Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities

We construct six unitary trace invariants for 2×2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We have discovered a curious trace identity for two unit-speed one-parameter subgroups of Sp(1). A modification gives an infinite family of trace identities for quaternions as well as for 2×2 complex matrices. We were not able to locate these identities in the literature. We prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field. Finally we consider the problem of describing the semi-algebraic set of pairs (X,Y) of quaternionic n×n matrices which are simultaneously triangularizable. Even the case n=2, which we analyze in more detail, remains unsolved.     

Das Atomgewicht des Cadmiums

Ohne Zusammenfassung     

Ueber die Destillation von Formaldehyd

Ohne Zusammenfassung     

Halogen

Ohne Zusammenfassung     

Zur Bestimmung des specifischen Gewichtes von D?mpfen und permanenten Gasen

Ohne Zusammenfassung     

Laser-induced fluorescence of seeded nitric oxide as a flame thermometer

Received: 10 June 1997/Revised version: 25 November 1997     

Fracture energy of automotive clearcoats—I. Experimental methods and mechanics

The fracture energy of five automotive clearcoats was measured using fracture mechanics methods originally developed for measuring the fracture energy of brittle inorganic thin films. The fracture energy of the clearcoats ranged from 12 J m⁻² to 140 J m⁻². A discrepancy between the actual fracture energy and the ‘apparent’ fracture energy was observed and attributed to differences in the tensile stress-strain behavior of the materials, in particular the yield strength. Correcting for differences in their tensile stress-strain behavior, the apparent fracture energy of the films ranged from 13 J m⁻² to 330 J m⁻². Contrary to expectations, the fracture energy of the clearcoats did not decrease with increasing cross-link density. This was ascribed to differences in chemical architecture and additives in these commercial materials. Measuring the toughness in this manner shows promise for evaluating the long-term cracking resistance of automotive clearcoats.