Phone:
617-552-3771I was born in Brooklyn, raised in Connecticut, from which my legacy
is my high
school Alma Mater which is surely the most often performed
of
my musical compositions. I was educated at Tufts (BS 1972) and
University of
Michigan (MA, PhD in Math 1980). My first full-time job was at Union
College in Schenectady, NY, for two years, and then I came to Boston
College in 1981, where I have been ever since. I began as a set
theorist,
studying ultrafilters on countable sets, spent some time in computer
graphics and dynamical systems, and lately have turned to mathematics
education.
In my leisure time, I spend a good deal of time at our cottage on Crescent Lake in Raymond, Maine. Over the past few years, we've done some remodeling up there, beginning with a new deck and kitchen (here's our construction of the sink peninsula), followed by a contracted project to put in a couple of solar tubes and a dormer to hold a new upstairs bathroom (the new roof and septic system don't make very exciting pictures.)
Extra-curricula activities:
Bridge: The greatest card game. Check this out for an account of
my favorite hand of the past few years.
Golf: Here
is where I play most often.
Windsurfing: I'm not very good, but that
doesn't stop me from doing it, or even from teaching it.
Piano: I play jazz and pop, here is the
Charlie Parker tune Confirmation
played by Sharp 11th, a jazz trio of which I am one third.
Sharp Eleventh web page
(with
some other tunes to listen to). And some solo piano tracks:
Autumn
Leaves, God
Bless the Child, Golden
Lady, Over the
Rainbow, Bonnie and
Clyde, Ring the Bell,
As Time Goes By
A vocal version of Ring the Bell
by Jim Nollman, who added the lyrics to the tune I wrote in 1969.
Research Description: The last few years have been spent pursuing results about pulse-coupled oscillators with my colleague Rennie Mirollo. Let G be a connected graph, each of whose nodes is a uniform oscillator with period one. When a node reaches the origin, it "fires" a message which is received by the nodes to which it is connected in G. Each such node instantly jumps from its current phase x to phase f(x), where f is a continuous, non-decreasing pulse response function for the system (identical for all the oscilllators). The fundamental question for the system (G,f) is global synchronization: is it the case that for almost all initial conditions, the sytem reaches the synchronous state in which all the oscillators fire together? Our focus has been on properties of the function f vis a vis synchronization of various classes of graphs (e.g. rings, chains, all connected graphs), for example we show that a any function which synchronizes all connected graphs cannot be C1. We had the pleasure of speaking about these results at the SIAM conference August 2000 on Maui.
Our previous collaboration concerned single wave form solutions of
the Josephson Junction equations; here is a copy of the paper.
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On a completely different note, a simple problem in a secondary math
class I was observing got me thinking about expected waiting times for
strings of outcomes in a sequence of coin flips, or slightly more
generally, in a sequence of identical, independent trials of some
experiment with finitely many outcomes. It turns out that if the
experiment has exactly L
outcomes of non-zero probability, and you randomly pick a string s of outcomes of length n by performing the experiment n times, and then start counting
trials until the sequence s
occurs again, then the expected waiting time E(n) for s to occur again is L^n
+ n -1, independent of the
individual probabilities of the L
outcomes. Here's my write-up of the
proof, which I
subsequently discovered is not new-- N. Johnson showed the same thing
by the same argument in 1968.
A funny consequence is the following: play the game with an ordinary
coin and take say, n = 3,
that is, flip the coin 3 times to obtain string s and then keep flipping, counting
flips until you get the same string s
on three consecutive flips. The expected length of this game is
2^3 +3 - 1 = 10 if
your coin flip has only two possible outcomes. But a real coin has some
very small, but non-zero, chance of landing on its edge, so in fact,
for real coin flips
there are L = 3 outcomes and
the expected length of this game is 3^3 + 3 -1 = 29, even though, for
all practical purposes, the two games are the same. Perhaps the median
is a better measure . . .
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Bye.
