Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

A sequence of i.i.d. matrix-valued random variables with probabilityp and with probability 1–p is considered. Leta() = a ₀ + O(), c() = c ₀ + O() lim ₀ b() = Oa ₀,c ₀, >0, andb()>0 for all >0. It is shown show that the top Lyapunov exponent of the matrix productX _n X _n-1...X ₁, = lim_n (1/n) n X _n X _n-1...X ₁ satisfies a power law with an exponent 1/2. That is, lim ₀(ln /ln ) = 1/2.     

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.     

Low-energy elementary surface excitations of symmetric films of liquid4He atT=0 K

It is known that low-energy elementary excitations of symmetric films of liquid⁴He atT=0 K are characterized by a momentum q parallel to the surface and may be described by bound states. We have evaluated wave functions and energies of these states for both best short-ranged and optimal long-ranged correlations. Quantities of physical interest may be expressed in terms of these eigenstates and, in particular, for very small momenta (q<0.2 Å^–1) they are mainly determined by the contribution due to the lowest-lying one. We propose analytic expressions for the lowest-lying excitations and fluctuations in the long-wavelength limit. It is proved that in this limiting case, the excitation energy _LW(q) and the averaged static structure functionS _LW(q) should go linearly to zero asq0, whereas the averaged direct correlationX _LW ^Dg (q) should diverge at the origin as 1/q. It is shown that numerical solutions exhibit the expected long-wavelength behavior provided that optimal correlations are used. All these results are displayed in a series of figures and are discussed in detail.     

The influence of compression on the absorption of AgBr single crystals

The influence of compression on the edge of self absorption of AgBr single crystals was studied. The measurements were performed at a temperature of –180°C. The shift of the edge of self absorption was studied both in the field of elastic and plastic deformations. The shift of the absorption edge towards the UV end of the spectrum was determined in the field of elastic deformations and towards the i.r. end in the field of plastic deformations. After the ending of the deformation and after unloading the crystal the return of the absorption edge towards the original position was observed.

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AgBr

AgBr. –180°C. , . , — . .

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The author thanks Prof. Dr. L. Zachoval and K. K. Vacek C. Sc. for their interest in this work and for many comments given during the work.     

Convective excitation of quasistatic waves in an inhomogeneous anisothermic plasma II. Nonlinear estimates

Nonlinear effects stabilizing the convective instabilities excited in an anisothermic plasma (T _etT _i) at the plasma boundaryaV_s/ _Bi) are discussed. Waves having in the linear theory (Part I) the highest growth rates ( _Bi) saturate at first. Being excited by a small part of slow plasma electrons ( _zTe) only, they saturate at a relatively low level. Further, surface waves with lower frequencies and higher phase velocities ( _ph/k_z) become dominant and a broadening of the plasma boundary occurs. For their saturation nonlinear interaction is more important than the quasilinear effects. During the time interval of several _Bi ^–1 the longest surface waves withk _yBi/V_s, _Bik_yV_s and _{ph Te} saturate at the absolutely highest level. The plasma boundary broadens in the meanwhile up toaV _s/_Bi. The wave energy is comparable to the whole energy connected with the longitudinal motion of the initially thermal electrons inside this boundary layer. The wave amplitude is large enough to trap the initially cold ions belonging to this layer and heat them up to energies comparable to those of the electron component. The heating process occurs again within several _Bi ^–1 and the Larmor radius of the ions is then comparable toV _s/_Bi. Further evolution of the system is governed by the unstable local perturbations.He leaves of absence from thePhysical and Technical Institute, Kharkov, USSR.     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

Quantum Pushdown Automata

Quantum automata are mathematical models for quantum computing. We analyze the existing quantum pushdown automata, propose a q quantum pushdown automata (qQPDA), and partially clarify their connections. We emphasize some advantages of our qQPDA over others. We demonstrate the equivalence between qQPDA and another QPDA. We indicate that qQPDA are at least as powerful as the QPDA of Moore and Crutchfield with accepting words by empty stack. We introduce the quantum languages accepted by qQPDA and prove that every -q quantum context-free language is also an -q quantum context-free language for any (0, 1) and (0, 1).     

Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems

We consider a ferromagnetic Ising spin system isomorphic to a lattice gas with attractive interactions. Using the Fortuin, Kasteleyn and Ginibre (FKG) inequalities we derive bounds on the decay of correlations between two widely separated sets of particles in terms of the decay of the pair correlation. This leads to bounds on the derivatives of various orders of the free energy with respect to the magnetic fieldh, and reciprocal temperature . In particular, if the pair correlation has an upper bound (uniform in the size of the system) which decays exponentially with distance in some neighborhood of (,h) then the thermodynamic free energy density (,h) andall the correlation functions are infinitely differentiable at (,h). We then show that when only pair interactions are present it is sufficient to obtain such a bound only ath=0 (and only in the infinite volume limit) for systems with suitable boundary conditions. This is the case in the two dimensional square lattice with nearest neighbor interactions for 0<₀, where ₀ ^–1 is the Onsager temperature at which (,h=0) has a singularity. For >₀, (,h)/h is discontinuous ath=0, i.e. ₀=_c, where _c ^–1 is the temperature below which there is spontaneous magnetization.Research supported by AFOSR Contract # F 44620-71-C-0013.     

The heat semigroup and integrability of Lie algebras: Lipschitz spaces and smoothness properties

We define and analyze Lipschitz spaces _,q associated with a representationxgV(x) of the Lie algebrag by closed operatorsV(x) on the Banach space together with a heat semigroupS. If the action ofS satisfies certain minimal smoothness hypotheses with respect to the differential structure of (,g,V) then the Lipschitz spaces support representations ofg for which productsV(x)V(y) are relatively bounded by the Laplacian generatingS. These regularity properties of the _,q can then be exploited to obtain improved smoothness properties ofS on . In particularC ₄-estimates on the action ofS automatically implyC -estimates. Finally we use these results to discuss integrability criteria for (,g,V).Dedicated to Res Jost and Arthur Wightman     

Symmetry properties of nonlinear barrier coefficients

This paper concerns the properties of a symmetric barrier between two reservoirs. The barrier can passK conserved quantities. The current of theith quantity is assumed to satisfy the nonlinear relationJ _i=A _ijj+B_ijkljkl where the _i's are the affinity differences across the barrier andA _ij andB _ijkl are functions of the average affinities of the reserviors. It is shown thatB _ijkl is symmetric in all indices.     

Dielectric constant characterization of large-area substrates in millimeter wave band

A possibility of the dielectric constant measurement for substrates with permittivity=+i without an essential restriction on their area has been shown experimentally. The method uses frequency measurement of quasioptical dielectric resonator (QDR) with two slots oriented along the QDR radius with a dielectric substrate in one of them. Taking QDR of teflon in 8mm waveband as an example it is found that measurable values of can ran up 15 _q , where _q is the QDR material permittivity. Absolute error of the measurements is determined by an accuracy with which the permittivity of calibrated (standard) samples is known. The relative measuring error is determined by the accuracy of the QDR frequency measurement and can be quite a small. As an example the method is demonstrated forLaAlO ₃ single crystals.     

Flux diffusion in high-T c superconductors

In type-II superconductors in the flux flow (J J _c ), flux creep (J _c J _c ), and thermally activated flux flow (TAFF) (J J _c ) regimes the inductionB(r,t), averaged over several penetration depths , in general follows from a nonlinear equation of motion into which enter the nonlinear resistivities (B, J ,T) caused by flux motion and (B, J ,T) caused by other dissipative processes.J andJ are the current densities perpendicular and parallel toB,B=|B|, andT is the temperature. For flux flow and TAFF in isotropic superconductors with weak relative spatial variation ofB, this equation reduces to the diffusion equation plus a correction term which vanishes whenJ =0 (this means B××B=0) or when – = 0 (isotropic normal conductor). When this diffusion equation holds the material anisotropy may be accounted for by a tensorial . The response of a superconductor to an applied current or to a change of the applied magnetic field is considered for various geometries. Such perturbations affect only a surface layer of thickness where a shielding current flows which pulls at the flux lines; the resulting deformation of the vortex lattice diffuses into the interior until a new equilibrium or a new stationary state is reached. The a.c. response, in particular the frequency with maximum damping, depends thus on the geometry and size of the superconductor.     

On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

The paper considers the wave equation, with constant or variable coefficients in ⁿ , with odd n3. We study the asymptotics of the distribution _t of the random solution at time t as t . It is assumed that the initial measure ₀ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of _t to a Gaussian measure as t , which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's room-corridor argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.     

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).     

Possible generalization of Boltzmann-Gibbs statistics

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS _q k [1 – _i=1 ^W p _i ^q ]/(q-1), whereq characterizes the generalization andp _i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.     

Optimal Gaussian solutions of nonlinear stochastic partial differential equations

We present a linearization procedure of a stochastic partial differential equation for a vector field (X _i (t,x)) (t[0, ),xR ^d ,i=l,...,n): _t X _i (t,x)=b _i (X(t, x)) +D, X _i (t, x) + _i f _i (t, x). Here is the Laplace-Beltrami operator inR ^d , and (f _i (t,x)) is a Gaussian random field with f _i (t,x)f _j (t,x) = _ij (t – t)(x – x). The procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations. The linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the Kullback-Leibler entropy. The procedure is applied to the scalar-valued Ginzburg-Landau model in R¹ withb ₁(z) =z - vz ³. Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones if 1.24 ( ² ₁ ⁴ /D ₁ ^1/3:= _c . When _c , there appear quasistationary states fluctuating around one of the bottoms of the potentialU(z) = b ₁(z)dz. The second moment at the quasistationary states almost agrees with the exact one. Transient phenomena are also discussed. Half-width at half-maximum of a structure function decays liket ^–1/2 for small t. The diffusion term _x ² X accelerates the relaxation from the neighborhood of an unstable initial stateX(0,x) 0.     

Quantum Observables,Lie Algebra Homology and TQFT

Let us consider a Lie (super)algebra G spanned by T where T are quantum observables in BV formalism. It is proved that for every tensor c^... that determines a homology class of the Lie algebra G the expression c^...T...T is again a quantum observable. This theorem is used to construct quantum observables in the BV sigma model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.     

Cooling-rate dependence of the residual energy of a one-dimensional configurational glass model

For a one-dimensional configurational glass model we have performed molecular dynamical calculations. The Newtonian equation of motion was solved numerically including a damping term. The residual energye _res() as a function of the damping constant , exhibits a power law behavioure _res(),0.061; in an intermediate range of . This behaviour can be explained as the freezing of a certain type of two level systems with an excitation energy and a barrier heightB. The exponent is approximately equal to /B. This relationship is justified analytically.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

Wavevector scaling,surface critical behavior,and wetting in the 2-d, 3-state chiral clock model

The controversial 2-d, 3-state chiral Potts model is studied using transfer matrix finite size scaling. at =0, we find dq _N/dN ^–4/5, whereq is the wavevector, the chiral field, andN the strip width (N=4–10). The result is consistent with den Nijs's crossover exponent =1/6. With surface fields on the infinite free boundaries, exponents associated with bulk magnetizationy _H, surface magnetizationy _H, and surface susceptibility are computed vs. ; results are similar for or to the infinite direction. Preliminary results are given for the bulk specific heat critical amplitudes, to test the universality of amplitude ratios. The interface wetting line is located for 01/4 using simple transfer matrix calculations of surface tensions in the solid-on-solid approximation. Overhangs or bubbles seem relatively unimportant at all temperatures.     

Painlevé test and energy level motion

From the eigenvalue H|_n()=E_n() |_n(), where HH₀+V, one can derive an autonomous system of first-order differential equations for the eigenvaluesE _n() and the matrix elements V_mn(), where is the independent variable. We perform a Painlevé test for this system and discuss the connection with integrability. It turns out that the equations of motion do not pass the Painlevé test, but a weaker form. The first integrals are polynomials and can be related to the Kowalewski exponents.