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     

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.     

Singular perturbations of piecewise monotonic maps of the interval

Let t: [0, 1] [0, 1] be a piecewise monotonic, C₂, and expanding map. In computing an orbit { ⁱ (x ₀)} _i=0 , we model the roundoff error at each iteration by a singular perturbation; i.e.,X _n+1=(X _n )+W , whereW is a random variable taking on discrete values in an interval (-&#x03B5;, ). The main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation 0.     

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|.     

Effect of ultrasonic vibrations on the parameters of the hardening curve for copper single crystals

The variations induced in the parameters of the hardening curve for copper single crystals by ultrasonic vibrations at a frequency of 20 kHz are investigated. It is shown that the application of ultrasound to the samples under tension significantly lowers the critical yield stress (₀) and increases the duration of the first hardening stage (₂). Preliminary ultrasonic irradiation is accompanied by an increase in ₀ and an appreciable increase in ₂. The magnitude of the variations oft ₀ and ₂ depends on the ultrasonic intensity. Experimental results are given for the time dependence of the recovery of the yield point for the irradiated copper samples at 20 and –50 °C.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 51–56, February, 1972.     

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).     

Determination of electrodynamical parameters of dielectric pipe-shaped materials using millimeter wave cavity

Employment ofE _0np modes of cylindrical cavity resonator for measurement of electrodynamical parameters of dielectric pipe-shaped materials in millimeter wavelength band is suggested. Dielectric permittivity and conductivity of a sample can be found by measuring the frequency shift and theQ-factor change of the resonator.Method of mode basis is used for the calculation of sample permittivity and conductivity . Measurements were carried out at 8mm wavelength and related calculations have demonstrated use and possibilities of the method in studying the dielectric properties of samples of cylindrical and other forms.     

Scattering states and bound states in λ℘(φ)2

By analyzing the Bethe-Salpeter equation for even ()₂ models we show that for weak coupling the mass spectrum is discrete and of finite multiplicity below 2m. Moreover on even states of energy less than 4(m–) we show that theS matrix is unitary. Herem is the physical mass and =()0 as 0. Our results rely essentially only on a simple assumption about the analyticity of the Bethe-Salpeter kernel which has been verified for weak coupling. For the interaction ⁴, (/m _o ² 1) we show that there are no even bound states of energy less than 4(m–).Work supported in part by NSF, Grant MPS 74-13252     

Retardation of electromagnetic waves by helices of large diameters

This paper is devoted to the study of the wave retardation by helices of large diameters, when the lengthL of one turn of the helix is larger than the vacuum wavelength, i.e.L/>1. The dispersion equation of the tape helix wound up on a dielectric cylinder was derived and the dependence of the phase velocity of slowed-down waves on the diameter of helix was calculated. The theory was verified experimentally at a frequencyf=2·385 GHz. Helices were wound up on polyfoam (=1·04) and plexiglass (=2·55) cylinders of various diameters. For the helix wound up on plexiglass cylinders it was found that the wave retardation depends on the form of conductor from which the helix is made. The retardation of waves is determined by an effective dielectric constant _ef which is equal either ( _i + _e)/2 for the tape helix or ( _{i e}) for the helix made from a wire of the circular cross-section; here _i is the dielectric constant of inner dielectric rod and _e is that of outer space.     

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,).     

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.     

Asymptotic theory of multidimensional chaos

A delay-differential equationu(t)+u(t)=f(u(t–1)), 0t < , and its generalization are investigated in the limit 0, when the attractor's dimension increases infinitely. It is shown that a number of statistical characteristics are asymptotically independent of. As for the attractor, it can be regarded as a direct product ofO(1/) equivalent subattractors, their statistical characteristics being asymptotically independent of . The results enable one to predict some characteristics of the attractor with fractal dimensionD 1 for the case 1, when they are inaccessible numerically. The approach developed seems to be applicable for a wide class of spatiotemporal systems.     

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.     

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.     

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.     

Electromagnetic waves in thin-film superconducting lines

Conclusions The distinctive features discussed about the propagation of electromagnetic waves in a structure containing a thin-film superconducting line are associated with the fact that the screening properties of the superconducting layer are determined by the distribution of superconducting currents in it. In the quasistatic case under discussion this distribution depends on the nature of the nonuniformity of the magnetic field along the normal to the layer. At specified angles of incidence of a wave the layer is equivalent to an ideal conductor or a diamagnetic; in the intermediate region the screening effect is far smaller. Due to this a thin-film superconducting line possesses some directionality of the leakage of the nonuniform waves when ₁ > ₂. The wave which is emergent at a specified angle which depends on the parameters of the layered structure (₁, ₂,l, _L) possesses the maximum amplitude.When ₁ < ₂, the structure in question does not have a sharply pronounced directionality of the leakage, and at angles of incidence on a thin layer corresponding to total internal reflection it supports the propagation of waves along a line with as small a film thickness as desired. The unscreening effect of a superconducting layer is exhibited here in the fact that the propagation of waves excited in the line occurs at specified angles of incidence onto the layer mainly along its outer surface.Institute of Applied Physics, Academy of Sciences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Vol. 25, No. 5, pp. 527–535, May, 1982.     

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     

Electronic absorption and emission spectral data and fluorescence quantum yields of bridgedp-oligophenylenes,bi- to deciphenyls,and related furans and carbazoles

Absorption and fluorescence emission spectral data, as well as fluorescence quantum yields (_f), were determined for 41p-oligophenylene compounds containing 2–6, 8, and 10 benzene rings. Of 29 compounds containing carbon-bridged rings (fluorenes), 28 were dialkylated on each bridge for improved solubility and photostability. Absorption maxima for oligophenylenes were observed at wavelengths as long as 366 nm, emission maxima to 437 nm, and molar extinction coefficients () as large as 153,000 L/mol-cm; all three exceeded predicted maximum values for the corresponding unbridged oligophenylenes. The substitution of furan for benzene or carbazole for a fluorene (two examples each) bathochromically shifted absorption and emission maxima. Dialkylated carbon bridges bathochromically shifted absorption and emission maxima, and lowered _f in biphenyl and in one terphenyl analogue, but appeared to cause no diminution of _f in higher oligophenylenes. Bis(2-methoxyethyl) substitution on the bridges, incorporated to provide solubility in polar solvents, lowered _f in all examples. Tertiary alkyl substituents on terminal rings bathochromically shifted the absorption and emission maxima and generally increased _f. The loose bolt effect, which lowers _f in mononuclear substituted benzenes, may operate in 9,9-dialkylfluorenes, but not in 2,7-di-t-butylfluorene or in higher oligophenylenes. Cyclic ether and methoxy substituents as auxofluors on terminal rings generally bathochromically shifted absorption and emission maxima and increased and _f. Cyano substituents bathochromically shifted absorption and emission maxima, and increased , but lowered _f slightly.     

Construction of a non-trivial planar field theory with ultraviolet stable fixed point

We study a ₄ ⁴ planar euclidean quantum field theory with propagator 1/p ^2–/2,>0. With the help of the tree expansion of Gallavotti and Nicolò [1], this non-renormalizable theory is shown to have a non-trivial ultraviolet-stable fixed point at negative coupling constant. The vicinity of the fixed point is discussed.     

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.