Semiclassical collision theory. Multidimensional Bessel uniform approximation

A multidimensional Bessel uniform approximation for the semiclassical S matrix is derived for the case of four real stationary phase points. A formula is also developed for the particular case when four stationary phase points may be considered to be well separated in pairs. The latter equation is then used in the treatment of two real and two complex stationary phase points

The Stationary Phase Approximation, Time-Frequency Decomposition and Auditory Processing

The principle of stationary phase (PSP) is re-examined in the context of linear time-frequency (TF) decomposition using Gaussian, gammatone and gammachirp filters at uniform, logarithmic and cochlear spacings in frequency. This necessitates consideration of the use the PSP on non-asymptotic integrals and leads to the introduction of a test for phase rate dominance. Regions of the TF plane that pass the test and don't contain stationary phase points contribute little or nothing to the final output. Analysis values that lie in these regions can thus be set to zero, i.e. sparsity. In regions of the TF plane that fail the test or are in the vicinity of stationary phase points, synthesis is performed in the usual way. A new interpretation of the location parameters associated with the synthesis filters leads to: (i) a new method for locating stationary phase points in the TF plane; (ii) a test for phase rate dominance in that plane. Together this is a TF stationary phase approximation (TFSFA) for both analysis and synthesis. The stationary phase regions of several elementary signals are identified theoretically and examples of reconstruction given. An analysis of the TF phase rate characteristics for the case of two simultaneous tones predicts and quantifies a form of simultaneous masking similar to that which characterizes the auditory system.Comment: Submitted to IEEE Trans Signal Processing 14th Aug 201

Gravitational waves from inspiraling compact binaries: Validity of the stationary-phase approximation to the Fourier transform

We prove that the oft-used stationary-phase method gives a very accurate expression for the Fourier transform of the gravitational-wave signal produced by an inspiraling compact binary. We give three arguments. First, we analytically calculate the next-order correction to the stationary-phase approximation, and show that it is small. This calculation is essentially an application of the steepest-descent method to evaluate integrals. Second, we numerically compare the stationary-phase expression to the results obtained by Fast Fourier Transform. We show that the differences can be fully attributed to the windowing of the time series, and that they have nothing to do with an intrinsic failure of the stationary-phase method. And third, we show that these differences are negligible for the practical application of matched filtering.Comment: 8 pages, ReVTeX, 4 figure

Countercurrent chromatography in analytical chemistry (IUPAC technical report)

© 2009 IUPACCountercurrent chromatography (CCC) is a generic term covering all forms of liquid-liquid chromatography that use a support-free liquid stationary phase held in place by a simple centrifugal or complex centrifugal force field. Biphasic liquid systems are used with one liquid phase being the stationary phase and the other being the mobile phase. Although initiated almost 30 years ago, CCC lacked reliable columns. This is changing now, and the newly designed centrifuges appearing on the market make excellent CCC columns. This review focuses on the advantages of a liquid stationary phase and addresses the chromatographic theory of CCC. The main difference with classical liquid chromatography (LC) is the variable volume of the stationary phase. There are mainly two different ways to obtain a liquid stationary phase using centrifugal forces, the hydrostatic way and the hydrodynamic way. These two kinds of CCC columns are described and compared. The reported applications of CCC in analytical chemistry and comparison with other separation and enrichment methods show that the technique can be successfully used in the analysis of plants and other natural products, for the separation of biochemicals and pharmaceuticals, for the separation of alkaloids from medical herbs, in food analysis, etc. On the basis of the studies of the last two decades, recommendations are also given for the application of CCC in trace inorganic analysis and in radioanalytical chemistry

The use of the stationary phase method as a mathematical tool to determine the path of optical beams

We use the stationary phase method to determine the path of optical beams which propagate through a dielectric block. In the presence of partial internal reflection, we recover the geometrical result obtained by using the Snell law. For total internal reflection, the stationary phase method overreaches the Snell law predicting the Goos-Haenchen shift.Comment: 11 pages, 2 figure