composite.

     

Spaces with Dual Order-Isomorphic to ℓ1

We establish that an ordered Banach space is order-isomorphic to c₀ if and only if it is a -Dedekind complete vector lattice and its norm dual is order-isomorphic to ₁.     

Effects of metal irises on the guided-mode properties in asymmetrical slab waveguides

The diffraction phenomenon caused by metal transverse irises placed into an asymmetrical slab waveguide is examined by using the integral equation method. We concentrate on the possibility of controlling the radiation characteristics of the structure by changing the irises positions and the slab waveguide asymmetry. The aperture electric-field distribution is expressed in terms of a finite series of Chebyshev polynomials. The dominant TE guided-mode reflection and transmission coefficients, the near-field distribution and the far-field radiation pattern are calculated, while numerical results are presented for several cases of asymmetrical slab waveguides and different irises’ positions.     

Partition coefficients of organic molecules in squalane and water/ethanol mixtures by molecular dynamics simulations

Accurate partition coefficient data of migrants between a polymer and a solvent are of paramount importance for estimating the migration of the migrant over time, including the concentration of the migrant at infinite time in the two solvents. In this article it is shown how this partition coefficient can be estimated for both a small hydrophilic and a hydrophobic organic molecules between squalane (used here to mimic low density poly ethylene) and water/ethanol solutes using thermodynamic integration to calculate the free energy of solvation. Molecular dynamics simulations are performed, using the GROMACS software, by slowly decoupling of firstly the electrostatic and then the Lennard-Jones interactions between molecules in the simulation box. These calculations depend very much on the choice of force field. Two force fields have been tested in this work, the TraPPE-UA (united-atom) and the OPLS-AA (all-atom). The computational cheaper TraPPE-UA force field showed to be more accurate over the whole range of systems compared to the OPLS-AA force field. Moreover, some of the calculations were done with five different water models to investigate the influence of the specific water model on the calculations. It was found that the combination of the TraPPE-UA force field and the TIP4p water model gave the best results. Based on the methodology proposed in this article, it is possible to obtain good partition coefficients only knowing the chemical structure of the molecules in the system.     

A Direct and Versatile Synthesis of 5-(2-Di-n-Propylaminoethyl)-7-methoxyindole

The Pummerer intermediate generated from 10 with TFAA was trapped intermolecularly in the presence of Lewis acid by the indoline 2 at position 5. Sequential treatment of the resulted compound 3 with Ra-Ni, LiAlH₄ and active MnO₂ gave compound 6 in high overall yield.     

Novel disposable microfabricated antimony-film electrodes for adsorptive stripping analysis of trace Ni(II)

This work describes a novel type of antimony electrode for adsorptive stripping voltammetry (AdSV). The electrode was microfabricated by coating a silicon chip with a thin antimony-film by means of sputtering and the active area of the electrode was defined by photolithography. The resulting antimony-film electrodes (SbFEs) were characterized by optical and electrochemical techniques. The sensors were tested for the detection of low concentrations of Ni(II) by AdSV in the presence of dimethylglyoxime. Well-formed stripping peaks and a linear dependence of the stripping peak current on the Ni(II) concentration were observed on the microfabricated SbFEs while comparative measurements attempted with electroplated SbFEs were unsuccessful. Utilizing thin-film technology for the formation of the antimony-film extends the scope of these devices as mercury-free sensors in AdSV.