A precise and wide-dynamic-range displacement-measuring homodyne quadrature laser interferometer

We present a fast, displacement-measuring, single-pass, two-detector homodyne quadrature laser interferometer and compare its performance with an arm-compensated, proportional, integral-derivative-controlled Michelson interferometer. Special attention is given to the extension of the dynamic range. The wide dynamic range is achieved by an accurate fringe subdivision based on an enhanced ellipse-specific fitting of the scattered Lissajous curve and by increasing the total displacement using the quadrature-detection technique. The common periodic deviations, i.e., the unequal AC amplitudes, the DC offsets, and the lack of quadrature are determined and reduced by data processing based on an ellipse-specific, least-squares fitting to obtain nanometric accuracy. The performance of the described interferometer is demonstrated through the measurement of high-amplitude and high-frequency laser-induced ultrasound.     

Dilatation monotonicity and convex order

We study, for functions and sets, the relation between law invariance, preserving the convex or uniform order, and dilatation monotonicity based on duality arguments.     

On the optimality of the Arf invariant formula for graph polynomials

We prove optimality of the Arf invariant formula for the generating function of even subgraphs, or, equivalently, the Ising partition function, of a graph.