Approximate quasi-Newton methods

We consider the effect of approximation on performance of quasi-Newton methods for infinite dimensional problems. In particular we study methods in which the approximation is refined at each iterate. We show how the local convergence behavior of the quasi-Newton method in the infinite dimensional setting is affected by the refinement strategy. Applications to boundary value problems and integral equations are considered.The research of this author was supported by NSF grant DMS-8601139 and AFOSR grant AFOSR-ISSA-860074.     

The precessional frequency of a gyroscope in the quaternionic formulation of general relativity

The precessional frequency of a gyroscope in a reference frame that orbits about a gravitational body is compared between Einstein's tensor formulation of general relativity and the author's quaternion generalization—obtained from a factorization of the tensor form. The difference in predictions then suggests an experiment that could choose which of these formulations of general relativity is more valid in the analysis of gyroscopic motion.     

Newton's method for singular constrained optimization problems

We consider an iterative procedure where at each iteration a constrained quadratic optimization problem is solved. A Goldstein type step length rule is incorporated in order to assure global convergence. We consider a class of minimization problems which are singular at the optimal point and show that locally the superlinear convergence rate is retained for a certain part of the iterations. These results are applied to problems with perturbed data. In the last section global convergence results are proven.     

Ein isotropes Analogon zu einem Satz von Abramescu und einige Grenzwertformeln

Ohne ZusammenfassungHerrn Prof. Dr. W.Meyer-König zum 60. Geburtstag gewidmet     

Classical effects from a factorized spinor representation of Maxwell's equations

It was demonstrated in earlier work that the vector representation of electromagnetic theory can be factorized into a pair of two-component spinor field equations (Sachs & Schwebel, 1962). The latter is a generalization of the usual formalism, in the sense that in addition to predicting all of the effects that are implied by the vector theory, it predicts additional observable effects that are out of the domain of prediction of the Maxwell formalism. The latter extra predictions were derived in previous publications (Sachs & Schwebel, 1961, 1963; Sachs, 1968a, b). In this paper, the spinor formalism is applied to effects that are expected to agree with the predictions of the standard formalism—the Coulomb force between point charges and the measured speed of a charged particle which moves in an electric potential. While there are no vector or tensor variables involved in this formalism, the results are found, as expected, to be in agreement with the conventional representation of electromagnetic theory. The analysis serves the role of demonstrating that in the appropriate limiting case, the factorized spinor formulation of electromagnetism does predict the explicit classical effects that are also predicted by Maxwell's field equations. The paper also presents a derivation of the general form of the solutions of the spinor field equations.     

Causally continuous spacetimes

Causally continuous general relativistic spacetimes are defined and analyzed. In a causally continuous spacetime, the past and future of a local observer behave continuously under small perturbations of the metric or small changes in his location. Causally simple spacetimes are causally continuous; causally continuous spacetimes are causally stable. Possible reasons for taking causal continuity as a basic postulate in macrophysics are briefly discussed.     

Geradenkomplexe im Flaggenraum I

In this paper we consider complexes of lines in an isotropic space of degree two, i. e. a three dimensional real affine space with the metricds ²=dx ². Using the method of differential forms we study the local differential geometry of first order and the theory of complex curves. Finally we give some applications in the theory of linear complexes.