Central Limit Theorems for Nonlinear Hierarchical Sequences of Random Variables

Let a random variable x ₀ and a function f:[a, b] ^k [a, b] be given. A hierarchical sequence {x _n :n=0, 1, 2,...} of random variables is defined inductively by the relation x _n =f(x _{n–1, 1}, x _{n–1, 2}..., x _{n–1, k} ), where {x _{n–1, i} :i=1, 2,..., k} is a family of independent random variables with the same distribution as x _n–1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.     

A stochastic derivation of the Sivashinsky equation for the self-turbulent motion of a free particle

Within the framework of the Kershaw approach and of a hypothesis on spatial stochasticity, the relativistic equations of Lehr and Park, Guerra and Ruggiero, and Vigier for stochastic Nelson mechanics are obtained. In our model there is another set of equations of the hydrodynamical type for the drift velocityv _i(x _j,t) and stochastic velocityu _i(x _j,t) of a particle. Taking into account quadratic terms in l, the universal length, we obtain from these equations the Sivashinsky equations forv _i(x _j,t) in the caseu _i →0. In the limit l →0, these equations acquire the Newtonian form.     

The Structure of Radiative Tunnel Recombination Sites in Emulsion Microcrystals of AgBr(I)

To identify the structure of emissive tunnel recombination sites in the emulsion microcrystals of silver bromide AgBr(I) with iodine contaminations and to determine the role of an emulsion medium in their formation, the temperature dependence of the luminescence spectra in the range from 77 to 120 K, the kinetics of the growth of the maximum luminescence intensity value at λ ≈ 560 nm, and the luminescence flash spectrum stimulated by the infrared light are investigated. Two types of the AgBr_{1 – x}(I _x ) (x = 0.03) microcrystals—namely, obtained in an aqueous solution and on a gelatin substrate—are used in the studies. It is established that the emissive tunnel recombination sites with a luminescence maximum at λ ≈ 560 nm in AgBr_{1 – x}(I _x ) (x = 0.03) are the {(I _a ^- I _a ^- )Ag _i ⁺ } donor–acceptor complexes with the I _a ^- iodine ions located in neighbor anionic sites of the AgBr(I) crystal lattice, next to which the Ag _i ⁺ interstitial silver ion is positioned. With an increase in the temperature, the {(I _a ^- I _a ^- )Ag _i ⁺ } sites undergo structural transformation into the {(I _a ^- I _a ^- )Ag_in⁺} sites, where n = 2, 3, …. Moreover, the {(I _a ^- I _a ^- )Ag _in ⁺ } sites (n = 2) after the capture of an electron and hole also provide the tunnel recombination with a luminescence maximum at λ ≈ 720 nm. The influence of an emulsion medium consists in that gelatin interacts with the surface electron-localization sites, i.e., the interstitial silver ions Ag _in ⁺ , n = 1, 2, and forms the complexes {Ag _in ⁰ G⁺} (n = 1, 2) with them. The latter are deeper electron traps with a small capture cross section as compared to the Ag _in ⁺ sites (n = 1, 2) and that manifest themselves in that the kinetics of the luminescence growth in AgBr(I) to a stationary level at λ ≈ 560 nm is characterized by the presence of “flash firing.” At the same time, the luminescence flash stimulated by IR light, for which the Ag _in ⁺ (n = 1, 2) electron-localization sites are responsible, is absent. It is supposed that the electrons localized on the {Ag _in ⁺ G⁺} complexes (n = 2) retain the capability for emissive tunnel recombination with holes localized on paired iodine sites with a luminescence maximum at λ ≈ 750 nm.     

A Level 1 Large-Deviation Principle for the Autocovariances of Uniquely Ergodic Transformations with Additive Noise

A large-deviation principle (LDP) at level 1 for random means of the type $$M_n \equiv \frac{1}{n}\sum\limits_{j = 0}^{n - 1} {Z_j Z_{j + 1} ,{\text{ }}n = 1,2,...}$$ is established. The random process {Z _n} _n≥0 is given by Z _n = Φ(X _n) + ξ _n , n = 0, 1, 2,..., where {X _n} _n≥0 and {ξ _n} _n≥0 are independent random sequences: the former is a stationary process defined by X _n = T ⁿ(X ₀), X ₀ is uniformly distributed on the circle S ¹, T: S ¹ → S ¹ is a continuous, uniquely ergodic transformation preserving the Lebesgue measure on S ¹, and {ξ_n} _n≥0 is a random sequence of independent and identically distributed random variables on S ¹; Φ is a continuous real function. The LDP at level 1 for the means M _n is obtained by using the level 2 LDP for the Markov process {V _n = (X _n, ξ _n , ξ _n+1)} _n≥0 and the contraction principle. For establishing this level 2 LDP, one can consider a more general setting: T: [0, 1) → [0, 1) is a measure-preserving Lebesgue measure, $\Phi :\left[ {0,\left. 1 \right)} \right. \to \mathbb{R}$ is a real measurable function, and ξ _n are independent and identically distributed random variables on $\mathbb{R}$ (for instance, they could have a Gaussian distribution with mean zero and variance σ²). The analogous result for the case of autocovariance of order k is also true.     

Analyticity properties and a convergent expansion for the inverse correlation length of the high-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(β) (= mass of the fundamental particle of the associated lattice quantum field theory) of the spin-spin correlation function 〈s _x s _y 〉,x, y εZ ^d , of thed-dimensional Ising model admits the representation $$m(\beta ) = - ln\beta + r(\beta )$$ for small inverse temperaturesβ > 0.r(β) is ad-dependent function, analytic atβ = 0.c _n , the nth β = 0 Taylor series coefficient of r(β) can be computed explicitly from the Z^d limit of a finite number of finite lattice A spin-spin correlation functions 〈s₀s_x〉t>Afor a finite number ofx = (x ₁,x₂, ..., x_d), ¦x¦ = ∑ _i ^d 1¦x_i¦< R(n), where R(n) increases withn. Furthermore, there exists aβ' > 0, such that for eachβ ε (0,β')m(β) is analytic. Similar results are also obtained for the dispersion curve ω(p), ω(p)=ω(0)=m, pε(-π, π]^d?1, of the fundamental particle of the associated lattice quantum field theory.     

First-Passage and Extreme-Value Statistics of a Particle Subject to a Constant Force Plus a Random Force

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point x ₁ of a particle which starts at x ₀ with velocity v ₀. The probability that the particle has not yet arrived at x ₁ after a time t, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement m=max  _t [x(t)].     

Produits tensoriels continus d'espaces et d'algèbres de Banach

The notion of tensor product of a family (A _i ) _i I of Banach algebras is generalized to the case whenI is a topological space; in this case A _i is generated by some elements x _i , the family (x _i ) being subjected to certain conditions: for instance the functioni x _i must be continuous. This notion is applied to Quantum Field Theory in the following sense: certain algebras of observables can be considered as continuous tensor products of simpler ones, namely of algebras of observables with one degree of freedom.     

Gamma rays and neutrons following preequilibrium 165Ho(p, xcnypγ) reactions at Ep = 60 MeV

Neutrons and γ-rays following 60 MeV proton bombardment of ¹⁶⁵Ho were measured in coincidence with discrete γ-rays characteristic of the reaction channels (residual nuclei). The cross sections for the (p, xcnγ) reactions with x = 2–6, the γ-ray multiplicities, and the energy and angular distributions of the emitted neutrons were analyzed in terms of the preequilibrium and equilibrium deexcitation processes. Characteristic behaviours of the preequilibrium process were found in the (p, 2nγ) and (p, 4nγ) reactions where the sum E_T, = ∑^xE_i of the energies E_i of the emitted neutrons was large, while those of the equilibrium process were typical for the (p, 6ny) reaction with small E_T. The reactions are well reproduced by the expression σ. ≈0.35∑_xσ (2, x?2) + 0.4∑_xσ(1, χ?1)+ 0.25∑_xσ(0, x), where σ(n_p, n_c) stands for emission of n_p neutrons at the preequilibrium stage followed by evaporation of n_c neutrons at the equilibrium stage.     

Optical properties of mechanically compressed K<Subscript>2</Subscript>SO<Subscript>4</Subscript> crystals

The effect of a uniaxial mechanical compression (σ_m ≤ 100 bar) on the spectral dependences (300–800 nm) of the birefringence Δn _i and refractive indices n _i of K₂SO₄ crystals is studied. The electronic polarizabilities, refractions, and parameters (λ_0i , B _1i ) of ultraviolet oscillators of mechanically compressed crystals are calculated. It is shown that the dispersions of Δn _i(λ) and n _i(λ) are normal and sharply increase near the absorption edge. It is found that the uniaxial compression changes the value of the dispersions dΔn _i/dλ and dn _i/dλ rather than their character. It is ascertained that the simultaneous action of the compressions σ_x and σ_z, as well as of σ_y and σ_z, leads to the appearance of new isotropic states in the K₂SO₄ crystal, which manifests itself in the equality of corresponding birefringences. It is shown that the baric dependences n _i(σ) are determined by the change in the density of oscillators (～30%), by the shift of the absorption edge and effective band maximum and by the change in the oscillator strength (～70%).     

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.     

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n×n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let x _k denote eigenvalue number k. Under the condition that both k and n?k tend to infinity as n→∞, we show that x _k is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues $(x_{k_{1}},\ldots,x_{k_{m}})$ from the GOE or GSE where k ₁, n?k _m and k _i+1?k _i , 1≤i≤m?1, tend to infinity with n. The result in each case is an m-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.     

Effect of the Ge content on the Schottky barrier height in structures based on Si1 − x Ge x solid solution

The subject of investigation is the influence of the Ge content on the Schottky barrier height in Au-p-i-n/Al structures based on Si_{1 ? x} Ge _x (0 < x < 0.26) solid solutions grown by electron-beam floating-zone melting. In Au-i-Si_{1 ? x} Ge _x , Au-αGe/Si(Li), Au-αGe/Si, and Au-n-Si_{1 ? x} Ge _x structures, the height of the barrier is about 1 eV, which exceeds the average value of this parameter for silicon available in the literature. It is shown that an excess Ge concentration in the near-surface region of the Si_{1 ? x} Ge _x crystal causes self-passivation of the surface, which leads to a rise in the barrier height. In the Au-i-Si_{1 ? x} Ge _x structures, the Schottky barrier height increases from 0.97 to 1.03 eV as Ge content x increases from 0 to 0.11.     

Concepts of convergence for a quantum law of large numbers

We introduce the notion of uniform convergence of an operator sequence relative to a state operator, and we compare it with the notion of almost everywhere convergence, introduced by Gudder. We show that a sequence (M_n) of mean-value operators

$\text{M}{}_{\mathrm{n}}\text{=}\text{1}\text{n}\text{∑n}\text{i=1}\text{1?…?A}{}_{\mathrm{i}}\text{?1?…,}$

defined on the infinite tensor product $\text{?}\text{i}\text{H}{}_{\mathrm{i}}$, converges relative uniformly to $\text{x}\text{1}$ if and only if it converges almost everywhere to $\text{x}\text{1}$.     

Spectra of random contractions and scattering theory for discrete-time systems

Random contractions (subunitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex eigenvalues of generic N × N random matrices  of such a type, corresponding to systems with broken time reversal invariance. Deviations from unitarity are characterized by rank M≤N and a set of eigenvalues 0<T _i≤1, i=1,..., M of the matrix $\hat T = \hat 1 - \hat A^\dag \hat A$ . We solve the problem completely by deriving the joint probability density of N complex eigenvalues and calculating all n-point correlation functions. In the limit N?M, n, the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.     

Physical anomalous dimensions at small x

I present a theoretical discussion of the uncertainties related to the QCD analysis of the proton structure function F ₂(x,Q ²) at small x. The role played by the ‘unphysical’ gluon density is pointed out. It is shown how the study of more observables can reduce the theoretical uncertainty and, in particular, an alternative method of analysis, based on the introduction of physical anomalous dimensions, is suggested.     

Degree Complexity of Birational Maps Related to Matrix Inversion

For a q × q matrix x = (x _{i, j} ) we let ${J(x)=(x_{i,j}^{-1})}For a q × q matrix x = (x _{i, j} ) we let J(x)=(x_i,j^-1){J(x)=(x_{i,j}^{-1})} be the Hadamard inverse, which takes the reciprocal of the elements of x. We let I(x)=(x_i,j)^-1{I(x)=(x_{i,j})^{-1}} denote the matrix inverse, and we define K=I°J{K=I\circ J} to be the birational map obtained from the composition of these two involutions. We consider the iterates Kⁿ=K°?°K{K^n=K\circ\cdots\circ K} and determine the degree complexity of K, which is the exponential rate of degree growth d(K)=lim_n?￥( deg(Kⁿ) )^1/n{\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}} of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case K may be represented by a simpler map. Here we show that for general matrices the value of δ(K) is equal to the value conjectured by Anglès d’Auriac, Maillard and Viallet.     

Improvement in magnetic properties of La substituted BaFe12O19 particles prepared with an unusually low Fe/Ba molar ratio

In this study, effect of lanthanum substitution on the phase composition, lattice parameters and magnetic properties of barium hexaferrite has been studied in samples synthesized in ammonium nitrate melt. Samples, prepared with different lanthanum amount and having various initial Fe/(Ba+La) ratios in between 12 and 2 {(Ba_1−xLa_x)·n Fe₂O₃, where 0≤x≤1 and 1≤n≤6)}, are sintered at temperatures from 800 to 1200 °C. The lattice parameters, both a and c, decreases with increasing La amount which results in a decrease of the unit cell volume. The scanning electron microscope micrographs show that the pure and La-substituted sample with x=0.3, both calcinated at 1000 °C, have grain sizes smaller than 1 μm. The coercivities of the La-substituted samples increase with increasing La amount and reaches to a maximum value of 5.73 kOe, when x=0.3. Sintering at higher temperatures (above 1000 °C) decreases the coercivity, resembling a transition from single to multi-domain behavior of the particles, while saturation magnetization of the samples continues to increase due to the increasing grain size. Magnetization measurements of the samples prepared with different Fe/(Ba+La) molar ratios, n's, revealed that the specific saturation magnetization slightly increases with decreasing n, while coercivities fluctuates around 5.5 kOe. However, a sharp increase in the saturation magnetization has been observed in the sample having n=1 and washed in HCl. It was measured as 59.2 emu/g at 15 kOe, which is higher than that of the pure sample (57.5 emu/g). Thus, the magnetic parameters are optimized in the sample Ba_0.7La_0.3Fe₁₂O₁₉ so as to maximize both coercivity and specific saturation magnetization in the HCl-washed sample synthesized by starting with an unusually low Fe/(Ba+La) molar ratio of 2 (or n=1).     

Vibrational spectra and force constants of symmetric tops: Rotational analysis of ν3 and 2ν3 of H3Si35Cl and H3Si37Cl

The gas-phase infrared spectra of natural and ³⁵Cl isotopically enriched H₃SiCl have been recorded with a resolution of 0.04 cm^?1 in the regions of ν₃ (～550 cm^?1) and 2ν₃ (～1100 cm^?1). The fundamentals of species with ²⁸Si, ²⁹Si, ³⁰Si, ³⁵Cl, and ³⁷Cl and several hot bands of the series nν₃ + mν_i ? mν_i (n, m = 1, 2; i = 3 and 6) have been detected. The parameters ν₀, x₃₃, x₃₆, B₀, α₃^A, D_J⁰ and D_J³ were determined by a combination of least-squares analysis of P and R branches resolved into J peaks (J ≤ 90) and a band contour simulation. Possible resonances of nν₃ and (n ? 1)ν₃ + ν₆ are discussed.     

Universal structural phase transition in network glasses

Raman and Mössbauer spectroscopy provide evidence for a transition from a molecular cluster network at x=0 to a continuous network at x=0.35 in Ge_1?xSn_xSe₂ alloy glasses. The nature of this morphological transition involves a reformation of molecular cluster surfaces in the heterogeneous phase to yield a homogeneous phase. The transition is believed to be a universal property of the easy glass formers and can be effected in one of several ways.