One-dimensional random walk with self-interaction

A standard random walk on a one-dimensional integer lattice is considered where the probability ofk self-intersections of a path =(0, (1),..., (n) is proportional toe ^–k . It is proven that for <0,n ^–1/3(n) converges to a certain continuous random variable. For >0 the formulas are given for the asymptotic Westerwater velocity of a generic path and for the variance of the fluctuations about the asymptotic motion.     

A Factorization of Determinant Related to Some Random Matrices

We consider the expectation of the determinant det(–X)^–1for Im >0 associated with some random N×Nmatrices and factorize it into NStieltjes transforms of probability measures. Moreover, using this factorization, we investigate the limiting behavior of the logarithm of the quantity as N.     

A solvable model exhibiting a glass-like transition

A nonlinear equation of motion of an overdamped oscillator exhibiting a glass-like transition at a critical coupling constant _c is presented and solved exactly. Below _c , in the fluid phase, the oscillator coordinatex(t) decays to zero, while above _c , in the amorphous phase, it decays to a nonzero infinite time limit. Near _c the motion is slowed down by a nonlinear feedback mechanism andx(t) decays exponentially to its long time limit with a relaxation time diverging as (1 – / _c )^–3/2 and (/ _c –1)^–1 for < _c and > _c respectively. At _c x(t) exhibits a power law decay proportional tot ^– with exponent -1/2.     

Existence of Resonances for Metric Scattering in Even Dimensions

Suppose _g is the (negative) Laplace–Beltrami operator of a Riemannian metric g on ⁿ which is Euclidean outside some compact set. It is known that the resolvent R()=(– _g –²)^–1, as the operator from L ² _comp( ⁿ ) to H ² _loc( ⁿ ), has a meromorphic extension from the lower half plane to the complex plane or the logarithmic plane when n is odd or even, respectively. Resonances are defined to be the poles of this meromorphic extension. We prove that when n is 4 or 6, there always exist infinitely many resonances provided that g is not flat. When n is greater than 6 and even, we prove the same result under the condition that the metric is conformally Euclidean or is close to the Euclidean metric.     

An IR-Radiometer with Internal Signal Modulation

An infrared (IR) radiometer electrical circuit on the basis of photoresistors and photodiodes made of silicon doped with zinc (Si) as well as the narrow bandgap semiconductor alloy Pb_0.78Sn_0.22Te is presented. In the circuit suggested a bridge with the photoreceiver connected to the radiometer input and immediately fed by signal generators functions as a radiation modulator. The threshold sensitivity turned out on a recorder is 2·10^–13 W·Hz^–1/2 (for the n⁺–n–n⁺ structures made of Si, =0.8–l.2m, T=300K); 1.4·10^–15W·Hz^–1/2 (for p⁺–n–n⁺ S-diodes on the basis of Si, =0.8–1.2m, T=300K) and 10^–12W·Hz^1/2 (for photodiodes on the basis of Pb_0.78Sn_0.22Te, =8–13m, T=77K).     

The statistics of Curie-Weiss models

LetS _n denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofS _n is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (S _n-nm)/n^1–1/2k has a limiting distribution with density proportional to exp[-_s ^2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.Research supported in part by National Science Foundation Grants MPS 76-06644 (to RSE) and MPS 74-04870 A01 (to CMN).     

Local observables and particle statistics I

We consider the family of those states which become asymptotically indistinguishable from the vacuum for observations in far away regions of space. The pure states of this family may be subdivided into superselection sectors labelled by generalized charge quantum numbers. The principle of locality implies that within this family one may define a natural product composition (leading for instance from single particle states ton-particle states). Intrinsically associated with then-fold product of states of one sector there is a unitary representation ofP ⁽ⁿ⁾, the permutation group ofn elements, analogous in its role to that arising in wave mechanics from the permutations of the arguments of ann-particle wave function. We show that each sector possesses a statistics parameter which determines the nature of the representation ofP ⁽ⁿ⁾ for alln and whose possible values are 0, ±d ^–1 (d a positive integer). A sector with 0 has a unique charge conjugate (antiparticle states); if =d ^–1 the states of the sector obey para-Bose statistics of orderd, if =–d ^–1 they obey para-Fermi statistics of orderd. Some conditions which restrict to ± 1 (ordinary Bose or Fermi statistics) are given.     

Effect of electron irradiation on the electrophysical properties and photoluminescence of nuclear-transmutation-doped gallium arsenide

The Hall effect, the electric conductivity, and the photoluminescence spectra of electron irradiated (E=1 MeV, D=1.1·10¹⁵–3.8·10¹⁸ cm^–2) nuclear-transmutation-doped n-GaAs crystals and crystals of n-GaAs doped by the standard metallurgical method were investigated. The energy spectrum of the radiation-induced defects, determined from the Hall effect and DLTS spectra [E1–E5 traps with ionization energy 0.08, 0.14, 0.31, 0.71, and 0.9 eV, respectively (from the bottom of the C band)], is the same in nuclear-transmutation-doped and standard GaAs and satisfactorily describes the experimental dependence n(D). The rate of introduction of traps E1, E2 decreases as n₀ increases (from 1.3 cm^–1 in GaAs with n₀ — 10¹⁷ cm^–3 to 0.7 cm^–1 in GaAs with n₀ 10¹⁸ cm^–3). The rate of removal of charge carriers () increases as n₀ increases, irrespective of the method of growth and doping of GaAs. The isovalent impurity In in nuclear-transmutation-doped gallium arsenide with N_In 10¹⁸ cm^–3 decreases .Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 45–51, April, 1991.     

q-Analogue of the Watanabe Unitary Transform Associated to the q-Continuous Gegenbauer Polynomials

We associate a family of Hilbert spaces H _q ^2;(D) of analytic functions on the unit disk D=z :|z|<1 the q-continuous Gegenbauer polynomials C _n (x;q) on the interval]–1;1[ and give a q-analogue of the unitary integral transform that Watanabe constructed from the Hilbert space L ²(]–1;1[;(1–x ²)^– dx onto the weighted Hilbert space H ^2;(D).     

Experimental determination of the rates of the excited muonic hydrogen charge exchange on helium

The data of the experiment with the H₂+⁴He mixture are analyzed in order to estimate muon transfer rates from the excited states of p atoms to helium. Experimental data turned out to be insensible to the transfer rates from the metastable 2s state. The rates forn=3, 4, 5 were found to be: ⁽³⁾=(2±7)×10¹⁰ s^–1, ⁽⁴⁾=(16±13)×10¹¹ s^–1, ⁽⁵⁾=(75±60) ×10¹¹ s^–1, respectively.     

The inverse scattering transformation in the angular momentum plane

The inverse scattering transformation (IST) with the angular momentum () as a spectral variable turned out to be a useful method to deal with rotationally invariant problems in field theory at higher spatial dimensions [Refs. (1) and (2)]. We derive the direct and inverse scattering problems for thev-dimensional radial Schrödinger equation for variable and fixed energy (negative or positive). We determine the scattering data (SD) in one to one correspondence with the potential and derive the corresponding Gelfand-Levitan equation. A family of exactly solvable potentials for any and fixed energy is obtained. The trace identities associated to this IST are derived for both signs of the energy. They relate integrals of local polynomials in the potential and its derivatives timesr ^2n–1(n1) with the SD. The presence of this power ofr makes these relations useful in higher dimensions.Laboratoire associé au CNRS     

Nonlocality of Equivariant Star Products on T*( RPn)

Lecomte and Ovsienko constructed SL _n+1(R)-equivariant quantization maps Q for symbols of differential operators on -densities on RP ⁿ . We derive some formulas for the associated graded equivariant star products on the symbol algebra Pol(T* RP ⁿ ). These give some measure of the failure of locality. Our main result expresses (for n odd) the coefficients C _p (·,·) of when = in terms of some new SL _n+1(C)-invariant algebraic bidifferential operators Z _p (·,·) on T* CP ⁿ and the operators (E + n/2 ± s)^–1 where E is the fiberwise Euler vector field and s {1, 2, ..., [p/2]}.     

On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations

Recently, considerable progress has been made in understanding the nature of the algebro-geometrical superposition principles for the solutions of nonlinear completely integrable evolution equations, and mainly for the equations related to hyperelliptic Riemann surfaces. Here we find such a superposition formula for particular real solutions of the KP and Boussinesq equations related to the nonhyperelliptic curve ⁴ = ( – E ₁) ( – E ₂) ( – E ₃) ( – E ₄). It is shown that the associated Riemann theta function may be decomposed into a sum containing two terms, each term being the product of three one-dimensional theta functions. The space and time variables of the KP and Boussinesq equations enter into the arguments of these one-dimensional theta functions in a linear way.On leave from Leningrad State University and Leningrad Institute of Aviation Instrumentation.     

Rigorous Spectral Analysis of the Metal–Insulator Transition in a Limit-Periodic Potential

We consider the limit-periodic Jacobi matrices associated with the real Julia sets of f (z)=z ²– for which [2, ) can be seen as the strength of the limit-periodic coefficients. The typical local spectral exponent of their spectral measures is shown to be a harmonic function in decreasing logarithmically from 1 to 0.     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

Parametric relaxation of spectra

We study the change of an quasienergy spectrum upon variation of the weight of a perturbation in the Floquet operatorF=F ₀e^–iV . Employing ideas from level dynamics and random matrix-theory we show that the distribution of nearest-neighbor spacings can display effectively irreversible behavior. Small deviations from equilibrium relax in a certain collision time which scales with the numberN of levels as _collN ^–3/2.     

Analyticity properties of the Feigenbaum function

Analyticity properties of the Feigenbaum function [a solution ofg(x)=–^–1 g(g(x)) withg(0)=1,g(0)=0,g(0)<0] are investigated by studying its inverse function which turns out to be Herglotz or anti-Herglotz on all its sheets. It is found thatg is analytic and uniform in a domain with a natural boundary.     

Fluorimetric Determination of Thyroxine Hormone with Eu(III)-(Pyridine-2,6-Dicarboxylate) Tris Complex

We describe a method for the determination of thyroxine (Thy) using its quenching effect on Eu(PDA)₃ ^3- tris complex fluorescence. The relative fluorescence intensities are measured at fixed _exc = 282 nm, _em = 615 nm by titrating the metal complex with Thy in piperazine buffer solution at pH 6.5. Data indicated an associative type of reaction of two molecules valid between 0.0 < R < 1.0, R being the mole ratio of Eu(PDA)₃ ^3- to Thy. Over this ratio and up to (R 1.0) collisional quenching of Eu(PDA)₃ ^3- complex ion emission is seen as a result of heavy atom effect, intermolecular energy transfer playing the main role. This is also confirmed by the Stern-Volmer equation. In optimized experimental conditions, the L- form of Thy is determined in a range of 15.5–551.6 g/ml (2.0 × 10^–5 – 7.1 × 10^–4 M) with relative error of ±1.17%.     

Purely absolutely continuous spectrum for almost Mathieu operators

Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl ²(), (l _Y, )(n) = (l+1)+(l–)+ cos(n+) (n) have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.     

Double-lambda compound nucleus and the formation of double hypernuclei

The formation probabilities of single- and double- compound nuclei from ^– following the (K^–, K⁺) reaction are calculated, showing that double- hypernuclei are formed with a fairly large probability through a hyperon compound process involving ^– directly after the (K^–, K⁺) reaction.We dedicate this paper to the late Professor Jan ofka with whom we enjoyed nice friendship and collaboration. We also thank the late Professor H. Bando and Professors K. Yazaki, Y. Akaishi and T. Fukuda for stimulating discussions.