More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter (r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential ^dJ(|x–y|), >0 x and y in ^d, in the limit 0. The macroscopic evolution is observed on the spatial scale ^–1 and time scale ^–2, i.e., the density (r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=x. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.     

Is the electron a composite particle? An experiment

The crucial problem of physics was first formulated by Democritus in the 5th century BC: to find the or smallest uncuttable bits of matter. That has not changed ever since. Among the current candidates are electron and positron, the least expensive of today's bona fide . Experimental tests of their non-compositeness by precisely measuring the gyromagnetic ratios of the individual trapped particles are reviewed.You know, it would be sufficient to really understand the electron.Albert Einstein     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Perturbed periodic Hamiltonians: Essential Spectrum and exponential decay of eigenfunctions

Spectral properties of – +V(x), whereV(x) lies in a neighbourhood of the periodic case and describes various models of disorder, are studied. We prove the exponential decay of generalized eigenfunctions corresponding to energies in the resolvent set of the unperturbed periodic Hamiltonian, as well as the stability of the essential spectrum for the dislocation disorder in two dimensions.     

ИЗЛУЧЕНИЕ ЭЛЕКТРИЧЕ СКОГО ДИПОЛЯ НАД ГРАН ИЦЕЙ ДВУХ СРЕД

, , . , - - . - . -, - , . , - ; , , .     

The influence of elastic deformation on the magnetostriction constant of nickel and iron-nickel alloys

The dependence of the magnetostriction constant of different materials in the form of a wire on the elastic elongation was measured. With pure nickel the magnetostriction constant is independent of the elastic deformation. All three alloys of iron and nickel and the triple alloy Fe-Ni-Co exhibit a linear dependence on the deformation, which differs for different crystallographic texture. In two iron-nickel alloys the dependence of the magnetostriction constants of the single crystal on the elastic deformation was calculated from measurements on polycrystals having different crystallographic textures. In the triple alloy Fe-Ni -Co a considerable linear decrease in the volume magnetostriction with the elastic elongation was also found. The observed changes can be explained by assuming the influence of the lattice deformation on the spinorbital interaction in alloys.

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. . - Fe-Ni-Co , . - . , Fe-Ni-Co . - .

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A preliminary report on some of the results of this paper was given at the conference on magnetism in Kyoto (Japan) in September 1961.

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In conclusion the authors thank Z. Frait C. Sc. and S. Libovický for remarks, J. Míová for help in the measurements, members of the chemical department of our institute for annealing and analyzing the samples and members of the Metal Research Institute in Panenské Beany for their exceptional helpfulness in preparing the wires.     

A matrix method for estimating the Liapunov exponent of one-dimensional systems

Let : [0, 1][0, 1] be a piecewise monotonie expanding map. Then admits an absolutely continuous invariant measure. A result of Kosyakin and Sandler shows that can be approximated by a sequence of absolutely continuous measures _n invariant under piecewise linear Markov maps _itn. Each _itn is constructed on the inverse images of the turning points of . The easily computable measures _n are used to estimate the Liapunov exponent of . The idea of using Markov maps for estimating the Liapunov exponent is applied to both expanding and nonexpanding maps.     

КРУЧЕНИЕ ПРИЗМАТИЧЕ СКИХ СТЕРЖНЕЙ ШВЕЛЛЕ РНОГО ПОПЕРЕЧНОГО СЕЧЕНИЯ

- [3], [4], U(x, y) . - U(x, y) -, . B § 10 - . , - , - D, . - .     

Inertial effects on the escape rate of a particle driven by colored noise: An instanton approach

A recent calculation, in the weak-noise limit, of the rate of escape of a particle over a one-dimensional potential barrier is extended by including an inertial term in the Langevin equation. Specifically, we consider a system described by the Langevin equation , where is a Gaussian colored noise with mean zero and correlator (t)(t')=(D/)exp(–|t–t'|/). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D0) limit. This yields an escape rateexp(–S/D), where the actionS is the minimum, over paths characterizing escape over the barrier, of a generalized Onsager-Machlup functional, the extremal path being an instanton of the theory. The extremal actionS is calculated analytically for smallm and for general potentials, and numerical results forS are displayed for various ranges ofm and for the typical case of the quartic potentialV(x)=–x ₂/2+x ₄/4.     

Local and non-local conserved quantities for generalized non-linear Schrödinger equations

It is shown how to construct infinitely many conserved quantities for the classical non-linear Schrödinger equation associated with an arbitrary Hermitian symmetric spaceG/K. These quantities are non-local in general, but include a series of local quantities as a special case. Their Poisson bracket algebra is studied, and is found to be a realization of the half Kac-Moody algebrak _R [], consisting of polynomials in positive powers of a complex parameter which have coefficients in the compact real form ofk (the Lie algebra ofK).     

Spectral properties of a class of operators associated with conformal maps in two dimensions

Iff is a rational map of the Riemann sphere, define the transfer operator by Let also be the Banach space of functions for which the second derivatives are measures. Ifg andg satisfies a simple integrability condition (implying thatg vanishes at critical points and multiple poles off) then is a bounded linear operator on . The essential spectral radius of can be estimated and, under suitable conditions, proved to be strictly less than the spectral radius. Similar estimates for more general operators are also obtained.     

On the Effect of Increasing the Radiative-Transition Rate Near a Nanosize Point

A short review of theoretical and experimental studies concerning the photoexcited florescence and Raman scattering of light for a substance in a space containing small material bodies is presented. Calculations of the radiativetransition probability for atoms (molecules) in the vicinity of bodies with a size much smaller than the light wavelength are performed. The probabilities of the singlephoton and doublephoton transitions are shown to increase by factors of 9 and 81 in the vicinity of a nanosize sphere with dielectric constant ||\ 1. The probability of a radiative transition in the vicinity of the vertex of a conic needle bearing up against a plane (both with || 1) increases by factors of (/R _in)² and (/R _in)⁴ for singlephoton and doublephoton transitions, respectively (R _in is the curvature radius for the needle vertex). This theoretical result is suggested as an explanation of the effect of increasing the radiation process intensity in the experiments carried out in the studies cited below.     

Double quantum transitions in paramagnetic resonance

The theory of double quantum transitions of the M=±2 type, with regard to inhomogeneously broadened spin systems is studied in this paper with the approximation ²T₂T₃ 1. We suppose that the inhomogeneous broadening is formed by an inhomogeneous crystal field. The obtained results describe the magnitude of absorption as a function of the h.f. power and also describe the shape of the absorption curve. It is demonstrated that in inhomogeneously broadened spin systems the absorption curve of double quantum transitions has the form of the difference of two different Lorentz's curves and that at the saturation ( ²T₂T₁ 1) the absorption increases with the cube of the h.f. field intensity. The shape of the curves is expressed by means of phenomenological relaxation constants of the system.

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M=±2 ²T₂T₃ 1. , - . . , ( ²T₂T₁ 1) . .

     

Natural boundaries for area-preserving twist maps

We consider KAM invariant curves for generalizations of the standard map of the form (x, y)=(x+y, y+f(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation +. In the complex plane, natural boundaries of different shapes are found. In the complex plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 as tends to the critical value.     

Particle-like configurations of the electromagnetic field: An extension of de Broglie's ideas

Localised configurations of the free electromagnetic field are constructed, possessing properties of massive, spinning, relativistic particles. In an inertial frame, each configuration travels in a straight line at constant speed, less than the speed of lightc, while slowly spreading. It eventually decays into pulses of radiation travelling at speedc. Each configuration has a definite rest mass and internal angular momentum, or spin. Each can be of electric or magnetic type, according as the radial component of the magnetic or electric field vanishes in the rest frame, and each has an antiparticle. Any such configuration, of electric or magnetic type, is characterized in part by a set of labels (, ₀, ,l, m), where ₀ is the mean of the angular frequencies of the plane waves making up the configuration, is the variance of those frequencies, is a positive constant with dimensions of action, andl, m are angular momentum quantum numbers withl a positive integer andm an integer such that ml. The rest energy of the particle is ₀, its spin is m, and its lifetime is of the order of 1/. Its antiparticle has ₀ replaced by –₀.     

On the Long Time Behavior of Infinitely Extended Systems of Particles Interacting via Kac Potentials

We analyze the long time behavior of an infinitely extended system of particles in one dimension, evolving according to the Newton laws and interacting via a non-negative superstable Kac potential (x)=(x), (0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order . We next study the motion of a fast particle interacting with a background of slow particles, and we prove that its velocity remains almost unchanged for a very long time (at least proportional to ^–1 times the velocity itself). Finally we shortly discuss the so called Vlasov limit, when time and space are scaled by a factor .     

Relaxation Processes and Fractional Differential Equations

Relaxation properties of different media (dielectrics, semiconductors, ferromagnetics, and so on) are normally expressed in terms of response function f(t) or of real and imaginary components of its Fourier transform dependent on the frequency . It had been recently recognized that most of real materials show deviation from classical Debye process. There exist a few empirical approximations of non-Debye response functions. One of them is the two-power approximation containing and , where and belong to the interval (0, 1). This formula gives the basis for introducing of fractional differential equation considered in this paper. A stochastic interpretation of this equation is offered; its solution is found and investigated. The results are in agreement with experimental data.     

Renormalized transport equations for the bistable potential model

Renormalized transport equations for general Fokker-Planck systems are derived and applied to the bistable potential model. The exact equation for the expectation value x _t can be evaluated in both domains Dx _± and xD ₀ outside and between the potential minima, leading to drastic differences of the dynamics prevailing inD _± andD ₀, respectively.