Moments and Distributions of Trajectories in Slow Random Monads

Given a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter, 0≤s≤1. We fix a point ⊙∈B and consider the sequence M ^t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis _n , Rec _n and Tra _n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis _n and Tra _n converge in law to geometric distributions and Rec _n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s=1 is trivial, so let 0≤s<1. For any natural number k there is a number such that the k-th moments of Vis _n , Rec _n and Tra _n do not exceed this number for all n. About Vis _n : for any natural k the k-th moment of Vis _n is an increasing function of n. So it has a limit when n→∞ and for all n it is less than this limit. About Rec _n : for any k the k-th moment of Rec _n tends to one when n tends to infinity. About Tra _n : for any k the k-th moment of Tra _n has a limit when n tends to infinity.     

Evolution of the structure of Rb2ZnCl4 over the temperature range 4.2–310 K

X-ray diffraction is used to study the temperature dependence of the lattice parameters and the sequence of structural realignments in crystalline Rb₂ZnCl₄ over temperatures of 4.2–310 K. The appearance of and changes in the system of satellite reflexes indicative of structural ordering are studied. Below 74 K, on going into the monoclinic phase (space group A11a), anomalies are observed in the behavior of the lattice parameters, and superstructural reflexes develop with wave vectors q=a ^*/3+b ^*/2+c ^*/2 corresponding to an increase by a large factor in initial parameters a, b, and c of the Pnma-phase. Fiz. Tverd. Tela (St. Petersburg) 41, 1084–1090 (June 1999)     

X-ray diffraction investigations of the crystallographic parameters and thermal expansion of β-BaB2O4 crystals

The lattice parameters a and c of β-BaB₂O₄ crystals have been measured in the temperature range 80–300 K by the x-ray diffraction method. The thermal expansion coefficients α are calculated from the measured values of the parameters. A substantial anisotropy of the thermal expansion is found. It is shown that the thermal expansion coefficient α _c along the c axis is an order of magnitude greater than the thermal expansion coefficient α _a in a plane perpendicular to this axis. It is established that α _a becomes negative in the temperature range 80–190 K. Fiz. Tverd. Tela (St. Petersburg) 39, 1038–1040 (June 1997)     

X-ray studies of magnetic phase transitions in the intermetallic compounds RMn2Ge2 (R=La, Sm, Gd, Nd, Tb, and Y)

The lattice parameters a and c of the tetragonal intermetallic compounds RMn₂Ge₂ (R=La, Sm, Gd, Nd, Tb, and Y) have been measured by x-ray diffraction in the temperature interval 10–800 K. Anomalies are observed in the temperature dependence of a and c due to phase transitions from the paramagnetic to the magnetically ordered state in the Mn subsystem, transitions between various magnetically ordered phases due to a change in the magnitude and sign of the Mn-Mn exchange interaction, and magnetic transitions caused by ordering of the rare-earth subsystem leading to a rearrangement of the magnetic structure of the Mn subsystem. It is concluded that, along with the lattice parameter a, the lattice parameter c also has an influence on the Mn-Mn exchange interaction. Fiz. Tverd. Tela (St. Petersburg) 41, 2053–2058 (November 1999)     

On the Discrete Spectrum of a Spatial Quantum Waveguide with a Disc Window

In this study we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disc window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We prove that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. We give also a numeric estimation of the number of discrete eigenvalue as a function of \fracad\frac{a}{d}. When a tends to the infinity, the asymptotic of the eigenvalue is given.     

Stochastic approach to the comparison of ellipsoidal and interval estimates

Two approaches to the multiplication of an uncertain vector in R ⁿ by an exactly known matrix are considered. Namely, uncertain vectors can be localized either in boxes or in ellipsoids. The accuracy of these approaches is compared by measuring the volume of the final localization domain. The main result is that, under additional assumptions, the probability of the event that the ellipsoidal localization is preferable tends to 1 as n → ∞.     

Superconducting transition temperature in mercury HTSC-cuprates under hydrostatic pressure

A study is made of the properties of the homologous series of mercury HTSC-cuprates HgBa₂Ca_n−1Cu_nO_2n +2+δ with n=1–8. Experiments are conducted under pressure for samples with n=1–5. The Hg-1223 and Hg-1234 phases were synthesized using a controlled high pressure chamber. The oxygen content of an initial mixture corresponding to the Hg-1234 phase was varied by changing the composition of the initial BaO/BaO₂ oxides. The dependence of the superconducting transition temperature T _c on the lattice constant a (and, therefore, on the oxygen content) and of T _c ^max and dT _c ^max /dp on n are convex upward up to n=4, 5. The maximum values always correspond to the Hg-1223 phase. Experimental T _c ^max (n) curves for the phases with n=1–6 and dT _c ^max /dp curves for n=1–5 are compared with Anderson’s theory (the so-called RVB model). A general analysis of these results indicates that the mercury cuprates have an ideal structure for HTSC. The Hg-1223 phase is the “champion” in this ideal structure and the critical temperature corresponding to this phase (T _c =135 K) is the highest at atmospheric pressure. Zh. éksp. Teor. Fiz. 113, 1474–1483 (April 1998)     

Some New Results for McKean’s Graphs with Applications to Kac’s Equation

The main goal of the present paper is to sharpen some results about the error made when the Wild sums, used to represent the solution of the Kac analog of Boltzmann’s equation, are truncated at the n-th stage. More precisely, in Carlen, Carvalho and Gabetta (J. Funct. Anal. 220: 362–387 (2005)), one finds a bound for the above-mentioned error which depends on (an ^Λ+ε). On the one hand, it is shown that Λ, the least negative eigenvalue of the linearized collision operator, is the best possible exponent. On the other hand, ε is an extra strictly positive number and a a positive coefficient which depends on ε too. Thus, it is interesting to check whether ε can be removed from the above bound. According to the aforesaid reference, this problem is studied here by means of the probability distribution of the depth of a leaf in a McKean random tree. In fact, an accurate study of the probability generating function of such a depth leads to conclude that the above bound can be replaced with (a′ n ^Λ).     

Cutoff effects in lattice actions at μ ≠ 0

We look at the cutoff dependence of several lattice actions, including two improved actions viz. Naik and p4, and and chirally-invariant ones, namely fixed-point, overlap and domain-wall, with the aim of understanding its behavior at μ ≠ 0. Apart from numerical results, we also derive a series expansion in N _r ⁻¹ for the free-gas pressure. We find that actions with O(a ⁿ )-improved rotational invariance produce O(a ⁿ )-improvement in the pressure. The series for unimproved overlap and domain-wall fermions are identical to the naive series, and hence using Naik or p 4 kernels should produce improvement in these formulations as well. Lastly, we find that actions that are improved at μ = 0 remain so as the chemical potential is turned on. The series coefficients become μ-dependent now, however their functional form at any given order is the same for all actions.     

Inverse resonance scattering for Jacobi operators

The Jacobi operator (Jf) _n = a _n−1 f _n−1 +a _n f _n+1 + b _n f _n on ℤ with real finitely supported sequences (a _n − 1) _n∈ℤ and (b _n ) _n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (a _n , b _n , n ∈ ℤ) → {the zeros of the reflection coefficient} and (a _n , b _n , n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.     

Physical properties of n-CdGeAs2 single crystals prepared by low-temperature crystallization

The first results obtained in studies of the temperature dependences of electrical conductivity and Hall constant of n-CdGeAs₂ single crystals prepared by low-temperature crystallization are reported. It has been established that the method developed permits growing single crystals with a free-electron concentration ⋍(1−2)×10¹⁸ cm⁻³ and a Hall mobility ⋍10000 cm²/(Vs) at T=300 K. It is shown that the temperature dependence of Hall mobility exhibits a behavior characteristic of electron scattering by lattice vibrations, whereas below 150 K a deviation from this law is observed to occur evidencing an increasing contribution of static lattice defects to scattering. The Hall mobility in the crystals prepared was found to reach ⋍36000 cm²/(Vs) at 77 K. Photosensitive heterojunctions based on n-CdGeAs₂ single crystals were prepared. The spectral response of the photosensitivity of these structures is analyzed. It is concluded that this method is promising for preparation of perfect CdGeAs₂ crystals. Fiz. Tverd. Tela (St. Petersburg) 41, 1190–1193 (July 1999)     

Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

Linear combinations x ₁ a ₁ ⋯ x _n a _n of n given natural numbers a _s (with nonnegative integral coefficients x _s ) attain all the integral values, starting from some integer N(a), called the Frobenius number of vector a (provided that the integers a _s have no common divisor, greater than 1). The growth rate of N(a) with the large value of σ = ta ₁ ⋯ a _n depends peculiarly from the direction α of the vector a = σα. The article proves the lower bound of order and the upper bound of order σ ². Both orders are reached from some directions α. The averaging of N(a) along all directions, performed for σ = 7, 19, 41 and 97, provides the values, confirming the rate σ ^p for some p between 3/2 and 2 (for n = 3), excluding neither 3/2 nor 2, for the asymptotic behaviour at large σ. One gets check p ≈ 1, 66 for σ between 100 and 200. These unexpected results, based on some strange relations of the Frobenius numbers to the higher-dimensional continued fractions geometry, lead to many facts of this arithmetic trubulence theory, discussed in this article both as theorems and as conjectures. ^★ Partially supported by RFBR grant 05-01-00104.     

Raman scattering in a near-surface n-GaAs layer implanted with boron ions

Structure in the Raman scattering spectra of near-surface n-GaAs layers (n=2×10¹⁸ cm⁻³) implanted with 100 keV B⁺ ions in the dose range 3.1×10¹¹–1.2×10¹⁴ cm⁻² is investigated. The qualitative and quantitative data on the carrier density and mobility and on the degree of amorphization of the crystal lattice and the parameters of the nanocrystalline phase as a result of ion implantation are obtained using a method proposed for analyzing room-temperature Raman spectra. Fiz. Tverd. Tela (St. Petersburg) 41, 1495–1498 (August 1999)     

Macroscopic Determinism in Interacting Systems Using Large Deviation Theory

We consider the quasi-deterministic behavior of systems with a large number, n, of deterministically interacting constituents. This work extends the results of a previous paper [J. Statist. Phys. 99:1225–1249 (2000)] to include vector-valued observables on interacting systems. The approach used here, however, differs markedly in that a level-1 large deviation principle (LDP) on joint observables, rather than a level-2 LDP on empirical distributions, is employed. As before, we seek a mapping _t on the set of (possibly vector-valued) macrostates such that, when the macrostate is given to be a ₀ at time zero, the macrostate at time t is _t (a ₀) with a probability approaching one as n tends to infinity. We show that such a map exists and derives from a generalized dynamic free energy function, provided the latter is everywhere well defined, finite, and differentiable. We discuss some general properties of _t relevant to issues of irreversibility and end with an example of a simple interacting lattice, for which an exact macroscopic solution is obtained.     

Pattern Theorems,Ratio Limit Theorems and Gumbel Maximal Clusters for Random Fields

We study occurrences of patterns on clusters of size n in random fields on ℤ ^d . We prove that for a given pattern, there is a constant a>0 such that the probability that this pattern occurs at most na times on a cluster of size n is exponentially small. Moreover, for random fields obeying a certain Markov property, we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value. This leads to an elegant and simple proof of the ratio limit theorem for these random fields, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges as n→∞. Implications for the maximal cluster in a finite box are discussed.     

Variations of lattice constants and thermal expansion coefficients of indium at high pressure and high temperature

ABSTRACT

The effects of pressure and temperature on the lattice constants and thermal expansion coefficients of Indium were studied up to 18.6?GPa and 506?K based on in situ X-ray diffraction method with an externally heated diamond anvil cell. The results show that the measured axial ratio (c/a) decreases with increasing temperature and its temperature dependence decreases with increasing pressure. The thermal expansion coefficient of the a-axis decreases with increasing pressure up to 7?GPa and remains almost constant above 7?GPa, whereas that of the c-axis increases monotonously with pressure and changes from negative to positive at around 7?GPa. The observed behavior suggests that temperature reduces the tetragonal distortion on the lattice, and its effect is dominant below 7?GPa; in contrast, pressure enhances lattice distortion, and tends to have a stronger effect above 7?GPa.     

Detection of (<Emphasis Type="Italic">C</Emphasis> + <Emphasis Type="Italic">A</Emphasis>)-type dislocation self-blocking in magnesium

The self-blocking effect predicted theoretically and then observed in intermetallides is established for a pure metal, namely, for magnesium. This effect can be observed only for magnesium single crystals whose axis is parallel to the c axis and whose yield stress behavior σ _y (T) has a temperature anomaly. For such single crystals, the self-blocking of the (c + a)-type edge dislocations is established during pyramidal slip of type II. The self-blocking is proved by dislocation extension along the preferred direction without external stress. In this case, the á 1[`1]00 ñ \left\langle {1\bar{1}00} \right\rangle directions appear preferred. TEM images of (c + a) dislocations extended along the preferred directions are presented. It is demonstrated that two effects – temperature anomaly of σ _y (T) and dislocation self-blocking – have the common nature: a two-valley potential relief of the dislocation. A model of two-valley relief of the (c + a) dislocations in Mg is proposed.     

Liesegang patterns: Complex formation of precipitate in an electric field

Formation of 1D Liesegang patterns was studied numerically in precipitation and reversible complex formation of precipitate scenarios in an electric field. The Ostwald’s supersaturation model reported by Buki, Kárpáti-Smidróczki and Zrínyi (BKZ model) was extended further. In the presence of an electric field the position of the first and the last bands (X _n) measured from the junction point of the outer and the inner electrolytes can be described by the functionX _n =a ₁τ _n ^1/2 +a ₂τ_n +a ₃, where τ_n is the time elapsed until the nth band formation,a ₁,a ₂ anda ₃ are constants. The variation of the total number of bands with different electric field strengths (ε) has a maximum. For higher ε one can observe a moving precipitation zone that becomes wider due to precipitation and reversible complex formation.