Student Theses and Dissertations

Date of Award

1967

Document Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Thesis Advisor

Mark Kac

Keywords

nonlinear systems, difference-differential equations, prediction theory, semistochastic matrices, learning dynamics, entropy

Abstract

We introduce several systems of nonlinear difference-differential equations and prove oscillation and global ratio limit theorems for some of them. These systems can be interpreted as a learning theory, or alternatively as a nonstationary prediction theory whose goal is to discuss the prediction of individual events, in a fixed order, and at prescribed times. They can also be interpreted as cross-correlated flows on networks, or as deformations of a probabilistic graph. Each system possesses an underlying geometry characterized by a semistochastic matrix, and we study the effect of this geometry on the system's limiting behavior as t→∞. We also investigate the effects which the ratios of solutions of our systems have on the outputs of each system. We show that the average output of each system is not a good index of the mechanism which characterizes its interactions, especially when this average is computed over long time intervals. In particular, the average output is linear whereas the interactions are nonlinear. A system is discussed whose interactions are always locally reversible but whose global interactions are irreversible or not depending on the inputs received by the system. We also find systems whose entropy decreases monotonically in time and connect this phenomenon with the process of learning in these systems.

Comments

A thesis presented to the faculty of The Rockefeller University in partial fulfillment of the requirements for the degree of Doctor of Philosophy

License and Reuse Information

Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

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