The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius >0 from the flat torus ², with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. Let (), and respectively R(), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when ² is being identified with [0,1)². We prove that the probability measures on [0, ) associated with the random variables and R are weakly convergent as 0⁺ and explicitly compute the densities of the limits. Research partially supported by ANSTI grant C6189/2000.     

Kinetic limits of the HPP cellular automaton

We study the Boltzmann-Grad limit in various versions of the two-dimensional HPP cellular automaton. In the completely deterministic case we prove convergence to an evolution that is not of kinetic type, a well-known phenomenon after Uchyiama's paper on the Broadwell gas, whereas the limiting equation becomes of kinetic type in the model with random collisions. The main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting, between any two successive HPP updatings, ^- stirring updatings, <1 being any fixed positive number and a parameter which tends to 0. The initial measure is a product measure with average occupation numbers of the order of (low-density limit) and varying on distances of the order of ^–1. The limit as 0 of the system evolved for times of the order of ^-1- corresponds to the Boltzmann-Grad limit. We prove propagation of chaos and that the renormalized average occupation numbers (i.e., divided by) converge to the solution of the Broadwell equation. Convergence is proven at all times for which the solution of the Broadwell equation is bounded.     

Correlations and spectra of an intermittent chaos near its onset point

A one-parameter family of piecewise-linear discontinuous maps, which bifurcates from a periodic state of periodm, (m=2, 3,...) to an intermittent chaos, is studied as a new model for the onset of turbulence via intermittency. The onset of chaos of this model is due to the excitation of an infinite number of unstable periodic orbits and hence differs from Pomeau-Manneville's mechanism, which is a collapse of a pair of stable and unstable periodic orbits. The invariant density, the time-correlation function, and the power spectrum are analytically calculated for an infinite sequence of values of the bifurcation parameter which accumulate to the onset point _c from the chaos side - _c > 0. The power spectrum near=0 is found to consist of a large number of Lorentzian lines with two dominant peaks. The highest peak lies around frequency=2/m with the power-law envelope l/¦-(2/m)¦⁴. The second-highest peak lies around o = 0 with the envelope l/¦¦². The width of each line decreases as, and the separation between lines decreases as/lg3^–1. It is also shown that the Liapunov exponent takes the form-/m and the mean lifetime of the periodic state in the intermittent chaos is given bym ^–1(ln ^–1+1).     

Accurate calculation of functional determinants

A dependence of the functional determinant of the operator from the family D() satisfying the condition D()=fD()+D()f on the parameter m² of infrared regularization is found in the regularization method using a generalized function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 27–32, June, 1988.     

Photopyroelectric spectroscopy (P2ES) of electronic defect centers in crystalline n-CdS

The electrical and dielectric properties of mercuric iodide were studied at room temperature under various intensities and colours of light in the frequency range 1 Hz–10 kHz. In the high-frequency region (>40 Hz), the real part of the dielectric constant () is almost constant with frequency (f), colour and intensity of light. At lower frequencies, varies nearly as 1/f and monotonically increases with intensity (I) of the yellow (or green) light, whereas it is almost constant with red light intensity. This behaviour is discussed in the view of the different polarization contributions. The imaginary part of the dielectric constant () was found to vary as 1/f over the frequency range studied. This behaviour was observed whether the crystal was in dark or illuminated implying that the roomtemperature ac dark- or photo-conductivity () is independent of frequency. The observed variation of with intensity of yellow (or green) light was found to follow anI ^1/2 dependence and a weaker dependence for the red light. the red light. The conductivity behaviour is discussed in the view of the current theories.     

P-polarized nonlinear surface polaritons near the surface of a saturable self-defocusing medium: Threshold case

We have obtained the solution of the stationary problem for P-polarized nonlinear surface polaritons (NLSP) near the interface of two media, one of which is optically linear and the other having an optical nonlinearity with saturation and a dielectric function of the form: =_xx=_zz=₀+a|E|²/(1+b|E|²), where E is the amplitude of the electric field. On this basis the NLSP energy and propagation properties, the polarization structure and the localization depth of the NLSP field as well as the physically allowed region in the parameter space have been investigated for the case in which the parameter ₀ is positive, b is non-negative, and a and ₁ are negative (₁ is the dielectric function of the linear medium); ₀>|₁|.     

Kinetic theory of adsorption and desorption

In this paper we use stochastic methods to discuss adsorption and desorption. The paper derives generalized coefficients of sticking and accomodation depending on surface temperatureT _s and gas temperatureT _g and shows, that for additive Markov processes, these kinetic coefficients are identical. Furthermore, exact solutions of the kinetic equations for certain simple transition probabilitiesP(, ) are found and an approximation method for more complicatedP(, ) is given. The comparison of the theory with experimental results for noble gas-metal systems indicate a quadratic relationship between the first moment of the transition probabilityP(, ) and the well depth of the physisorption system.     

Spectral behavior of quasi periodic potentials

Spectral properties of Schrödinger operators of the typeH =–+V, where is the Laplacian,V a quasiperiodic potential and a coupling constant, are developed.V is taken to be finite sum of exponentials with generic frequencies. For small a strong stability is shown. On the other hand, examples (in the finite diffeence case) are given, for which a transition in the type of spectrum occurs, as is increased.Research supported in part by grant NSF MCS 7900813     

Exact solution of a three-component system on the honeycomb lattice

A model three-component system is considered in which the bonds of a honeycomb lattice are covered by rodlike molecules of typesAA, BB, andAB. The ends of molecules near a common lattice site interact with energies _AA, _BB, and _AB. The model is equivalent to an Ising model on the 3–12 lattice. Exact results are obtained for the two-phase coexistence curves in the isothermal composition plane.     

Truncated Navier-stokes equations: continuous transition from a five-mode to a seven-mode model

A two-parameter family of nonlinear differential equations x=F(x, R, ) is studied, which allows one to connect continuously, as varies from zero to one, the different phenomenologies exhibited by a model of 5-mode truncated Navier-Stokes equations and by a 7-mode one extending it. A critical value is found for, at which the most significant phenomena of the 5-mode system either vanish or go to infinity. New phenomena arise then, leading to the 7-mode model.Supported by G.N.F.M., C.N.R.     

One-dimensional harmonic lattice caricature of hydrodynamics

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ¹ It is known that equilibrium (i.e., time-invariant attractive) states for this model are translationally invariant Gaussian ones, with the mean 0, which satisfy some linear relations involving the interaction quadratic form. The natural parameter characterizing equilibrium states is the spectral density matrix function (SDMF)F(), [– , ). Time evolution of a space profile of local equilibrium parameters is described by a space-time SDMFF(t;x, ) t, xR ¹. The hydrodynamic equation forF(t; x, ) which we derive in this paper means that the normal mode profiles indexed by are moving according to linear laws and are mutually independent. The procedure of deriving the hydrodynamic equation is the following: We fix an initial SDMF profileF(x, ) and a familyP ,>0 of mean 0 states which satisfy the two conditions imposed on the covariance of spins at various lattice points: (a) the covariance at points close to the value ^–1 x in the stateP is approximately described by the SDMFF(x, ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly in. Given nonzerotR ¹, we consider the states P _–1 ,>0, describing the system at the time moments ^–1 t during its harmonic time evolution. We check that the covariance at lattice points close to ^–1 x in the state P _–1 is approximately described by a SDMFF(t;x, ) and establish the connection betweenF(t; x, ) andF(x,).     

A new solution of Schmutzer's projective relativistic cosmology

Schmutzer's cosmological model on the basis of his projective unified field theory leads to a system of differential equations equivalent to a set of highly complicated Abel differential equations [1, 2]. This paper is concerned with the properties of this coupled system of differential equations. The main results of this paper are as follows, (i) We have found the explicit general solution of the cosmological equations for the case=±5/2,= ±1. (ii) The procedure of solving the general case arbitrary=0 is easily within this formalism, (iii) After having found the solution with arbitrary,=±1, one gets the corresponding solution= -, =±1. (iv) In general the coupled system of differential equations describing Schmutzer's cosmological model is equivalent to a single generalized Emden differential equation and it is identical to Emden's differential equation for special values of the parameter.     

Mean field theory for Coulomb systems

We study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero. We prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field(x) where is given by-+ 2z sinh() =q This is the mean field equation of Debye and Hückel. The proof uses the sine-Gordon transformation, the Mayer expansion, and a correlation inequality.Work partially supported by NSF Grant MCS 82-02115.     

Lattice and continuum wavelets and the block renormalization group

We obtain a resolution of the identity operator, for functions on a latticeZ ^d, which is derived from the block renormalization group. We use eigenfunctions of the terms of the decomposition to form a basis forl ₂(Z^d) and show how the basis is generated from lattice wavelets. The lattice spacing is taken to zero and continuum wavelets are obtained.     

БЕТАТРОННЫЕ КОЛЕБАН ИЯ В УСКОРИТЕЛЕ С ОБЩИМ ПОЛЕМ I

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Betatron oscillations in an accelerator with a general field I

The paper gives a linear theory of equilibrium trajectories in an accelerator with a generalized magnetostatic field, the components of which are defined on a general rotation surface. Equations of motion of the particles in natural coordinates are derived with respect to the change in energy and dissipative force. A system of equilibrium trajectories is found in the general form. Conditions for the field components on the reference surface, necessary for the existence of equilibrium trajectories, for the conservation of their geometric similarities and for maintaining the constancy of the frequencies of the betatron oscillations, are derived. A condition is also derived which must be satisfied by the reference surface in order to conserve constant circular frequency of the particles. It is seen that it is not possible to find a field for an accelerator with an exactly constant circular frequency and with constant frequencies of the betatron oscillation in the relativistic energy region. An ultra-relativistic cyclotron with such properties is realizable.

     

Spectral superbroadening of self-focused picosecond laser pulses in D2O

Intense picosecond light pulses of a mode-locked Nd: glass laser at _L =1054 nm (fundamental wavelength) and _SH =527 nm (second harmonic wavelength) are passed through a sample of D₂O under self-focusing conditions. Spectrally structured superbroadened, spatially bell-shaped emission in the forward direction is obtained. Primary generation processes are pump-pulse-degenerate stimulated parametric four-photon interaction (₁ + _{1 3} + ₄) and stimulated Raman scattering (_{1 R} + ), which occur concurrently (₁= _L or _SH angular pump frequency, #x03C9; _R first Stokes frequency, #x03C9; signal frequency, #x03C9;₃ signal frequency, #x03C9;₄ idler frequency). The parametric four-photon interaction occurs under collinear non-phase-matched conditions and under longitudinally phase-matched, transversally non-phase-matched (erenkov-like) conditions. Subsequent interaction processes are pump-pulse-nondegenerate four-photon interaction of the type ₁ + _{R 3} + ₄, coherent antiStokes Raman scattering (CARS, ₁ + ₁ – _{4 3}), inverse Raman scattering ( _A + ₁ + ), and cascading light up-conversion of the type ₁ + _(i) – _{R (i+1)}.     

A lower bound for the memory capacity in the Potts-Hopfield model

We consider Potts-Hopfield networks of sizeN. We prove the result: _c >0 such that for all 0<< _c we can find, >0 in such a way that, whenN, we can store N patterns, all of them being sorrounded by -energy barriers at distance.     

Field-theoretical approach to static critical phenomena in semi-infinite systems

The critical behaviour of a semi-infinite system withO(n) spin symmetry is studied in 4- dimensions near the ordinary transition using renormalization-group methods of field theory and -expansion techniques. It is found that, to all orders in , all surface exponents can be expressed in terms of two bulk exponents and a single surface exponent which follows from the anomalous dimension of the derivative (x ,0) of The order parameter (x,x ) at the surface (x =0). As a byproduct, Barber's scaling law 2₁ – ₁₁ = + is obtained. The surface exponents are calculated to second order in . Our results show that the scaling relation = ^–1 proposed by Bray and Moore is incorrect. The behaviour of various scaling functions close to the surface (i.e. forx correlation length) is determined with the help of short-distance expansions. We also treat corrections to scaling and logarithmic corrections in four dimensions. Our results for the logarithmic corrections of the layer and local susceptibilities disagree with those obtained by Guttmann and Reeve.A brief account of some of the results presented here was given in [1]. The surface exponents were independently calculated to order ² by Reeve and Guttmann [2] using an alternative method     

Renormalization group approach for the site-bond percolation in structured stochastic environments

The quenched averaged percolation problem of a lattice with a given structure is analyzed. The structure is described by the static structure factorS(q)q ^–ain the regionq 0. As a result of the renormalization group, it follows that the critical behavior fora < 2 is the same as in the random percolation. In the case ofa=2 second universality class with=0 and=1/2+/8+ ²/32 is predicted.     

Equilibrium fluctuations for interacting Brownian particles

We consider an infinitely extended system of Brownian particles interacting by a pair force-gradV. Their initial distribution is stationary and given by the Gibbs measure associated with the potentialV with fugacityz. We assume thatV is symmetric, finite range, three times continuously differentiable, superstable, and positive and that the fugacity is small in the sense that 0z0.28/edq(1-e ^–V(q)). In addition a certain essential self-adjointness property is assumed. We prove then that the time-dependent fluctuations in the density on a spatial scale of order ^–1 and on a time scale of order ^–2 converge as 0 to a Gaussian field with covariance dqg(q)(e ^(/2)|t| f)(q) withp the density and the compressibility.