Some remarks on invariant subspaces in real Banach spaces (revised version)

It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators on a real Hilbert space

Parametric frequency mixing in the magneto-elastically driven FMR-oscillator

We demonstrate the nonlinear frequency conversion of ferromagnetic resonance (FMR) frequency by optically excited elastic waves in a thin metallic film on dielectric substrates. Time-resolved probing of the magnetization directly witnesses magneto-elastically driven second harmonic generation, sum- and difference frequency mixing from two distinct frequencies, as well as parametric downconversion of each individual drive frequency. Starting from the Landau-Lifshitz-Gilbert equations, we derive an analytical equation of an elastically driven nonlinear parametric oscillator and show that frequency mixing is dominated by the parametric modulation of FMR frequency

The spall strength limit of matter at ultrahigh strain rates induced by laser shock waves

New results concerning the process of dynamic fracture of materials (spallation) by laser-induced shock waves are presented. The Nd-glass laser installations SIRIUS and KAMERTON were used for generation of shock waves with pressure up to 1 Mbar in plane Al alloy targets. The wavelengths of laser radiation were 1.06 and 0.53 μm, the target thickness was changed from 180 to 460 μm, and the laser radiation was focused in a spot with a 1-mm diameter on the surface of AMg6M aluminum alloy targets. Experimental results were compared to predictions of a numerical code which employed a real semiempirical wide-range equation of state. Strain rates in experiments were changed from 106 to 5 × 107 s−1. Two regimes of spallation were evidenced: the already known dynamic regime and a new quasi-stationary regime. An ultimate dynamic strength of 80 kbar was measured. Finally, experiments on targets with artificial spall layers were performed showing material hardening due to shock-wave compression

PROBLEMS OF MEASUREMENT OF DENSE PLASMA HEATING IN LASER SHOCK-WAVE COMPRESSION

Experimental results of heating measurements of matter carried out in a study of laser-driven shock waves in aluminum (Batani et al. 1997) are discussed. The measured temporal evolution of the "color" temperature of the rear surface of the target is compared with that computed by a numerical code. It has been established that the target preheating can substantially affect optical signal features recorded from the rear side of the target, and consequently influence upon the accuracy of temperature and pressure measurements of the material behind the shock wave front