The codimension of degenerate pencils

Let d_n[d_n(r)] denote the codimension of the set of pairs of n×n Hermitian [really symmetric] matrices (A, B) for which det(λI?A?xB)=p(λ,x) is a reducible polynomial. We prove that d_n(r)?n?1, d_n?_n?1 (n odd), d_n?n (n even). We conjecture that the equality holds in all three inequalities. We prove this conjecture for n=2,3.     

Global principle for free-endpoint problems in optimal control and differential games

It is shown that the maximum principle in optimal control and the min-max principle in differential games are consequences of a certain global principle which is stated here.     

Matrices satisfying the van der Waerden conjecture

It is shown here that any n×n doubly stochastic matrix whose numerical range lies in the sector from -π/2n to π/2n satisfies the van der Waerden conjecture.     

Die Reaktion11B(d, α)9Be im Energiebereich von 0,7 bis 2,2 MeV

Excitation functions were measured from 0.7 to 2.2 MeV for the α₀ -, α₁-, α₂- and α₃-groups at an angle ofθ _L=90°. Two broad maxima were found in all four cases, suggesting the existance of resonances in the¹³C-compound nucleus. A problem, however, arises, as the first maximum appears at a bombarding energy of 1.15 MeV for the α₀- and α₁-groups, whilst it is shifted to 1.35 MeV in the excitation curves for the α₂- and α₃-groups. The second maximum appears at 1.85 MeV in all curves. Angular distributions were measured at 7 energies between 1.0 and 2.2 MeV for the α₀- and α₂-transitions. The coefficients of the first four Legendre polinomials, describing these distributions, are given.

     

Elastische Streuung von Deuteronen an10B bei niedrigen Energien

The excitation curve for the elastic scattering of deuterons from¹⁰B in the energy range from 1.0 to 2.0 MeV has been measured atθ _L =135°. Strong indications for compound nucleus resonances in¹²C at bombarding energies of approximately 1.0 and 1.9 MeV were observed. Angular distributions from 80 to 175° were determined in 200 keV steps from 1.0 to 2.0 MeV. Optical model analysis were made both for volume absorption and surface absorption potentials. Reasonable fits were obtained at energies up to 1.6 MeV with one set of parameters in both cases, whilst for the higher energies strong deviations were found with the same parameters.     

Das Titsgebäude von Siegeischen Modulgruppen vom Geschlecht 2

Zusammenfassung  We describe the Tits buildings of the Siegel modular groups and fort squarefree. These give the configuration of boundary components in the moduli spaces of abelian surfaces with polarization of type (1,t) and polarization of type (1,t) with canonical level structure. We also do the same calculation for the Siegel modular groups forp,q prime,p odd, corresponding to moduli of abelian surfaces with polarization of type (1,p) and full level-q structure. Colour pictures are available at: http ://www.bath.ac.uk/~masgks/Buildings Dieser Artikel beruht im wesentlichen auf der Hannoverschen Diplomarbeit des ersten Autors, die unter Begleitung von Prof. K. Hulek zustande gekommen ist. Die Verfasser danken dem DAAD und dem British Council für finanzielle Unterstützung im Rahmen des ARC-Projekts 313-ARC-XIII-99/45.     

Magneto infrared absorption in high electron density GaAs quantum wells

Magneto infrared absorption measurements have been performed in a highly doped GaAs quantum well which has been lifted off and bonded to a silicon substrate, in order to study the resonant polaron interaction. It is found that the pinning of the cyclotron energy occurs at an energy close to that of the transverse optical phonon of GaAs. This unexpected result is explained by a model taking into account the full dielectric constant of the quantum well.     

Images of the phonon propagation across twist-bonded crystals

We investigate identical but twist-bonded crystals using phonon imaging techniques. As in homogeneous crystals, very anisotropic flux patterns are observed. However, the shape of the pattern depends dramatically on the respective twist angle. The observed phonon images in wafer bonded GaAs/GaAs and Si/Si samples are essentially consistent with the predictions of the acoustic mismatch model for defect-free interfaces, with the exception of GaAs wafers twist bonded at a 45 degrees angle where modes with large shear stress are missing, which indicates strong dislocation scattering.     

Extremal eigenvalue problems for convex sets of symmetric matrices and operators

LetA ₁,...,A_n andK bem×m symmetric matrices withK positive definite. Denote byC the convex hull of {A ₁,...A_n}. Let {λ _p (KA)} ₁ ⁿ be then real eigenvalues ofKA arranged in decreasing order. We show that maxλ _p (KA) onC is attained for someA ^*=Σ _i ^{= 1/n} for which at mostp(p+1)/2 of α _i ^* do not vanish. We extend this result in several directions and consider applications to classes of integral equations. This paper is based mainly on the author’s doctoral dissertation written at the Technion—Israel Institute of Technology, March 1971, under the direction of Professor B. Schwarz. I wish to thank Professor Schwarz for his advice and encouragement. I am also grateful to Professor S. Karlin for supplying simplifications of several of my arguments. Some extensions discussed here are joint results of Karlin and the author.     

A characterization of normal operators

LetA be a bounded linear operator in a Hilbert space. IfA is normal then log ‖e ^Ai u‖ and log ‖e ^A·1 u‖ are convex functions for allu≠0. In this paper we prove that these properties characterize normal operators. Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.