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Necessary and Sufficient Conditions for the Two-Point Phase Probability Function of Two-Phase Random Media

John A. Quintanilla
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 464, No. 2095 (Jul. 8, 2008), pp. 1761-1779
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/20209497
Page Count: 19
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Necessary and Sufficient Conditions for the Two-Point Phase Probability Function of Two-Phase Random Media
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Abstract

Constructing realizations of random media with a specified two-point phase probability function S₂ has attracted considerable attention in the recent literature. However, little is known about conditions under which a prescribed S₂ is realizable. The only known necessary and sufficient condition, due to McMillan, involves a class of square matrices, called corner-positive matrices, about which almost nothing is known except their definition. As a result, McMillan's theorem has gone mostly unused in the literature for over 50 years. In this paper, we present a general decomposition formula for corner-positive matrices, which allows McMillan's theorem to be written in a significantly more tractable and testable form. We also connect McMillan's theorem with many known but heretofore unrelated necessary conditions on S₂, extending many of these conditions.

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