Numerical experiments with an inexact Jacobian trust-region algorithm

We present the adaptation and implementation of a composite-step trust region algorithm, developed in (Walther, SIAM J. Optim. 19(1):307–325, 2008), that incorporates the approximation of the Jacobian of the equality constraints with a specialized quasi-Newton method. The forming and/or factoring of the exact Jacobian in each optimization step is avoided. Hence, the presented approach is especially well suited for equality constrained optimization problems where the Jacobian of the constraints is dense.     

Donor-like defects in ZnO substrate materials and ZnO thin films

We have investigated donor-like defects in ZnO substrate material grown by three different methods, and in epitaxial ZnO thin films grown on sapphire by pulsed laser deposition. Temperature dependent Hall effect measurements yield information about dominant donors. The thermal activation energies lie in a wide range from about 20 meV to about 370 meV. Deep level transient spectroscopy is used to obtain parameters of deep donor-like defects. For that, a high-speed diode contact configuration was laid out for the epitaxial thin films in order to determine the defect parameters with high precision. The identified levels are E1, E3 and E4, though the level E4 is observed only in single crystals grown by seeded chemical vapor transport. PACS 72.10.Fk; 72.80.Ey; 73.50.-h     

Die Umkehrung des Satzes von Brauer-Berman-Witt über induzierte Charaktere endlicher Gruppen

Ohne Zusammenfassung     

On the Arf invariant in historical perspective, part?2

This is a continuation of our previous article (Lorenz and Roquette in Math. Semesterber. 57(1):73?C102, 2010) in this journal, where we discussed the paper of Cahit Arf (J. Reine Angew. Math. 183:148?C167, 1941). There he introduced what today is called the ??Arf invariant?? of a quadratic form over a field of characteristic?2. After our article had appeared we obtained some new information about the present state of the theory. When we say ??new?? then this means that this was new to us, i.e., we had not been aware before of the literature on the subject. (See our list of references.) It seems worthwhile to us to report here about it, as far as it is relevant for the assessment of Arf??s paper in its historical perspective.     

Shape evolutions under state constraints: A viability theorem

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite-dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space).In this morphological framework, the evolution of compact subsets of is described by means of flows along differential inclusions with bounded and Lipschitz continuous right-hand side. This approach is a generalization of using flows along bounded Lipschitz vector fields introduced in the so-called velocity method alias speed method in shape analysis.Now for each compact subset, more than just one differential inclusion is admitted for prescribing the future evolution (up to first order)—correspondingly to the step from ordinary differential equations to differential inclusions for vectors in the Euclidean space.We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in , but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally the viability condition is applied to constraints of nonempty intersection and inclusion, respectively, in regard to a fixed closed set .     

Modelling by a planar map: Lyapunov-type numbers in the presence of spiral points

Lyapunov-type numbers are usually defined for diffeomorphisms with a smooth invariant manifold. We consider here the case of a planar diffeomorphism with an invariant curve that contains spiral points. The limits defining the Lyapunov-type numbers are shown to exist. Numerical results for the delayed logistic map illustrate the analysis.     

Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1st-Order Geometric Evolutions

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ ^N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.      

Zur Schurgruppe eines lokalen Körpers

Ohne Zusammenfassung