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In this paper we investigate the question: When does the direct product of partial orders each satisfying the normalized matching condition also satisfy it? A proof of Harper's result that a sufficient condition for this is that factor partial orders have rank populations that are multiplicatively convex, is presented. A general necessary and sufficient condition is described, and conditions which occur when one factor is a chain are obtained. In particular we show that under these circumstances it is necessary that the product order have multiplicatively convex population rank.

We investigate the proportion of graphs on n-vertices each of which is m-valent, that are "x-reducible". An asymptotic formula is developed for large nand fixed values of the other variables, in terms of an n-independent parameter that is evaluated for a number of (m, k) values.

This work investigates the high-accuracy phase discriminator in 1-bit software-defined receiver (SDR) for three signal-to-noise ratio (SNR) ranges: low SNR, high SNR and moderate SNR.

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