Invariant scale matrix hypothesis tests under elliptical symmetry

The usual assumption in multivariate hypothesis testing is that the sample consists of n independent, identically distributed Gaussian m-vectors. In this paper this assumption is weakened by considering a class of distributions for which the vector observations are not necessarily either Gaussian or independent. This class contains the elliptically symmetric laws with densities of the form f(X(n × m)) = ψ[tr(X ? M)′ (X ? M)Σ^?1]. For testing the equality of k scale matrices and for the sphericity hypothesis it is shown, by using the structure of the underlying distribution rather than any specific form of the density, that the usual invariant normal-theory tests are exactly robust, for both the null and non-null cases, under this wider class.     

Alkylation and silylation of directed enolates from diethylketene and organometallic reagents

Diethylketene and organometallic reagents form enolates capturable by methyl iodide or trimethylsilyl chloride with greater than 95% positional selectivity.     

A facile synthesis of α-chloroboronic esters via the base induced reaction of α,α-dichloromethyl methyl ether with hindered borinic esters

Treatment of hindered borinic esters, R₂ BOR′; with an equimolar quantity of α,α-dichloromethyl methyl ether and lithium triethylcarboxide yields the α-chloroboronic esters, R₂ CHClB(OCH₃)OR′, in excellent yields. In cases where the steric requirements of R are not sufficient, the steric requirements of R′ can be increased to achieve the synthesis.     

Two-dimensional convexities are join-hull commutative

It is shown that two-dimensional convex structures with certain natural properties are join-hull commutative. The main intermediate step is the computation of the so-called exchange number. We also give two examples of three-dimensional convexities which are not join-hull commutative. The second one has certain additional properties showing that the main theorem is sharp in many other respects. These properties are obtained from a study of convex hyperspaces.     

Padé-type approximants with orthogonal generating polynomials

The interpolation of the function x → 1/(1 ? xt) generating the series f(t) = ∑^∞_{i = 0}c_itⁱ at the zeros of an orthogonal polynomial with respect to a distribution d α satisfying some conditions will give us a process for accelerating the convergence of f_n(t) = ∑ⁿ_{i = 0}c_itⁱ. Then, we shall see that the polynomial of best approximation of x → 1/(1 ? xt) over some interval or its development in Chebyshev polynomials T_n or U_n are only particular cases of the main theorem.At last, we shall show that all these processes accelerate linear combinations with positive coefficients of totally monotonic and oscillating sequences.