Normal subgroups of are not finitely generated

As a generalization of Wedderburn's classic theorem, it is shown that the multiplicative group of a noncommutative finite dimensional division algebra cannot be finitely generated. Also, the following conjecture is investigated: An infinite non-central normal subgroup of cannot be finitely generated.

     

Vibrational structure of quasiline fluorescence and absorption spectra of bisanthene molecules

The quasiline spectra of the fluorescence and absorption of bisanthene in n-hexane are obtained at 77 and 4.2 K, and their vibration analysis is carried out. Substantial violation of the mirror symmetry of the fluorescence and S₁ ← S₀-absorption bands with respect to both the frequencies and intensities is found. This allowed the conclusion that the S₂-level is localized in the region of vibronic sublevels of the S₁-state with . Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 67, No. 4, pp. 488–491, July–August, 2000.     

The use of the tetrachoric series for evaluating multivariate normal probabilities

The tetrachoric series is a technique for evaluating multivariate normal probabilities frequently cited in the statistical literature. In this paper we have examined the convergence properties of the tetrachoric series and have established the following. For orthant probabilities, the tetrachoric series converges if $\text{|;?}{}_{\mathrm{ij}}\text{|; <}\text{1}\text{(k ? 1)}$, 1 ≤ i < j ≤ k, where ?_ij are the correlation coefficients of a k-variate normal distribution. The tetrachoric series for orthant probabilities diverges whenever k is even and $\text{?}{}_{\mathrm{ij}}\text{>}\text{1}\text{(k ? 1)}$ or k is odd and $\text{?}{}_{\mathrm{ij}}\text{>}\text{1}\text{(k ? 2)}$, 1 ≤ i < j ≤ k. Other specific results concerning the convergence or divergence of this series are also given. The principal point is that the assertion that the tetrachoric series converges for all k ≥ 2 and all ?_ij such that the correlation matrix is positive definite is false.     

Limit distributions for risk processes in case of claim amounts of finite expectation

We prove limit theorems for the distribution of $\text{Z}{}_{\mathrm{l}}\text{if {Z}{}_{\mathrm{l}}\text{:l ?}\text{R}{}_{\mathrm{+}}\text{}}$ is a risk process with claim amounts of finite mean. The results are illustrated by several examples and counterexamples.     

Normal families of holomorphic functions and mappings on a Banach space

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented.     

Well-posed optimization problems for integral functionals

A necessary and sufficient condition for the correctness of minimum problems defined by integral functionals over pointwise constrained summable functions is that the integrand defines a well-posed minimum problem over the constraint region.This work was supported by the Laboratorio per la Matematica Applicata, Consiglio Nazionale delle Ricerche (CNR).     

Some first passage problems related to cusum procedures

Let $\text{X}{}_1\text{,}\text{X}{}_2\text{,…}$ be independent random variables, and set W_n = max(0,W_n-1 + X_n), W₀ = 0, n ? 1. The so-called cusum (cumulative sum) procedure uses the first passage time T(h) = inf{n ? 1: W_n?h}for detecting changes in the mean μ of the process. It is shown that lim_h→∞ μET(h)/h = 1 if μ > 0. Also, a cusum procedure for detecting changes in the normal mean is derived when the variance is unknown. An asymptotic approximation to the average run length is given.     

Planar semi-martingales

The concept of a semi-martingale is extended to processes with index set in the plane. The definitions of planar semi-martingales are similar to those of two parameter bounded variation. Necessary and sufficient conditions for a Doob-Meyer decomposition are obtained, and a maximal inequality and almost everywhere convergence theorem is given for planar semi-martingales in LlogL.     

Products of roots of the identity

It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five -th roots of the identity for every 2$">. For invertible normal operators four factors suffice in general.