It is well known that mammals and other animals cycle between periods of sleep and wakefulness. Sleep and wake states are each maintained by activity in a corresponding network of neurons in the brain, with mutually inhibitory connections between the networks. In infants, the durations of both sleep and wake states follow an exponential distribution, but in adults the wake states have a heavy-tailed distribution. Is it the altered network structure or a change in neuronal dynamics that drives

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... is transformation during development? We use mathematical models and random graph theory to explore this issue and the mechanisms of transition between sleep and wake states. Abstract: Archimedes' use of Eudoxos' method of exhaustion to determine the area bounded by a parabolic arc and a line segment was a crowning achievement in Greek mathematics. The promise of the method, so apparent to us now, seems to have died with Archimedes, only to rise again in different form some 1900 years later with the modern calculus. Archimedes' result though is not just about computing an area. It is about comparing a parabolic area with a related triangular area. That is, there is a geometric content in the comparison that is interesting in its own right. In this talk we would like to make the case that Archimedes' area comparisons deserve more attention, not so much because of his methods, but rather because of the interesting geometric content of the comparisons and the new questions they suggests. We feel that there are more results to be had, and present a few here with some speculation on further research directions. December 6, 2013: Stefaan Delcroix (CSUF)