I 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.
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.
Some Sharp Eleventh originals: Sea Breeze,
A few solo piano tracks: Autumn Leaves,
God Bless the
Child, Over the Rainbow,
Bonnie and Clyde
A vocal version of Ring
the Bell by Jim Nollman, who added the lyrics to the tune I
wrote in 1969.
Research Description: Define the function C(n) for n > 0 by C(1) = 0, C(2n) = 2 C(n), C(2n+1) = 1 + C(n) + C(n+1). C(n) counts the number of unbalanced nodes in the most balanced full binary tree you can build with n leaves. Then C(n) is 0 when n is a power of 2, and we define the oddness of a positive integer n by D(n) = C(n) / n. It is easy to check that D(n) < 1/2 and D(2n ) = D(n) for all n. It follows that D has a sort of periodicity: it is the same on each interval between consecutive powers of 2, except that each successive interval has twice as many values of D as the previous one. So by rescaling each such interval down to [0,1], we get a universal oddness function f defined on binary rationals by f( J / 2^K ) = D( 2^K + J ). It turns out that f is continuous on binary rationals and hence can be extended to a continuous function on [0,1], with many interesting properties. For example, it has a local minimum at every binary rational, it has a unique global maximum value of 1/2 at x = 1/3, and f (q) is rational for all rational q. The closely related function h(x) = f (x)(1+x), which extends the local oddness function mapping J / 2^K to C(2^K + J ) / 2^K, has many similar properties as well as being easier to understand to due its graph having a straightforward symmetry and fractal structure. For example, the local maxima of h occur precisely at all reals in [0,1] whose base 4 expansion contains an equal (finite) number of 0's and 3's.
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. I subsequently
discovered that my proof was
not new-- N. Johnson showed the same thing by the same argument in
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 ...
Back in the early part of the century, I pursued 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