A discrete mathematical model of unlabelled granulocyte kinetics. A preliminary study of feedback control.

The equations used in formulating the continuous model of granulocyte kinetics developed by O'Fallon et al. (1971) were analyzed to see if they could be altered to simulate a feedback mechanism operating on the production and development of granulocytes. After extensive study and modification of the continuous model, it was found that a discrete model based on a Leslie matrix procedure was more effective for simulating the feedback system. This discrete model was used to show experimentally, from a mathematical view point, that a feedback mechanism of some kind must be operating on the production and development of granulocytes. Further, the discrete model was subjected to preliminary tests (simultaneous and cascading feedback) to demonstrate that it has the capability of responding to feedback control.

A note on life tables and nonlinear death processes.

This note is viewing survival data of a natural cohort as being generated by a possibly nonlinear, nonhomogeneous death process. It proves that the usual conditional distributions of the number of survivors at a certain age are binomial if and only if the death process is linear. Thus the customary statistical methods for the analysis of life table data are, strictly speaking, invalid whenever the underlying death process is nonlinear. For example, if a contagious disease is the cause of some or all of the deaths, the deaths will not be independent and the death process, not linear. One should then base the statistical analysis on a model for the spread of the disease rather than the routine binomial model.

Conditions under which a number of sinusoids may be instantaneously in phase.

The literature on the phase relationships between frequency components of a Fourier analysis is reviewed, with examples and theories from acoustics and neurophysiology. Given n sinusoids of different frequencies omega1, omega2, .., omegan and phase angles phi1, phi2, .., phin, it is shown that for n greater than or equal to 2 the set of initial phase angles allowing the n sinusoids to be in phase at some time t0 consists of one or more planes of constant dimension 2 and that for n = 2 such a time t0 always exists. The conditions under which the common phase of n sinusoids at one time t0 will be the same as the common phase at another time t0 are also investigated. The importance of incommensurately related frequency components is emphasized by proofs which do not depend on harmonic relationships. Proofs are formulated in a linear algebra format to demonstrate the versatility of the method for analysing long sequences of frequencies and phases.