Electronic Mail: rosen@bc.edu

Phone

Associate Professor of Mathematics (retired)

Boston College, Chestnut Hill, MA 02467

Office: Maloney 521

Office Hours Fall 2015:

Monday 10-12; Wednesday 12-1; Friday 10-11 and by appointment anytime.

Math 2210-01 Linear Algebra

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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.

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.

Some Sharp Eleventh originals: Sea Breeze,
Nairobi
Incident, Sun Sparks,
Ring the
Bell

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 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 C^{1}. 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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**Bye.**