On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

Quantum relativistic action at a distance

A well-known relativistic action at a distance interaction of two unequal masses is altered so as to yield purely Newtonian radial forces with fixed particle rest masses in the system center-of-momentum inertial frame. Although particle masses experience no kinematic mass increase in this frame, speeds are naturally restricted to less than the speed of light. We derive a relation between the center-of-momentum frame total Newtonian energy and the composite rest mass. In a new proper time quantum formalism, we obtain an L²(R⁴ R⁴, C) Hilbert space by varying individual particle rest masses. We propose the use of density operators, recognizing that the auxiliary proper time parameter is not an observable. The quantum formalism is applied to our altered version of the relativistic harmonic oscillator. Our generalized coherent states yield four-dimensional wave packets which follow the correct classical world lines. Appendices contain reviews of classical Hamiltonian reparametrization (incorporating our notion of manifest covariance), and a comparison of this work with the literature.     

Diastereoselective [4+2] annulation of vinyl carbodiimides with N-alkyl imines. Asymmetric synthetic access to the batzelladine alkaloids

A diastereoselective [4+2] annulation of vinyl carbodiimides with chiral N-alkyl imines has been developed to access the stereochemically rich tricyclic core of the batzelladine alkaloids. Its application to the asymmetric synthesis of batzelladine D permitted the use of long-range, directed hydrogenation and stereoselective intramolecular iodoamination as additional key steps to establish the remaining stereocenters within the natural product with excellent stereocontrol.     

Chemical Reactions in the Manufacture of Waveguides for Long Distance Optical Data Transmission

One of the modern high technologies which has advanced enormously in the last few years is glass fiber technology. This is used in the manufacture of glass fibers for lighting purposes and for the optical transfer of analog and digital data with a high transfer density. The technical demands made on the glass fibers required for data transfer, the optical waveguides, are extremely high and are already fulfilled to a large extent by industry. At present about four million kilometers of fiber, worth ca. 800 million DM are produced worldwide (10% in the Federal Republic of Germany). Numerous chemical processes take place during the manufacture of optical waveguides. However, in contrast both to the high and constantly growing demands on the quality and to the increasing production volume of such fibers, little is in fact known about the reactions involved. The present article will attempt to develop a picture of the multifarious reactions occurring in the course of this technical process on the basis of literature data and our own studies.     

CuAl2Cl8. Darstellung und Kristallstruktur

CuAl₂Cl₈. Preparation and Crystal Structure. Crystals of CuAl₂Cl₈ have been prepared. Their remission spectrum agrees with that of gaseous CuAl₂Cl₈. The crystal structure (P2₁/c;a = 6.614, b = 7.376, c = 12.319 Å; β = 94.11°) shows a molecular lattice with Cu in square planar coordination, which is completed to a stretched octahedron (as in solid CuCl₂) by two Cl atoms from neighbour molecules. Al has tetrahedral coordination.     

Zur Katalysierten Verflüchtigung von rotem Phosphor und von Claudetit

The Catalysed Volatilization of Red Phosphorus and of Claudetite The catalysis of the volatilization of P_red (as P₄) by AlCl_3,g, and of the volatilization of claudetite (as As₄O₆) by I_2,g is observed by mass spectrometry.