Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps

We first treat multidimensional nonlinear noisy maps. We assume that the variables can be split into two classes of variablesu ands so that the linearized equations would give rise to growth or decay foru ands, respectively. We show how the slaved variabless can be explicitly expressed by the order parametersu by making use of the fully nonlinear equations. By taking the limit of vanishing time steps and using a Wiener process and the Îto calculus we derive the corresponding formulas for stochastic differential equations (including multiplicative noise). In this way a high-dimensional problem can be reduced to a problem of much lower dimensions described again by stochastic equations of theÎto type. A similar procedure holds for theStratonovich calculus.     

The lie algebra of invariant group of the KdV,MKdV, or Burgers equation

Let (E): u _t=H(u) denote the KdV, MKdV or Burgers equation, and U(s)=(D^j _s)/u _j, where D=d/dx, u _i=Dⁱ _u, s=s(u, u ₁, ..., u _n) is a polynomial of u _i with constant coefficients, be the generator of invariant group of equation (E). We prove in this paper that all such generators form a commutative Lie algebra, from which it follows that for any symmetry s(u, ..., u _n) of (E), the evolution equation u _t=s(u, ..., u _n) possesses an infinite number of symmetries (or conservation laws in the case of KdV and MKdV equations).     

Aeolian transport of sand

The airborne transport of particles on a granular surface by the saltation mechanism is studied through numerical simulation of particles dragged by turbulent air flow. We calculate the saturated flux q_s and show that its dependence on the wind strength u_* is consistent with several empirical relations obtained from experimental measurements. We propose and explain a new relation for fluxes close to the threshold velocity u_t, namely, q_s=a(u_*-u_t)^α with α≈2. We also obtain the distortion of the velocity profile of the wind due to the drag of the particles and find a novel dynamical scaling relation. We also obtain a new expression for the dependence of the height of the saltation layer as function of the strength of the wind.     

Chaotic Expansions of Elements of the Universal Enveloping Superalgebra Associated with a Z2-graded Quantum Stochastic Calculus

Given a polynomial function f of classical stochastic integrator processes whose differentials satisfy a closed Ito multiplication table, we can express the stochastic derivative of f as

非晶TbCo/Si多层膜的磁性和磁光特性

利用射频磁控溅射方法制备了非晶TbCo/Si多层膜,并对多层膜的磁性和磁光特性进行了测量.实验发现,随着Si层厚度的增加,非晶TbCo/Si多层膜的饱和磁化强度M_s、垂直各向异性常数K_u、磁光克尔角θ_K都显著下降.分析认为这是由于在TbCo层与Si层之间的层间互扩散形成了非磁性的Co₂Si所致. 关键词：     

Long-Time tails in a random diffusion model

Letw = {w(x)xZ^d} be a positive random field with i.i.d. distribution. Given its realization, letX _t be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w^–1(X_t). The processX={X _t:t0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantities _t =(d/dt) E (X _t ² –M ^–1 t and _t(w) = (d/dt)E_w(X _t ² – M^– 1t). Here E_w. is expectation overX at fixedw and E = E_w (dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) lim_t t^d/2_t= V²M^d/2–3(d/2)^d/2 and (2) lim_t t^d/4 _st(w)= Z_s weakly in path space, with {Z_s:s>0} the Gaussian process with EZ_s=0 and EZ_rZ_s= V²M^d/2–4(d)^d/2 (r + s)^–d/2. HereM and V² are the mean and variance of w(0) under . The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since , with ₀ the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.     

Exact stationary probability distribution for a simple model of a nonequilibrium phase transition

We derive the exact stationary probability distribution for the coupled system of Langevin equationsd _t u=u–u s,d _t s=–s+d ²+F(t).     

Growth of Sobolev Norms in Linear Schrödinger Equations with Quasi-Periodic Potential

In this paper, we consider the following problem. Let iu _t +Δu+V(x,t)u= 0 be a linear Schr?dinger equation ( periodic boundary conditions) where V is a real, bounded, real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector λ. Denote S(t) the corresponding flow map. Thus S(t) preserves the L ²-norm and our aim is to study its behaviour on H ^s (T ^D ), s> 0. Our main result is the growth in time is at most logarithmic; thus if φ∈H ^s , then

Bifurcations in fluctuating systems: The center-manifold approach

Bifurcations in fluctuating dynamical systems are studied using the ideas of center-manifold reduction. The method provides not only a systematic procedure for the reduction of the system to a small number of variables-but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points. The joint probability density factorizes into a stationary Gaussian densityp(v/u) in the fast variablesv, and a time-dependent densityP(u, f) in the slow variablesu describing the dynamics on the center manifoldv=v₀ (u). P(u, t) obeys a reduced Fokker-Planck equation that can be written in a normal form by means of local nonlinear transformations. Both additive and multiplicative white noise are considered, as is colored noise. The results extend and formalize Haken's concept of adiabatic elimination of fast variables.     

Vortex rings moving into turbulence

We report on an experimental study of turbulent vortex rings injected with velocity U _v0 into a grid-generated turbulent flow (with RMS streamwise velocity u _*) and followed relative to the mean flow. The initial Reynolds number of the vortices varies from 4500 to 11,500. The turbulence was characterised by an intensity I_t =u _*/U _v0, which varied over the range 0<I_t <0.03. A mathematical model based on a stochastic model of the vortex core is developed to explain and interpret the results. The vortex radius grows diffusively in time with the rate of increase of the square of the vortex radius increasing linearly with I_t . As the vortices grow, they slow down sufficiently rapidly in a manner that they penetrate a finite distance into the turbulence. The vortex velocity, averaged over many experiments, showed an initial t ^?1 decay, consistent with Maxworthy’s experiments. The analysis and experiments show that such vortices ultimately only move a finite distance from their point of generation and this distance varies inversely with I_t .     

Rigorous lower and upper bounds for the slope parameter

It is proved within the framework of axiomatic field theory that the logarithmic derivative of the absorptive part of the scattering amplitude with respect to momentum transfer is bounded from above by $\text{(15}\text{log}\text{s)}\text{[4√t(2 ? √t)]}$ for a sequence of s→+∞, and from below either in the s-channel by const. × s^?5 log^?4s or in the u-channel by const. × u^?5 log^?4u for at least one sequence of s or u →+∞, respectively. In the particular case of the s?u even-symmetric amplitude, a stronger lower bound is obtained; namely, const. × s^?5 log^?4s for at least one sequence of s→+∞. Here s, t, and u are the usual Mandelstam variables, and all bounds are obtained in the forward and the unphysical regions: 0?t<4 (in units of pion mass).It is observed that the Regge amplitude β(t)s^α(t) of high-energy scattering gives the same energy dependence as the above upper bound, and, furthermore, that the slope of the Regge trajectory is bounded from above by $\text{15}\text{[4√t(2 ? √t)]}\text{for}\text{0 < t < 4}$.     

Minimal representations of supersymmetry and 1<Emphasis Type="Italic">D N</Emphasis>-Extended <Emphasis Type="Italic">σ</Emphasis> models

We discuss the minimal representations of the 1D N-Extended Supersymmetry algebra (the Z ₂-graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time. Their knowledge allows to construct onedimensional sigma-models with extended off-shell supersymmetries without using superfields.     

Ab initio calculation of excited state dipole polarizability - Application to the first 1, 3 Σ + g,u states of Li 2

Firstly, imaginary frequency-dependent dipole polarizabilities of Li in its ² S ground state and the first ² P excited state are calculated from a time-dependent gauge-invariant method using effective core pseudo-potentials and the multi-configuration one-electron wave function. C₆ dispersion coefficients of the 2 s + 2 s and 2 s + 2 p dissociations are deduced and also compared with all-electron values. Then, static polarizabilities of Li₂ in its ground and first excited ^{1, 3} Σ ⁺ _g,u states are obtained at interatomic distance corresponding to the energy minimum of each state. Received 10 May 2001 and Received in final form 14 September 2001     

Positron lines from subcritical heavy ion-atom collisions

Positron lines were observed in heavy ion-atom collisions at bombarding energies close to the Coulomb barrier in subcritical systems with the sum of the atomic numbers of the colliding nucleiZ _u =Z ₁+Z ₂ being smaller thanZ _u =172. Each collision system, studied,²⁰⁸Pb +²⁰⁸Pb(Z_u=164),²³⁸U+¹⁸¹Ta(Z_u=165), and²³⁸Au(Z_u=171), exhibits the emission of two positron lines withZ _u -independent c.m. energies of ～ 258 keV and ～ 340 keV, and with widths of about 30 keV, superimposed on continuous positron spectra from nuclear pair decay and pair emission induced by the time changing Coulomb field of the collision. The production cross section of thee ⁺-lines rises with a high power ofZ _u (ocZ _u ²²), which is comparable to theZ _u ²⁰-dependence for the collision induced positrons.     

Asymptotic attenuation coefficients in random perturbations of Angular Correlations

The time-differential attenuation coefficients for randomly fluctuating interactions in isotropic systems are shown to be of the simple exponential type (well known from the first order treatment of Abragam and Pound) fort?Τ _c , notwithstanding the strength of 〈Ω²〉^1/2Τ _c . Explicit expressions for the relaxation constant are given which are strongly dependent on the strength of 〈Ω²〉^1/2Τ _c and on the type of interaction but prove to be virtually independent of the stochastic model assumptions.     

Measurements ofK-shell ionization with132Xe and208Pb projectiles in the energy range of 1.4–5.9 MeV/u

Projectile and targetK-shell ionization cross sections induced by 3.6-, 4.7-, and 5.9 MeV/u¹³²Xe ions and 1.4-, 3.6-, 4.7-, and 5.9 MeV/u²⁰⁸Pb ions from the UNILAC in thin solid targets between C and U are measured. The cross sections are discussed in terms of the molecular model of innershell vacancy production in heavy ion-atom collisions. The sharing of 2p _1/2σ vacancies between theK shells of the two collision partners in these very heavy ion-atom collisions is found to deviate from the Meyerhof-Demkov formula forR≦10^?2. The measured ionization cross sections are compared with theoretical calculations for 1sσ and 2p _1/2σ excitation cross sections. AZ _UA=Z₁+Z ₂ dependence is found independent ofZ ₁/Z ₂. Outer-shell vacancy configurations measured in these close encounters are reported.     

Completely analytical interactions: Constructive description

An interactionU is called a completely analytical (CA) interaction, if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions Z_v(u), or correlation decay, or truncated correlation bounds, or asymptotic behavior of ln Z_v(u), v→∞. The 12 conditions are presented, together with part of the proof of their equivalence. The main result of the paper is that each condition is constructive in the following sense: instead of checking it in all finite volumesv??^v, it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions Z _v (u+?) for the complex perturbationsu+? ofu do not vanish for all V? ^v and all ? with ∥?∥<?, provided this is true only forv with diam v?C(?) and ∥?∥<?′ (but with?<?′).     

Tails in harnesses

We consider space- and time-uniformd-dimensional random processes with linear local interaction, which we call harnesses and which may be used as discrete mathematical models of random interfaces. Their components are rea random variablesa _s ^t , wheres ∈ Z ^d andt=0, 1, 2.,... At every time step two events occur: first, every component turns into a linear combination of itsN neighbors, and second, a symmetric random i.i.d. “noise”v is added to every component. For any σ ∈Z _d ⁺ define Δ_σ a′ _s as follows. If σ=(0,...,0), σ=(0,...,0), Δ_σ a _s ^t =a _s ^t . Then by induction, wheree _i is thed-dimensional vector, whoseith component is one and other components are zeros. Denote |σ| the sum of components of σ. Call a real random variable ϕ symmetric if it is distributed as −ϕ. For any symmetric random variable ϕpower decay or P-decay is defined as the supremum of thoser for which therth absolute moment of ϕ is finite. Convergence a.s., in probability and in law whent→∞ is examined in terms of P-decay(v): Ifd=1, σ=0 ord=2, σ=(0,0), Δ_σ a _s ^t diverges. In all the other cases: If P-decay(v)<(d+2)/(d+|σ|), Δ_σ a _s ^t diverges; if P-decay(v)>(d+2)/(d+|σ|), Δ_σ a _s ^t , converges and P-decay(ν) For any symmetric random variable ϕexponential decay or E-decay is defined as the supremum of thoser for which the expectation of exp(|x|^r) is finite. Let E-decay(v)>0. Whenever Δ_σ a _s ^t converges (that is, ifd>2 or |σ|>0: Ifd>2, E-decay(lima _s ^t )=min(E-decay(v),d+2/2); if |σ|=1, E-decay (lim Δ_σ a _s ^t )=min(E-decay(ν),d+2); if |σ| ⩾, E-decay (lim Δ_σ a _s ^t )=E-decay(ν).     

Polarized pK<Superscript>-</Superscript> scattering in Unitary Baryon Chiral Perturbation Theory

We study pK^- scattering in the energy range from threshold through the L \Lambda(1520) peak in UBChPT, taking into account O(q) vertices from meson-baryon contact interactions and s - and u -channel ground-state baryon exchange, s - and u -channel decuplet- and nonet-baryon exchange and t -channel vector-meson exchange, as well as O(q ²) flavor-breaking vertices. Detailed fits to data are presented, including a substantial body of differential cross-section data with meson momentum q _lab > 300 MeV not considered in previous treatments.