David G. Cantor

On non-blocking switching networks

A switching network may be informally described as a collection of single-pole, single-throw switches arranged so as to connect a set of terminals called inputs to another set of terminals called outputs. It is non-blocking if, given any set of connections from some of the inputs to some of the outputs, and an idle input terminal x and idle output terminal y, then it is possible to connect x to y without disturbing any of the existing connections. Denote by σ(a, b) the minimal number of switches necessary to connect a inputs to b outputs using a non-blocking network. We are interested in studying the growth of σ(a, a) as a ∞. Results of C. Clos show that σ(a, a) ⩽ C ae2√log a·log 2. We show that σ(a, a) ⩽ 8a(log2a)2.

Sequences of integers with missing differences

Abstract This paper deals with the following problem posed by Professor T. S. Motzkin: Suppose M is a given set of positive integers. How dense can a set S of positive integers be, if no two elements of S are allowed to differ by an element of M? The problem is solved for |M| ⩽ 2, and some partial results are obtained in the general case.

The dead space in a compartmental lung model.

A compartmental lung model with any number of synchronously filling and emptying functional chambers and a common dead space or conducting region is considered. It is shown that the model gives rise to an output, in an open circuit washout determination, which is a weighted sum of exponentials. From estimates of these weights and exponential components, estimates of the model parameters can be recovered. Relations giving the unique correspondence between the output parameters and the model parameters are derived and the existence and uniqueness of solutions established.

On direction-preserving maps of graphs*

Abstract We study direction-preserving maps from the vertices of a directed graph α to the edges of a directed graph α .

A correction for the effect of the dead space in pulmonary gas washout tests

Under the assumption that there is complete mixing in the dead space a series of equations are given which give the values for a washout test of the functional component of the pulmonary bellows (without a dead space) in terms of the washout values of the bellows with a dead space.