**A symposium devoted to the mathematical
interests of David Rosen**

Saturday, October 25

Beginning at 9:00 am (note updated start time and schedule)

DuPont Science Building, Room 199.

What is a Rosen Continued Fraction? Here are three responses by Thomas Schmidt:

1.) The Rosen Continued Fractions are an alternate means for expressing real numbers.

2.) The Rosen Continued Fractions are actually an infinite collection of closely related algorithms, each of which expresses real numbers in terms of certain `algebraic integers'.

3.) A continued fraction algorithm gives an alternate manner to express real numbers. For example, 8/3 = 2+1/(1+1/2); more generally, x = a +1/(b + 1/(c + ... . In the Rosen continued fractions the numbers a, b, c, ... are certain special types of numbers (specifically, they are certain `algebraic' numbers). These algorithms have both number theoretic and geometric significance.

Schedule: (click the title to go to an abstract for the talk)

Abstracts:

Thomas Schmidt, University of Oregon, keynote speech:

*The Rosen continued fractions: arithmetic and geometry.*

Abstract: In his Ph.D. dissertation, published in 1954, David Rosen introduced his λ-continued fractions in order to study a certain infinite family of two-by-two matrix groups, the Hecke groups.

Dave Rosen had various favorite questions about the Hecke groups and his continued fractions. Over the some 50 years that he enthusiastically shared his work and these questions, some were answered, but not all.

Beginning with a brief overview of the basics of continued fractions, in this talk we indicate a selection of results by various authors, as well as some of the remaining open questions about the Hecke groups and the Rosen continued fractions.

Mark Sheingorn, Baruch College of CUNY:

*How David Rosen set me up for life in 1958 without even meeting me.*

Abstract: How Dave's Duke paper of that year led to my life-long struggle with continued fractions and the Modular group.

Helen Grundman, Bryn Mawr College:

*Continued fractions and units in cubic orders.*

Abstract: Paying homage to Dave's great work with continued fractions, I will present results concerning the use of continued fractions in the determination of fundamental units of orders in number fields. I'll review the well-known methods for quadratic orders, then discuss my work on using continued fractions to determine fundamental units in cubic orders.

Christopher Towse, Scripps College:

*Rosen continued fraction representations of certain units.*

Abstract: We consider the units in Z[λ_{q}]
where λ_{q} = 2 cos(π/q). We show that when q = 4 or 6, the
units are infinite pure periodic λ_{q}-fractions. The case q
= 7 is quite different; examples of units that are finite λ_{7}-fractions
and units that are infinite λ_{7}-fractions are given.

George Andrews, Penn State:

*Rademacher's Conjecture, q-partial fractions, and Munagi's Theorem.*

Abstract: In the 1973 book, Topics in Analytic Number Theory, page 302, H. Rademacher conjectured a very natural relationship between the partial fraction expansion of the generating function for p(n) (the number of partitions of n) and the partial fraction expansion of the generating function for p(n,m) (the number of partitions of n into parts each not exceeding m). In the intervening 30 years, no theoretical progress has been made on this conjecture, and the little empirical evidence has been uncertain at best. Recently, Augustine Munagi has developed an analog of the partial fration decomopsition which he calls q-partial fractions. In this setting there is a natural analog of the Rademacher conjecture, and one nontrivial special case has been proved by Munagi. This talk will describe these developments.

Todd Drumm, U. of Pennsylvania;

*Lorentzian visions of the modular group.*

Abstract: We will discuss how the modular group, and other subgroups of PSL(2,R) near and dear to a number theorist's heart, can be deformed into the group of Lorentzian transformations. For these deformations, we will define a new spectrum with nice isospectrality results.

Wladimir Pribitkin, College of Staten
Island, CUNY:

*New functions associated with cusp forms.*

Abstract: We introduce some functions intimately linked to classical modular forms. In particular, we present a natural generalization of Eichler integrals.

Sinai Robins:

*Hecke operators on rational functions.*

Abstract: We define Hecke operators U_{m} that
sift out every m-th Taylor series coefficient of a rational function in one
variable, defined over the reals. We prove several structure theorems concerning
the eigenfunctions of these Hecke operators, including the pleasing fact that
the point spectrum of the operator U_{m} is simply the set {±
m^{k}, k Î N} È
{0}. It turns out that the simultaneous eigenfunctions of all of the Hecke operators
involve Dirichlet characters mod L, giving rise to the result that any arithmetic
function of m that is completely multiplicative and also satisfies a linear
recurrence must be a Dirichlet character times a power of m. We also define
the notions of level and weight for rational eigenfunctions, by analogy with
modular forms, and we show the existence of some interesting finite-dimensional
subspaces of rational eigenfunctions (of fixed weight and level), whose union
gives all of the rational functions whose coefficients are quasi-polynomials.

Marvin Knopp, Temple University:

*Multiplier systems.*

Abstract: An interest of Dave Rosen was multiplier systems on the Hecke groups [Cont. Math.143(1993)]. I have found multiplier systems of interest since my dissertation work (1957-58) and, indeed, my first published paper is in this area [Duke Math. J.1960]. Until quite recently the study of multiplier systems has been confined to those of absolute value one. In a recent paper [J. of Number Theory, 2003] Geoffrey Mason and I have found that multiplier systems of arbitrary nonzero absolute value are worth investigating. These are the subject of my talk.

Charles L. Bennett, MD PhD,
Northwestern:

*Practical uses of probability and statistics coursework at Swarthmore: Identification of potentially fatal
side effects of medications.
*

Abstract: Adverse drug reactions account for 100,000 deaths annually in the United States. Using techniques derived from probability and statistics as well as pharmacology and medicine, Dr. Bennett and colleagues have developed an novel program that seeks to identify potentially fatal and previously unreported adverse drug reactions. The program, called RADAR (Research on Adverse Drug Events and Reports) is funded by the National Institutes of Health and reviews detailed information from FDA sources to identify adverse drug events that result in death or severe organ failure. Between 1998 and 2002, seven potentially fatal adverse drug reactions associated with seven medications that were used by between 17,000 and 5 million individuals annually in the United States were identified by the RADAR program. Six of these medication reactions occurred in "off-label" clinical settings- i.e. clinical settings where a medication was used, but not formally approved by the FDA for this use. The toxic events generally occurred between three days and three months after drug initiation. Moreover, dissemination of this information by the pharmaceutical industry has been substandard, as four of the events are not described in the package inserts of the relevant medication.

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