Comments on the widom Hall effect current formula

Recently Widom has derived a formula for the Hall current. The aim of the following comments is to clarify the relation between Widom's formula and the usual one.     

Optimal systems of fundamental S-units for LLL-reduction

We show that a particular parameter plays a vital role in the resolution of S-unit equations, at the stage where LLL-reduction is applied. We define the notion of optimal system of fundamental S-units (with respect to this parameter), and prove that such a system exists and can be effectively constructed. Applying our results and methods, one can obtain much better bounds for the solutions of S-unit equations after the reduction step, than earlier. We also briefly discuss some effects of our results on the method of Wildanger and Smart for the resolution of S-unit equations.     

Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given k 4 and L 3 there are only finitely many arithmetic progressions of the form with x_i , gcd(x₀, x_l) = 1 and 2 l_i L for i = 0, 1, …, k − 1. Furthermore, we show that, for L = 3, the progression (1, 1,…, 1) is the only such progression up to sign. Our proofs involve some well-known theorems of Faltings [9], Darmon and Granville [6] as well as Chabauty's method applied to superelliptic curves.     

Linear response,persistent currents and related problems

Based on a general linear response approach we provide a systematic and unified survey of existing theories on persistent currents. The central notions in this context are equilibrium and dynamic persistent currents which are analyzed with respect to their similarities and differences in the canonical and grand canonical ensemble. We present criteria which relate the existence of persistent currents to the equipartition law and ergodicity for current correlators. We find that in additive Fermion systems at low temperatures both kinds of persistent currents coincide in the canonical ensemble whereas they differ in the grand canonical ensemble. Comparing different works on averaged persistent currents in diffusive mesoscopic rings within our framework and discussing several methods of calculating canonical currents with the help of grand canonical ensembles, we clarify some misunderstandings which have arisen in methodologically different approaches to the phenomenon of persistent currents. Finally, we relate the presence of dynamic persistent currents to the Hall conductivity on a finite cylinder and the center coordinate Kubo formula for the Hall conductivity.     

Mössbauer study of phase separation in Ni80 57Fe1P19 amorphous alloys

Mössbauer spectroscopy, X-ray diffractometry, scanning calorimetry and electrochemical measurements were used to study the crystallization process of Ni₈₀ ⁵⁷Fe₁P₁₉ amorphous alloys kept in melt at different temperatures before quenching. Samples were heated up to 430°C and 720°C at a rate of 20°C/min, in order to reach characteristically different stages of crystallization. Even at the same crystallization, stage the room temperature Mössbauer spectra and the X-ray differactograms were different depending on the temperature (1050°C or 1400°C) at which the samples were kept before quenching the melt. The Mössbauer spectra showed a paramagnetic component and two sextets (H=267 kOe and H=245 kOe) at 430°C while at 720°C there was only one sextet (H=267 kOe) besides the paramagnetic component. The changes in the Mössbauer spectra of different samples due to crystallization are consistent with the explanation that phase separation occurs in Ni₈₀ ⁵⁷Fe₁P₁₉ rapidly quenched from the melting temperature of 1050°C.     

A new method for the stereospecific synthesis of ether phospholipids. Preparation of the amide analog of platelet-activating factor and related derivatives

A novel stereospecific synthesis of biologically active ether-phospholipids is reported.     

On a Diophantine Equation Concerning the Number of Integer Points in Special Domains

In this paper a family of diophantine equations concerning the number of integer points in special domains is investigated. Some of the equations are completely solved.