On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed.     

Synthesis,structural, electrical and magnetic studies of La0.5Ca0.45−xSrxBa0.05MnO3

Half doped mixed valence manganite system La_0.5Ca_0.45?xSr_xBa_0.05MnO₃ (with x=0.1, 0.2, and 0.3) synthesized through a low temperature nitrate route is systematically investigated in this paper. The electronic transport and magnetic properties are analyzed and compared apart from the study of unit cell structure and composition. The system is found to crystallize only in orthorhombic structure (Pnma) and the electronic phase transitions are observed to be of second order. The charge and orbital ordering have been observed to coexist with ferromagnetism in x=0.1 compound. Application of small polaron and variable range hopping models to resistivity data of the system corresponding to high temperature range shows increasing mobility of e_g electrons with x, with the later model describing the electronic transport very closely than the former. The temperature dependent magnetization of the compounds shows monotonic increase of paramagnetic to ferromagnetic transition (T_C) with x. Ferromagnetism is exhibited for the complete temperature range down from respective T_C in contrast to antiferromagnetism usually exhibited by half-doped compounds in the low temperature range. The plots of magnetization versus magnetic field reveal a transition from soft to hard magnetic character for all the compounds as the temperature is lowered.     

On the Glivenko–Cantelli theorem for generalized empirical processes based on strong mixing sequences

Given \s{X_i, i 1\s} as non-stationary strong mixing (n.s.s.m.) sequence of random variables (r.v.'s) let, for 1 i n and some γ ε [0, 1],

F₁(x)=γP(X_i<x)+(1-γ)P(X_ix)

and

I_i(x)=γI(X_i<x)+(1-γ)I(X_ix)

. For any real sequence \s{C_i\s} satisfying certain conditions, let

.

In this paper an exponential type of bound for P(D_n ), for any >0, and a rate for the almost sure convergence of D_n are obtained under strong mixing. These results generalize those of Singh (1975) for the independent and non-identically distributed sequence of r.v.'s to the case of strong mixing.     

Tests for sphericity under correlated multivariate regression equations model

Summary In this paper, the authors investigated the asymptotic distribution theory connected with the likelihood ratio test (LRT)-like test statistic for sphericity under correlated multivariate regression equations (CMRE) model. An asymptotic expression is obtained for the null distribution of the above test statistic. Asymptotic nonnull distribution of the above test statistic under fixed alternatives is also derived. The above results are derived when the underlying distribution is multivariate normal. It was also shown that the above results are valid even when the joint distribution of the observations is elliptically symmetric. The authors also derived the asymptotic null distribution of the LRT-like test statistic when the observations on each variable are elliptically symmetric. This work was supported by the Air Force Office of Scientific Research under Contract F49620-82-K-0001. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Limiting properties of the occurrence/exposure rate and simple risk rate

In this paper, we study the asymptotic distributions of the functions of the occurrence/exposure rates of several groups of patients as well as Berry-Esseen bound on the distribution function of the occurrence/exposure rate. Asymptotic distributions of functions of the simple risk rates are also derived. The results are useful in not only medical research but also in the area of reliability.The work of the first two authors was supported by Contract F49620-C5-C-0008 of the Air Force Office of Scientific Research. The United States Government is authorized to reproduce and distribute reprints for governmental purposes.     

Efficient Regio‐ and Stereoselective Conversions of Oxiranes and Aziridines into β‐(Nitrooxy)‐Substituted Alcohols and Amines by Using Bismuth Nitrate

Oxiranes and aziridines efficiently undergo ring opening with bismuth nitrate at room temperature to furnish the corresponding β‐(nitrooxy)‐substituted alcohols and amines respectively. The conversions are highly regio‐ and stereoselective and afford the nitrooxy‐compounds in excellent yields within a short period of time.     

Facile one pot multicomponent synthesis of novel 4-(benzofuran-2-yl)-2-(3-(aryl/heteryl)-5-(aryl/heteryl)-4,5-dihydro-1H-pyrazol-1yl)thiazole derivatives

An efficient base catalyzed one pot multicomponent reaction of aryl/hetryl chalcones, thiosemicarbazide and 1-(benzofuran-2-yl)-2-bromoethan-1-one was developed to synthesize the novel 4-(benzofuran-2-yl)-2-(3-(aryl/heteryl)-5-(aryl/heteryl)-4,5-dihydro-1H-pyrazol-1yl)thiazole derivatives.