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,
Incident, Sun Sparks,
A few solo piano tracks: Autumn
Bless the Child, Over the
A vocal version of Ring the Bell
by Jim Nollman, who added the lyrics to the tune I wrote in 1969.
Department Home Page.
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 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 ...
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 the paper.