Consistency of a system of equations: What does that mean?

The concept of (structural) consistency also called structural solvability is an important basic tool for analyzing the structure of systems of equations. Our aim is to provide a sound and practically relevant meaning to this concept. The implications of consistency are expressed in terms of explicit density and stability results. We also illustrate, by typical examples, the limitations of the concept

Non-resonant background suppression in preresonance CARS spectra of flavin adenine dinucleotide: Demonstration of a background suppression technique using phase mismatching and comparison with the polarization-sensitive CARS technique

Polarization-sensitive CARS spectra of a 5.7 × 10-3 mol dm-3 flavin adenine dinucleotide (FAD) solution were recorded under preresonance conditions at a pump wavelength of 532 nm. The depolarization ratios of the vibrations are shown to be close to the depolarization ratio of the non-resonant background. This results in a severe reduction of the vibration resonant signal (a factor of 700-900) in the polarization CARS spectrum, and a poor improvement in the ratio of the resonant signal and the non-resonant background (<10). \ud In this context, a non-resonant background suppression technique is discussed and demonstrated for 5.7 × 10-3 and 1.4 × 10-3 mol dm-3 FAD solutions excited at 532 nm; the non-resonant susceptibility of the walls of the cuvette, which contains the FAD solution, is used to compensate the non-resonant signal contribution of the solution. An improvement in the signal-to-noise ratio of ca. 50 is achieved at the cost of a factor of 30 in the resonant signal strength. Lorentzian-shaped spectral bands are obtained, facilitating the determination of band position, width and intensity. Line shape parameters and depolarization ratios for FAD are extracted from the presented spectra by curve fitting. The signal strength and background suppression achieved with these techniques and the resonance CARS technique (at a pump wavelength of 480 nm) are compared and discussed