Adiabatic behavior of sine-Gordon solitons in the presence of perturbations

The behavior of sine-Gordon solitons in the presence of weak perturbations is considered. The procedure is based on the exact inverse scattering solution of the unperturbed sine-Gordon equation. Accounting for perturbations such as those arising from impurities, external forces as well damping and spatially inhomogeneous frequencies the corresponding perturbed operator equation can be solved by the Green's function technique if one expands the Green's operator in terms of a set of biorthogonal eigenfunctions. Ordinary linear differential equations prescribing the time evolution of the scattering data are obtained. Instead of solving the inverse scattering problem completely the adiabatic assumption is then used anticipating the result that solitons maintain their integrity to a high degree. Explicit solutions for the one-soliton dynamics are presented.Work supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich Nr. 162 Plasmaphysik Bochum/Jülich     

Some remarks concerning the M.I.T. public-key cryptosystem

Let a messageM be encrypted by raisingM to a powere moduloR, whereR ande are integers which are made public. The recipient of this encrypted form ofM can decipher it by raising the cipher text to a powerd moduloR. Only the recipient knows the values of the two large primesp ₁,p ₂ such thatR=p ₁ p ₂; consequently, only he knowsd, ase is preselected such that (e, (p ₁ – 1)(p ₂ – 1))=1 anded 1 (mod (p ₁ – 1)(p ₂ – 1)).Recently several attacks have been made on the proposed security of this cryptosystem under iteration of the encryption procedure. In this paper we discuss methods of selecting the primesp ₁,p ₂ and the encryption exponente such that the possibility of breaking this cryptosystem by using an iteration procedure is minimized. Several numerical results are also presented.     

Smooth distribution group and Schrödinger equation in Lp(Rn)

In College de France, Seminaire E.D.P. II, Nov. 1963–May 1964, Peetre has introduced the smooth distribution semi-group which is discussed here. This notion is redefined by introducing a functional space $\text{T}$, which measures the regularity of such a distribution. This allows us to give a spectral characterization of such groups. We show that the iterated resolvent of the infinitesimal generator of a smooth distribution group satisfies the relation

$\text{6(λ?A)}{}^{\mathrm{?q}}\text{6L(X)?cq}{}^{\mathrm{k}}\text{|λ|}{}^{\mathrm{k}}\text{|}$

, Re λ ≠ 0. The application of this notion is illustrated by the study of the Schrödinger equation in L^p($\text{R}$ⁿ).     

Numerical Integration of Systems of Stiff Ordinary Differential Equations with Special Structure

Algorithms for the solution of general systems of stiff differentialequations commonly use implicit integration formulae. The associatednon-linear equations at each step of the integration are efficientlysolved by an iteration such as the parallel chord method, wherethe matrix is an approximation for the Jacobian at a calculatedpoint of the solution. This iteration frequently gives sufficientlyrapid convergence over a number of integration steps beforeupdating and re-inversion of the matrix is required. When thedifferential equations have a special structure, satisfactoryconvergence may be maintained by updating a partition of theJacobian less frequently than the remainder and an efficientcomputational procedure consists in calculating the correspondingupdate of the inverse. Sufficient conditions for local convergencemay be expressed in terms of the difference between the iterationmatrix and the derivative at the solution or in terms of thedifference of the corresponding inverses. Similarly the asymptoticrate of convergence is estimated in terms of the norms of theseperturbations. To assess the effectiveness of updating a partitionof the Jacobian or its inverse we set the corresponding perturbationto zero and evaluate the estimate of the rate of convergence.Variable transformation and "weighting" of equations may beused to obtain more accurate computable estimates of convergencerates and tighter conditions for convergence. This is particularlyrelevant in non-linear stiff systems arising in applicationsfrom physics, chemistry and engineering and associated withfast and slow motions. Such systems exhibit special structurein the Jacobian and higher derivatives related to the sensitivityof the system to the components of the fast motion which makesthem particularly amenable to matrix updating techniques. Anumber of illustrative problems from the literature are cited.A worked example has been solved numerically using an inversewhose partitions are updated irregularly as required for convergenceand a comparison is made of iteration counts and inversion statisticswith updating of the full inverse. Computational savings maysometimes belarge.     

Electron tunneling into ion implanted PbMn and SnMn alloys

Tunnel junctions ofPb-I-Al andSn-I-Al were implanted at low temperatures with Mn ions in the ppm region. The tunnel characteristicdI/dV was measured on the pure sample and on the alloy with different concentrations. A bound state is found in both systems at an energy 0.7 ₀. Comparison with theory confirms, that the Kondo effect is not necessary for the existence of a bound state.     

Mössbauer effect in187rhenium

The Mössbauer effect of the 134 keV transition in¹⁸⁷Re has been investigated in order to determine the Debye-Waller factors of rhenium in different environments. The Debye temperatures of Re (Re) and Re (W) were determined to be (9 K)=(291±10) K and=(298±5) K, respectively. These data are compared with results obtained by other methods.     

Elastic scattering of α-particles on10B forEα = 5?50 MeV

Experimental data of the elastic scattering of-particles on¹⁰B forE = 30–50.6 MeV are presented. They are analysed together with the data of a previous measurement forE = 5–30 MeV in the frame of the optical model including spin-orbit coupling. The interaction radii of the-¹⁰B-systems are determined with the Inopin-Ericson model forE = 5–50 MeV. The mean free path of-particles in¹⁰B is calculated.     

Die Präparation von GaAs-Substraten,untersucht mit Reflexionselektroneninterferenzen

The RHEED-method is used for the examination and characterization of gallium arsenide substrates. It is shown the use of both beam focussing at the object and at the photoplate in studying this subject. It was found that the samples after abrasive cutting and machine lapping give interference patterns of mechanical damaged surfaces with characteristical features. Lap polished surfaces give Kikuchi pattern and a rel-rod pattern of perfect single crystal which is nearly smooth in the atomic level. Critical remarks are given on the Kikuchi pattern as a feature of perfection of the crystal. Results on the penetration depth of rough defects due to machine lapping are in good agreement with data from literature.     

Investigation of InxGa1−xAs films grown from thin solution layer

In_xGa_1−xAs films with x⩽0.12 were grown from a thin solution layer between substrates. The calculation of final film thickness as a function of liquid phase composition, based on supposition of film deposition only on the substrate, is in a good agreement with experimental results. The dependences between compositions of liquid and solid phases at 800°, 750° and 700°C were determined. The morphology of the film surface was investigated as a function of liquid phase composition and (100), (111) A, (111) B substrate orientations. Dislocation density increases from 10⁴ cm⁻² to 10⁷ cm⁻² with change of x from 0 to 0.12.     

Nucleation and growth kinetics for the crystallization of ice from dextrose solutions in a continuous stirred tank crystallizer with supercooled feed

The crystallization of ice in aqueous dextrose solutions is studied in an adiabatic continuous stirred tank crystallizer with a supercooled feed stream. The effective diameter of the ice crystals was determined for various values of mean crystal residence time, feed supercooling, magma density, stirring rate, and dextrose concentration. For all process conditions the supercooling was measured at 9-12 different locations in the crystallizer. These local supercoolings were averaged algebraically to yield the bulksupercooling. From the experimental results growth and nucleation rates have been calculated. By comparing the experimental growth rates to growth rates calculated by means of a mathematical model kinetics for the inbuilding of water molecules into the ice lattice have been determined. The growth rate appears to be directly proportional to the interface supercooling. The rate constant decreases exponentially with increasing weight percentage of dextrose in the solution. The nucleation rate was found to be directly proportional to total crystal surface per unit volume of suspension and proportional to the bulksupercooling to the power 2.1. Nucleation is believed to occur by breakage of dendrites from the surface of parent crystals.