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By imbedding a given complex physical process within an appropriate class of processes and expressing the functional relationships among the members of the class, it is possible to obtain insights into the structure of the original process which would not be possible by considering this process alone. Not only may analytic expressions be obtained, but frequently computational tools are forged which make possible the exploitation of modern digital computing machines. By way of illustration, this paper is devoted to a discussion of the functional equation techniques of dynamic programming and invariant imbedding in the study of some problems arising in the theory of adaptive control processes and in the transmission of signals through random media. Still other applications which have been made are briefly sketched.