Well‐posedness,smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamics

We prove existence, uniqueness, regularity and smooth dependence of the weak solution on the initial data for a semilinear, first order, dissipative hyperbolic system with discontinuous coefficients. Such hyperbolic systems have successfully been used to model the dynamics of distributed feedback multisection semiconductor lasers. We show that in a function space of continuous functions the weak solutions generate a smooth skew product semiflow. Using slow fast structure and dissipativity we prove the existence of smooth exponentially attracting invariant centre manifolds for the non‐autonomous model. Copyright © 2006 John Wiley & Sons, Ltd.     

Constitutive relation of an orthorhombic polycrystal with the shape coefficients

An orthorhombic polycrystal is an orthorhombic aggregate of tiny crystallites. In this paper, we study the effect of the crystalline mean shape on the constitutive relation of the orthorhombic polycrystal. The crystalline mean shape and the crystalline orientation arrangement are described by the crystalline shape function (CSF) and the orientation distribution function (ODF), respectively. The CSF and the ODF are expanded as an infinite series in terms of the Wigner D–functions. The expanded coefficients of the CSF and the ODF are called the shape coefficients s ^l _m0 and the texture coefficients c ^l _mn , respectively. Assuming that C ^eff in the constitutive relation depends on the shape coefficients s ^l _m0 and the texture coefficients c ^l _mn , by the principle of material frame-indifference we derive an analytical expression for C ^eff up to terms linear in s ^l _m0 and c ^l _mn , and the expression would be applicable to the polycrystal whose texture is weak and whose crystalline mean shape has weak anisotropy. C ^eff contains six unspecified material constants (λ, μ, c, s ₁, s ₂, s ₃), five shape coefficients (s ² ₀₀ , s ² ₂₀,s ⁴ ₀₀ , s ⁴ ₂₀ , s ⁴ ₄₀ ), and three texture coefficients (c ⁴ ₀₀ , c ⁴ ₂₀ , c ⁴ ₄₀ ). The results based on the perturbation approach are used to determine the five material constants approximately. We also find that the shape coefficients s ² _m0 and s ⁴ _m0 are all zero if the crystalline mean shape is a cuboid. Some examples are given to compare our computational results. The project supported by the National Natural Science Foundation of China (10562004), and the Oversea Returning Grant of China. The English text was polished by Keren Wang.     

On the tvGARCH(1,1) model: Existence, CLT, and tail index

We study the properties of the tvGARCH(1, 1) model (1.8) with logistic coefficients. We obtain conditions for the existence of an L ^p -bounded solution (p ⩾ 0) of Eq. (1.8). For p ⩾ 4, we prove an exponential decay of the lagged correlation function and the central limit theorem for partial sums of the squared tvGARCH(1, 1) process under similar conditions as in the stationary case. In the second part of the paper, we study the (weak) tail index of the tvGARCH(1, 1) model.     

On the evaluation of Bernoulli numbers using binomial coefficients

A new algorithm is presented for calculating the Bernoulli numbers valid for all values of their argument. We introduce a simple, new analytical computer method for calculating the Bernoulli numbers on the basis of their integral expressions. The formula given in this study for the evaluation of Bernoulli numbers shows good rate of convergence and numerical stability with existing formulas.     

差分方程解的稳定性、有界性及概周期解的存在性

作者通过Liapunov泛函建立了一类高维差分方程解一致稳定、一致渐近稳定及指数渐近稳定的充要条件. 此外, 作者还证明了解的一致渐近稳定性蕴含解的有界性, 同时也给出了概周期差分方程存在概周期解的一个充分条件.     

Piezoelectricity in quasicrystals: A group-theoretical study

Group-theoretical methods have been accepted as exact and reliable tools in studying the physical properties of crystals and quasicrystalline materials. By group representation theory, the maximum number of non-vanishing and independent second-order piezoelectric coefficients required by the seven pentagonal and two icosahedral point groups — that describe the quasicrystal symmetry groups in two and three dimensions — is determined. The schemes of non-vanishing and independent second-order piezoelectric tensor components needed by the nine point groups with five-fold rotations are identified and tabulated employing a compact notation. The results of this group-theoretical study are briefly discussed.      

Study of ion–solvent interactions and activation energy of LiBr in DMSO,H<Subscript>2</Subscript>O and DMSO–H<Subscript>2</Subscript>O mixtures at various temperatures

The Jones–Dole B coefficients of the electrolyte Lithium bromide (LiBr), reference salts tetra butyl ammonium tetra phenyl borate (BU₄NBPh₄), tetra butyl ammonium bromide (BU₄NBr), and potassium chloride (KCl) in dimethylsulfoxide (DMSO), water, and DMSO–water mixtures were obtained at different temperatures range from 25 to 45 °C For this, the relative viscosities were measured for Lithium bromide (LiBr) and reference salts in DMSO, water, and DMSO–water mixtures at above-mentioned temperatures. The B coefficients of these electrolytes were behaved as structure makers in DMSO, while in H₂O and DMSO–H₂O mixtures, the B-coefficient values were less positive showing the weak structure-making effect. Ionic viscosity B coefficients allow us to assess the behavior of ions in the solvent mixtures. In this study it was observed that all the values of ionic B coefficient of (Li⁺) were positive and small showing the weak structure-making effects. It was also observed that Br⁻ ions maintain negative B coefficient values in all DMSO–H₂O mixtures, except in 60% DMSO mole fraction. From this it can be concluded that Br⁻ ion behaved as a structure breaker in water and in all DMSO–H₂O mixtures except in 60% DMSO mole fraction mixtures. The low B _± values of alkali metal ions and Br⁻ ions in water are due to the breakdown of the tetrahedral structural of water and the formation of strongly structured solvated ion. It is also observed that the values of the energy of activation of the flow for LiBr are greater in DMSO–water mixtures and in pure water than in DMSO. This may be due the presence of a network of hydrogen bonds which cause the hindrance in the flow of the solution of LiBr in DMSO–water mixtures and in pure water than in DMSO.     

Dual properties of orthogonal polynomials of discrete variables associated with the quantum algebra <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(<Emphasis Type="Italic">su</Emphasis>(2))

We show that for every set of discrete polynomials y _n (x(s)) on the lattice x(s), defined on a finite interval (a, b), it is possible to construct two sets of dual polynomials z _k (ξ(t)) of degrees k = s-a and k = b-s-1. Here we do this for the classical and alternative Hahn and Racah polynomials as well as for their q-analogs. Also we establish the connection between classical and alternative families. This allows us to obtain new expressions for the Clerbsch-Gordan and Racah coefficients of the quantum algebra U _q (su(2)) in terms of various Hahn and Racah q-polynomials. Dedicated to the memory of our teacher and friend Arnold F. Nikiforov (18.11.1930–27.12.2005).     

Experimental study of H2O spectroscopic parameters in the near-IR (6940-7440 cm) for gas sensing applications at elevated temperature

Tunable diode laser (TDL) absorption sensors of water vapor are attractive for temperature, gas composition, velocity, pressure, and mass flux measurements in a variety of practical applications including hydrocarbon-fueled combustion systems. Optimized design of these sensors requires a complete catalog of the assigned transitions with accurate spectroscopic data; our particular interest has been in the 2ν₁, 2ν₃, and ν₁+ν₃ bands in the near-IR where telecommunications diode lasers are available. In support of this need, fully resolved absorption spectra of H₂O vapor in the spectral range of 6940-7440 cm⁻¹ (1344-1441 nm) have been measured as a function of temperature (296-1000 K) and pressure (1-800 Torr), and quantitative spectroscopic parameters inferred from these spectra compared to published data from Toth, HITRAN 2000 and HITRAN 2004. The peak absorbances were measured for more than 100 strong transitions at 296 and 828 K, and linestrengths determined for 47 strong lines in this region. In addition to reference linestrengths S(296 K), the air-broadening coefficients γ_air(296 K) and temperature exponents n were inferred for strong transitions in five narrow regions, near 7185.60, 7203.89, 7405.11, 7426.14 and 7435.62 cm⁻¹ that had been targeted as attractive for future diagnostics applications. Most of the measured results, determined within an accuracy of 5%, are found to be in better agreement with HITRAN 2004 than with earlier editions of this database. Large discrepancies (>10%) between measurements and HITRAN 2004 database are identified for some of the probed transitions. These new spectroscopic data for H₂O provide a useful test of the sensor design capabilities of HITRAN 2004 for combustion and other applications at elevated temperatures.