An alternate proof of Cohn's four squares theorem

While various techniques have been used to demonstrate the classical four squares theorem for the rational integers, the method of modular forms of two variables has been the standard way of dealing with sums of squares problems for integers in quadratic fields. The case of representations by sums of four squares in was resolved by Götzky, while those of and were resolved by Cohn. These efforts utilized modular forms. In previous work, the author was able to demonstrate Götzky's theorem by means of the geometry of numbers. Here Cohn's theorem on representation by the sum of four squares for is proven by a combination of geometry of numbers and quaternionic techniques.     

Do spherical polyelectrolyte brushes interdigitate?

The structure of spherical micelles of the diblock copolymer poly(styrene-block-acrylic acid) in water was investigated with small angle neutron scattering and contrast matching. We have monitored intermicelle correlation and the extension of the polyelectrolyte chains in the coronal layer through the overlap concentration. Irrespective of ionic strength, the corona shrinks with increasing packing fraction. Furthermore, at high charge and minimal screening conditions, the corona layers interpenetrate once the volume fraction exceeds the critical value 0.53+/-0.02.     

Enhanced fusion-evaporation cross sections in neutron-rich 132Sn on 64Ni

Evaporation residue cross sections have been measured with neutron-rich radioactive 132Sn beams on 64Ni in the vicinity of the Coulomb barrier. The average beam intensity was 2 x 10(4) particles per second and the smallest cross section measured was less than 5 mb. Large sub-barrier fusion enhancement was observed. Coupled-channel calculations taking into account inelastic excitation significantly underpredict the measured cross sections below the barrier. The presence of several neutron transfer channels with large positive Q values suggests that multinucleon transfer may play an important role in enhancing the fusion of 132Sn and 64Ni.     

Reduction of ion thermal diffusivity associated with the transition of the radial electric field in neutral-beam-heated plasmas in the large helical device

Recent large helical device experiments revealed that the transition from ion root to electron root occurred for the first time in neutral-beam-heated discharges, where no nonthermal electrons exist. The measured values of the radial electric field were found to be in qualitative agreement with those estimated by neoclassical theory. A clear reduction of ion thermal diffusivity was observed after the mode transition from ion root to electron root as predicted by neoclassical theory when the neoclassical ion loss is more dominant than the anomalous ion loss.     

Observation of the "self-healing" of an error field island in the large helical device

It was observed that the vacuum magnetic island produced by an external error magnetic field in the large helical device shrank in the presence of plasma. This was evidenced by the disappearance of flat regions in the electron temperature profile obtained by Thomson scattering. This island behavior depended on the magnetic configuration in which the plasmas were produced.     

An anionic siloxane polymer as a pseudostationary phase for electrokinetic chromatography

A novel polymeric pseudostationary phase for electrokinetic chromatography is introduced and characterized. Siloxane polymers are of interest for this application because of the range of chemistries that could be developed based on these backbones, and because successful development of siloxane polymers would make it possible to employ much of the stationary phase chemistry developed in the past thirty years. A commercially available water-soluble siloxane with a hydroxy-terminated alkyl group was converted to the sulfate derivative. This siloxane polymer is water-soluble, effectively eliminating this limitation associated with siloxane polymers. When employed as a pseudostationary phase, this compound provided rapid, efficient, and selective separations. The electrophoretic mobility of the polymer was less than sodium dodecyl sulfate (SDS) and poly(sodium 10-undecenylsulfate), providing a compressed migration time range, which is the main limiting factor for this polymer. The chemical selectivity of the siloxane sulfate was somewhat different than SDS micelles. The siloxane was employed in buffers modified with a large amount of acetonitrile to separate a number of polynuclear aromatic hydrocarbons. The addition of acetonitrile caused an apparent discontinuity in the electrophoretic mobility of the polymer, which may indicate a change in the structure with increasing organic solvent content.     

Observation of power-Law scaling for phase transitions in linear trapped ion crystals

We report an experimental confirmation of the power-law relationship between the critical anisotropy parameter and ion number for the linear-to-zigzag phase transition in an ionic crystal. Our experiment uses laser cooled calcium ions confined in a linear radio-frequency trap. Measurements for up to ten ions are in good agreement with theoretical and numeric predictions. Implications on an upper limit to the size of data registers in ion trap quantum computers are discussed.     

Entangled state quantum cryptography: eavesdropping on the ekert protocol

Using polarization-entangled photons from spontaneous parametric down-conversion, we have implemented Ekert's quantum cryptography protocol. The near-perfect correlations of the photons allow the sharing of a secret key between two parties. The presence of an eavesdropper is continually checked by measuring Bell's inequalities. We investigated several possible eavesdropper strategies, including pseudo-quantum-nondemolition measurements. In all cases, the eavesdropper's presence was readily apparent. We discuss a procedure to increase her detectability.