On a characterization of the normal distribution by means of identically distributed linear forms

Let X₁, X₂,…, be independent, identically distributed random variables. Suppose that the linear forms L₁ = Σ_j=1^∞a_jX_j and L₂ = Σ_j=1^∞b_jX_j exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X₁ is normally distributed are presented. The result is an extension of a theorem of Linnik (Ukrainian Math. J.5 (1953), 207–243, 247–290) concerning the case that the linear forms L₁ and L₂ have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.     

FOURIER TRANSFORM FAR INFRARED SPECTRA AND NORMAL COORDINATE ANALYSIS OF FIFTEEN RARE-EARTH OXALATE HYDRATES

The Fourier transform far infrared spectra of oxalate hydrates of whole rare-earth series except Pm and Sc are investigated in the range of 100-400 cm~(-1). The assignment of vibrational frequencies is made on the basis of normal coordinate analysis using our NORVIB program. The model used for the hghter elements of the lanthanide series (La to Eu) is based on the crystal structure of Nd_2(C_2O_4)_3. 10H_2O, in which each lanthanide ion is surrounded by nine O atoms, six from three oxalate ions and three from Water molecules.The model adopted for the heavier elements of the lanthanide series (Gd to Lu) and Ytterbium is based on the crystal structure of Yb_2(C_2O_4)_3· 6H_2O, in which each ion is surrounded by eight atoms, six from three oxalate ions and two from water molecules. The variation of the metal-ligand frequencies and force constants with the atomic numbers of lanthanides is plotted and discussed.     

二组分混合物的最小流化特性 Ⅰ.混合/分离状态与最小流化空隙率

分析了二组分混合物的３种混合／分离状态及在完全混合时发生连续相转变的极限组成。提出一个判别非等密度体系混合／分离状态的法则与确定混合物空隙率简易可行的随意松散堆积方法。     

Quality evaluation of fingerprints of herbal medicine with chromatographic data

In this study, local least squares (LLS) and principal component analysis (PCA) were applied to deal with the disturbances in a data set of chromatographic fingerprints after necessary data transformations. It has been demonstrated that PCA with standard normal variate (SNV) transformation of data led to meaningful classification of 33 different Erigeron breviscapus herbal samples. The result was also corroborated by variance squares discriminant method. The quality of herbal objects was further evaluated, and the causes of this fact have been explained from a chemical point of view. At the same time, it implied an idea for qualitative evaluation of the herbal objects with a common class pattern of chromatographic fingerprints.     

Rheology of polystyrene solutions around the coil overlapping concentration: A phenomenological description of stresses in simple shear flow

Linear viscoelasticity behavior is described with the sum of two terms for polystyrene solutions in tricresyl phosphate around the coil overlapping concentration (K. Osaki, T. Inoue, & T. Uematsu, J Polym Sci Part B: Polym Phys 2001, 39, 211). One is a Rouse–Zimm (RZ) term represented by the Zimm theory with arbitrarily chosen values of the hydrodynamic interaction parameter and the longest relaxation time (τ_RZ). The other (the L term) consists of a relaxation mode with a single relaxation time (τ_L > τ_RZ) and a high‐frequency limiting modulus proportional to the square of the concentration. In this study, we describe the viscosity (η) and first normal stress coefficient (Ψ₁) in steady shear with simple formulas. The stress due to the L term is assumed to be given by a Kaye, Bernstein, Kearsley, and Zapas (K‐BKZ) equation with the damping function h(γ) = (1 + 0.2γ²)^?1/2, where γ is the magnitude of shear. Contributions to η and Ψ₁ from the RZ term are derived from the RZ model, in which the relaxation time in steady flow is given by τ_st = τ + (τ_RZ ? τ)/(1 + 0.35τ_RZ γ˙) instead of τ_RZ. Here, γ˙ is the rate of shear, and τ is the τ_RZ value at the infinite dilution limit. η and Ψ₁ at various concentrations for two polystyrene samples (with molecular weights of 2890 and 8420 kg mol^?1) are well described with parameters derived from dynamic viscoelasticity. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1038–1045, 2002     

Refining adsorption parameters of n-butyl alcohol at the interface of mercury electrode with aqueous NaF and Na2SO4 solutions by correcting the adsorbate concentration

As an adjunct to the regression analysis of differential capacitance curves, which allows refining the adsorption parameters, a program, which takes into account possible errors in volume concentrations of organic substances, is developed. Using this procedure, the earlier data on the differential capacitance of a mercury electrode in aqueous solutions of normal butanol (1-BuOH) containing either 0.1 M NaF or 0.05 and 0.5 M Na₂SO₄ as the supporting electrolyte are analyzed. This allows obtaining the most accurate values of adsorption parameters for the systems mentioned above within the framework of the model of two parallel capacitors and the Frumkin isotherm. It is shown that, when a linear dependence of the intermolecular interaction parameter on the electrode potential is taken into account, the standard deviation of experimental capacitance values from those calculated using the mentioned model is 6.8–8.8%, which points to very high accuracy of this phenomenological model.     

正规BCI-代数的“拟-广结并”构造

本文进一步讨论BCI—代数的拟结合部分的特征性质,引入正规BCI—代数的概念。同时给出一种构造正规BCI—代数的“拟—广结合并”方法。     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).