Surface Quadrature

The first BioCD took its inspiration from the compact disc. The compact disc was invented in 1970 by Claus Campaan of Phillips Laboratory. The concept is purely digital and uses null interferometers that are far from quadrature, as appropriate for the readout of two binary intensity states. The interferometers were common-path and stable, as required for the mechanical environment of portable compact disc readers. The original BioCD used the same physics as the compact disc, but modified the on-disc microstructures to change from the digital readout to an analog readout that operated in quadrature for sensitive detection of surface-bound proteins7,8. Because the quadrature condition is established by diffraction off of microstructures on the disc, this is called the microdiffraction-class (MD-Class) of BioCD. 

A drawback of the MD-class BioCD is the microfabrication required to pattern gold spokes on the disc to set the quadrature condition. To remove the microfabrication, an alternative means to establish a quadrature condition uses adaptive optical mixers in the far field to establish and lock phase quadrature. In this case, the disc can be a high-reflectance antinode surface with protein patterned directly on the disc surface without any need for surface structuring. As the disc spins, the immobilized protein causes phase modulation that is detected in the quadrature condition set up by the adaptive mixer. 

It is very practical to work with the depth as an independent variable, for, if we need it, we can find the time of rise from the solution of the equations by quadrature. The pressure is given by the gas law and is, in fact, equal to either side of the differential equation for . The depth was calculated from the point at which the extrapolated hydrostatic pressure is zero, so that if P is the pressure at the surface, the equations must be integrated until... 

To calculate the surface integral in (12.48) accurately by a high order quadrature, the integrand / must be known at a large number of points on the surface A. This information is generally not available, rather only the... 

An alternative closed Newton-Cotes quadrature formula of second order can be obtained by a polynomial of degree 1 which passes through the end points. This quadrature formula is called the trapezoid rule. In 2D this surface integral approximation requires the integrand values at the GCV corners. 

The quadratic interpolation has a third order truncation error on both uniform and non-uniform grids [114, 49]. However, when this interpolation scheme is used in conjunction with the midpoint rule approximation of the surface integral, the overall approximation is still of second order accuracy (i.e., the accuracy of the quadrature approximation). Although the QUICK approximation is slightly more accurate than CDS, both schemes converge asymptotically in a second order manner and the difference are rarely large [49]. 

But mi is usually not zero when the internal coordinate represents particle mass, surface area, size, etc. In these cases the PD algorithm can be safely used. The case of null mi occurs more often when the internal coordinate is a particle velocity that, ranging from negative to positive real values, can result in distributions with zero mean velocity. Another frequent case in which the mean is null is when central moments (moments translated with respect to the mean of the distribution) are used to build the quadrature approximation. These cases will be discussed later on, when describing the algorithms for building multivariate quadratures. 

The process of finding the area of any surface is called, in the regular text-books, the quadrature of surfaces, from the fact that the area is measured in terms of a square—sq. cm., sq. in., or whatever unit is employed. In applying these principles to specific examples, the student should draw his own diagrams. If the area bounded by a portion of an ellipse or of an hyperbola is to be determined, first sketch the curve, and carefully note the limits of the integral. 

A surface of revolution is a surface generated by the rotation of a line about a fixed axis, called the axis of revolution. The quadrature of surfaces of revolution is sometimes styled the complanation of surfaces. Let the curve APQ (Fig. 114) generate a surface of revolution as it rotates about the fixed axis Ox. It is required to find the area of this surface. 

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