NOVEMBER 10: Homeira Pajoohesh, CCNY, "Topological and categorical
properties of binary trees", At Queens College; tea 3:15 Kiely 508, talk
4:00 Kiely 281. For parking contact Jerry Itzkowitz, (718)380-1832, gitzkowitz@prodigy.net.
ABSTRACT:
Binary trees are very useful tools in computer science for
estimating the running time of comparison based algorithms, that is,
algorithms in which every action is ultimately based on a prior
comparison between two elements. For two given algorithms A and B,
where the decision tree of A is more balanced than that of B, it is
known that the average and worst case times of A will be better than
those of B. Thus the most balanced and the most imbalanced binary
trees play a major role. Here we consider them as semilattices and
characterize the most balanced and the most imbalanced binary trees
by topological and categorical properties. Also we define the
composition of binary trees as a commutative binary operation, *,
such that for binary trees A and B, A*B is the binary tree obtained
by attaching a copy of B to each leaf of A. We show that for the
collection T of binary trees, (T,*) is a commutative po-monoid and
investigate its properties.
NOVEMBER 17: Michael Mislove, Tulane University,
"Models for Probability and Nondeterminism". At CUNY Graduate Center Computer
Science Colloquium. Talk 4:15-5:30 in 9206/9207, wine and cheese reception to
follow in 4102. More information about this and other such colloquim meetings is at:
http://web.gc.cuny.edu/Computerscience/news/colloquium.htm.
ABSTRACT: In many computational settings
both the nondeterministic behavior of the system and the random effects of the
environment are apparent, and it is important to have models that support reasoning
about both phenomena. In this talk I'll discuss how to devise models that
support both these phenomena within the context of domains. These
structures are fundamental for defining the semantics of programming
languages, but the application of domains has spread far afield of
semantics. The models for nondeterminism are the now-classic power domains
due to Hennessy, Plotkin and Smyth, while the models of probabilistic
choice include sub-probability measures and now random variables defined
on domains. I'll explain what domains are all about, and how these various
models are devised, so no prior knowledge of the area is required.
This work was supported by the Office of Naval Research and the NSF.
DECEMBER 1: Aaron Todd, Baruch College, "Continuous function spaces
and types of tightness". At C. W. Post cmpus of LIU. Tea 3:15, Pell Hall 240; talk
4:00, Pell Hall 204. For parking and more information, contact Susan Andima,
sandima@liu.edu.
ABSTRACT:
If, for each subset A of a space and each a in the closure of
A, there is a countable subset B of A with a in the closure
of B, the space is said to be of countable tightness. If B may always be
taken to be a sequence converging to a, it is Frechet-Urysohn. J. Kakol
et al. define bounding (bounded) tightness with B required only to be
bounding (bounded for tvs). For continuous function spaces, we discuss
simple relationships, some well-known characterizations and the recent
identification of bounded tightness with the Frechet-Urysohn property.
DECEMBER 8: Larry Narici, St. John's Univ., "On the Hahn-Banach Theorem".
At Baruch College Vertical Campus, Lexington Av. and 24 St.; Tea 3:15 room 6-215,
talk 4:00 room 6-215. For local information contact A. Todd,
artbb@cunyvm.cuny.edu.
ABSTRACT:
I love the Hahn-Banach theorem. I love it the way I love
"Casablanca" and the Fontana di Trevi. It is something not so much to be
read as fondled. What is the Hahn-Banach theorem?" Let f be a continuous
linear functional defined on a subspace M of a normed space X.
Take as the Hahn-Banach theorem the property that
f can be extended to a continuous linear functional on X without changing its norm.
Innocent enough, but the ramifications of the theorem pervade functional analysis and
other disciplines (even thermodynamics!) as well. Where did it come from? Were
Hahn and Banach the discoverers? The axiom of choice implies it, but what about
the converse? Is Hahn-Banach equivalent to the axiom of choice? (No.) Are
Hahn-Banach extensions ever unique? They are in more cases than you might think,
when the unit ball of the dual is "round," as for lp with 1 < p < infinity,
for example, but not for l1 or l{infinity}.
Instead of a linear functional, suppose we substitute
a normed space Y for the scalar field and consider a continuous linear map
A:M --->Y. Can A be continuously extended to X with the same norm?
Well, sometimes. Unsurprisingly, it depends on Y, more specifically, on the "geometry"
of Y: If the unit ball of Y is a "cube," as for Y=R^{n} with sup norm or
Y=real l{\infinity}, for example, then for any subspace M
of any X, any bounded linear map A:M-->Y can be extended to X
with the same norm. This is not true if Y=R^{n}, for any of the p-norms
other than 1 or infinity for any n > 1, despite the topologies being identical.
The cubic nature of the unit ball does not suffice, however---if Y=c{0},
the extendibility dies. This article traces the evolution of the analytic
form as well as subsequent developments up to 2004.
For more information, contact:
CCNY (212-650-5346): R. Kopperman
College of Staten Island
(718-982-3626): P. R. Misra
Baruch College (212-387-1463): A. Todd
LIU C.
W. Post Center (516-299-2447): S. Andima
Marymount College (914-631-3200): M.
Hastings
Queens College, Math. (718-380-1832): G. Itzkowitz
Queens
College, Comp. Sci. (718-997-3478): T. Y. Kong