The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H³, and solve both of them. We construct the unique mean curvature one surface in H³ that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.     

Weak invariance of the multidimensional rank statistic for nonstationary absolutely regular processes

In this paper, we study the weak invariance of the multidimensional rank statistic when the underlying random variables are nonstationary absolutely regular.     

Reference Electrode for Ionic Liquids

Reference electrodes for room temperature ionic liquid (RTIL) applications were constructed that have a known and reproducible potential versus the ferrocene/ferrocenium couple. They are based on reference electrodes of the first kind, Ag/Ag⁺ couple type, or of the second kind, based on Ag/AgCl in M⁺Cl^?. The former uses AgNO₃ salt and the latter tetrabutylammonium chloride, Bu₄NCl, dissolved in acetonitrile which are then introduced to the ionic liquid of choice for a final concentration of 0.1 M. The reference electrodes can be easily and reproducibly constructed. An ionic contact of these reference systems with the test electrolyte was made using an asbestos fiber liquid junction. The internal compartment of the reference system was filled with the same ionic liquid as used for the electrochemical experiment. The performance of these reference electrodes was tested in selected ionic liquids using the ferrocene/ferrocenium redox couple. The stability, reproducibility, and temperature behavior of the two reference systems have been characterized in the following ionic liquids: 1‐butyl‐3‐methylimidazolium tetrafluoroborate (BMIBF₄), 1‐butyl‐3‐methylimidazolium bis(trifluoromethanesulfonyl)imide (BMI(CF₃SO₂)₂N), and 1‐butyl‐3‐methylimidazolium hexafluorophosphate (BMIPF₆). It has been found that the formal potentials of the examined reference systems are stable over several days. There is a linear relationship for the temperature studied in the range from 25 to 60 °C.     

Secure optical system that uses fully phase-based encryption and lithium niobate crystal as phase contrast filter for decryption

In this paper, the implementation of a secure optical system using fully phase encryption is described. A two-dimensional phase image obtained from an amplitude image is encrypted and decrypted by using a spatial light modulator working in phase mode. The fully phase encryption is achieved using double random phase encoding. The encrypted image is holographically recorded in a photorefractive crystal and is then decrypted by generating, through phase conjugation, the conjugate of the encrypted image. A lithium niobate crystal has been used as a phase contrast filter to change the decrypted phase image into an amplitude image, thus alleviating the need for alignment of the phase contrast set-up in the Fourier plane. Simulation results are provided to verify the proposed study. The mean square error between the primary image and decrypted image has been calculated to study the sensitivity of the system.     

Application of a momentum method to some rarefied gas flow problems

Zusammenfassung Eine Kombination der Lees-Methode und des Mott-Smith Ansatzes wird auf zwei Probleme der Couette-Strömung zwischen zwei Zylindern angewandt. Es werden Ausdrücke hergeleitet für das Moment im Fall, dass ein Zylinder rotiert, und für den Wärmefluss im Fall, dass beide Zylinder stationär sind, aber verschiedene Temperatur haben; diese Ausdrücke sind für alle Werte der Knudsenzahl gültig.     

Some Applications of ABHYANKAR's Lemma

ABHYANKAR 's Lemma provides a simple method of producing unramified extensions of number fields. By application of the lemma we simultaneously simplify and strengthen some recent results of ISHIDA concerning unramified abelian extensions and class numbers.     

On a theorem of M. Deuring and I. R. Šafarevič

Let F/K be a field of algebraic functions of one variable, K algebraically closed of characteristic p≠0. Let E/K be a cyclic extension of F/K of degree ?, a prime not necessarily different from p. Let ξ, ρ denote the ?-ranks of the null class groups of E and F respectively. If E/F is ramified, Deuring proved that ξ=?ρ+(t?δ)(??1) where t is the number of ramified primes and δ is 1 or 2 according as ? equals p or not. ?afarevi? proved the same relation in the unramified case for ?=p where ξ, ρ denote the ranks of the Hasse-Witt matrices. Subrao gave a unified proof for ?=p. Rosen and Sullivan have proved the theorem in special cases. In this paper, using galois cohomology, Deuring's method of proof is modified so as to give a proof of the theorem in all cases.     

On non-linear periodic differential equations and associated point mappings. Application to stability problems

In this paper a method is given allowing the determination of the lower degree terms of a point mapping (i.e. recurrent relationship) associated to a non-linear differential equation with time periodic coefficients. This permits the study of some resonance phenomena. An example of application is given.     

Galois module structure of Tate modules

Dedicated to Professor Peter Roquette Received 1 February 1996; in final form 26 April 1996