Differential delay

The external geometric differential delay (see below) of an off axis source is exactly balanced within a Fizeau interferometer, resulting in fringes with the same phase on top of each source in the field. The position of a source may differ from the position of zero OPD in a Michelson interferometer depending on how dissimilar entrance and exit pupils are. The fringe contrast of off-axis sources also depend on the temporal degree of coherence of the detected light. 

Dominant female mice in groups do not release the delay substance in diestrus the urine of dominant females has no effect on female puberty relative to water-treated control mice, while urine from dominant females in estrus accelerates puberty in test females (Drickamer, in press a). Urine from subordinate females in diestrus delays puberty, but urine collected from subordinate females within groups that are in estrus does not accelerate or delay puberty (Drickamer, in press a). There are no differential delays in puberty for mice that are related or unrelated to the donor females (Drickamer, 1984b). 

As suggested by Fig. 25.4, the echoes from a point source will have a spherical wavefront. The center elements of the array will receive these echoes at first while the outer elements will receive them last. To compensate for this and to achieve constructive interference at the summer, the center elements will be given the longest delays, as suggested by the length dimension of the delay lines. The calculation to determine the differential delays among the received echoes is straightforward. 

The equations of interest are differential delay or differential difference equations (DDEs), equations in which the time derivatives of a function depend not simply on the current value of the independent variable t, but on one or more earlier values t — t — X2, as well. In this chapter, we deal primarily with problems involving a single delay, since they are the most common and the most tractable mathematically. As our first example, we consider the prototype equation... 

Equation (10.1) is perhaps the simplest nontrivial differential delay equation. It is the analog of the rate equation that describes unimolecular decay. As we shall see, eq. (10.1) offers a far richer range of behavior. 

Since most readers will be unfamiliar with the mathematics of DDEs, we devote this section and the next to some fundamental notions of how to solve and assess the stability of solutions of such systems. Readers who are already conversant with this material may safely skip to section 10.3. Those who seek more detail should consult any of several excellent works on the subject. Macdonald (1989) is probably the most accessible for the reader without extensive mathematical background. The monograph by Bellman and Cooke (1963), while more mathematically oriented, contains a number of useful results. Hale s (1979) review article and several sections of Murray s (1993) excellent treatise on mathematical biology also contain readable treatments of differential delay equations. 

A very valuable technique, useful in the solution of ordinary and partial differential equations as well as differential delay equations, is the use of Laplace transforms. Laplace transforms (Churchill, 1972), though less familiar and somewhat more difficult to invert than their cousins, Fourier transforms, are broadly applicable and often enable us to convert differential equations to algebraic equations. For rate equations based on mass action kinetics, taking the Laplace transform affords sets of polynomial algebraic equations. For DDEs, we obtain transcendental equations. 

Epstein, I. R. 1989. Differential Delay Equations in Chemical Kinetics Some Simple Linear Model Systems, J. Chem. Phys. 92, 1702-1712. 

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