Abstracts Fall 2009 – Summer 2010
Fall 2009- Summer 2010
Interesting questions about the powers of 2 will be addressed, such as:
• Which powers of 2 begin with a 7?
• How do we find those powers?
• How can the powers of two give you strategy in a game?
• And more!
Also discussed will be Benford’s Law and Mersenne Primes.
A basic question in number theory asks “Which primes are of the form
p=x2+ny2,where x, n, and y are integers?” This leads naturally to the study of quadratic forms, which are expressions of the form
f(x,y) = ax2 + bxy + cy2.In 1801, a group law on equivalence classes of quadratic forms was published by Gauss in which the class of x2 + ny2 is the identity element. In his 2001 Princeton PhD thesis, Manjul Bhargava developed a series of far-reaching generalizations of Gauss composition that give new information about class groups in number fields of degree < 6. The first of these generalizations is a composition law based on geometric cubes with an integer assigned to each corner. This thesis defense introduces the first generalization developed by Bhargava and applies it by introducing his theory of composition of binary cubic forms.
Odd primes of the form p=x2+y2 are easy to characterize. As written by Fermat in the 1600’s, they are precisely the primes that “surpass by one a multiple of four.” But what about primes of the form p=x2+ny2? It turns out that the solution to this question is much more difficult, requiring sophisticated 20th century number theory. In this talk I will focus on one aspect of the classical theory developed by Gauss in 1801 and refined by Dirichlet in 1851. Gauss defined an operation on equivalence classes of quadratic forms, creating a group structure in which the equivalence class of x2+ny2 is the identity element. Although this alone doesn’t answer the question “which primes are of the form x2+ny2,” the insights provided by Gauss composition form an important step towards the solution to this problem.
Monday, December 14, 2009
Zezheng Li, UMaine Master’s candidate. Advisor: Ramesh Gupta.
Thesis defense: Estimating the Minimum Effective Dose in dose response studies
Lecture at 2:10 pm, 421 Neville Hall
The proper understanding and characterization of the dose-response
relationship for a new compound is a fundamental step in the clinical
development process of any medicinal drug.
A major goal of dose-finding studies is to estimate target doses of
interest. Among the many target doses that can be estimated the minimum
effective dose (MED) is of particular interest. Here the MED is defined as
the smallest dose that produces a
practically relevant response. Thus, any dose smaller than the MED can be
discarded for future studies or for potential release in the market.
In this presentation, we will study various optimal designs for estimating
the MED. More specifically, we will study designs that minimize the
asymptotic variance of the MED estimator under a particular dose response
Friday, November 6, 2009
Laboratoire Joliot Curie et Laboratoire de Physique, Ecole Normale Supérieure de Lyon 46 Allée d’Italie, 69364 Lyon Cedex 07, France
What is the Role of “JUNK DNA”
Lecture at 2:10 pm, 101 Neville Hall
What is the role of 95% of the human genome that does not code for proteins? Recent technical progress in live cell imaging have confirmed that the structure and dynamics of chromatin play an essential role in regulating many biological processes, such as gene activity, DNA replication, recombination and DNA damage repair. In the first part of this talk, we use the space-scale decomposition provided by the continuous wavelet transform (WT) to characterize the scale invariance properties of genomic sequences. We show the existence of long-range correlations (LRC) over distances up to 20 -30 kbp. To understand to which extent the observed LRCs could influence the compaction and accessibility of genomic information in the cells, we perform a fractal analysis of DNA structural profiles, e.g., DNA bending profiles based on nucleosome positioning data. In the second part of this talk, we investigate the thermodynamical properties of 2D elastic chains submitted locally to mechanical/topological constraint as loops. We show that a possible key to understanding this is that the LRC structural disorder induced by the sequence may favor the autonomous formation of small (few hundreds bp) DNA loops and, in turn, the propensity of eukaryotic DNA to interact with histones to form nucleosomes. We further compare the model predictions to genome-scale nucleosome positioning data recently obtained by Yuan et al. for S. cerevisiae chromosome III (Science 309, 2005). The statistical analysis of the experimental profile of nucleosome occupancy displays striking similarities to the energy landscape of nucleosome formation computed from the sequence.including the nucleosome-free regions observed at gene promoters These results constitute a first experimental evidence of the influence of LRCs on the nucleosomal organisation. The third part of this talk is devoted to the recent experimental observation of LRC in the 2D equilibrium conformations of eukaryotic DNA by AFM visualization techniques.
In her talk she will mention how she came to work as NSA, the different kinds of work mathematicians do there, including a brief description of public key cryptography, and the different summer programs and entry-level programs for mathematicians.
Tuesday, October 20, 2009 – Mathematics “Career Day” Lecture
Marylou Murphy, FSA MAAA, Vice-President & Actuary, Unum Group
Paul Correia, ASA , Associate Actuary, Unum Group
Lecture at 2:10 pm, 100 Neville Hall
Marylou Murphy and Paul Correia from Unum will discuss career opportunities in the actuarial profession. Marylou will describe the role actuaries play and the skill sets needed to excel. In addition, she will discuss opportunities at Unum, a leading provider of employee benefits and the market leader in providing disability and long term care insurance in the U.S.
Paul Correia will then describe his experiences to date in the career path and how his educational background has helped him succeed.
The purpose of this paper is to determine an appropriate variance function (mean- variance relationship) which can be used in the semi-parametric analysis of over- dispersed count data (for example, for the analysis of count data by extended quasi- likelihood and double extended quasi-likelihood). We use hypothesis testing approach through a broader class of models and data analytic approach. The models considered are the three parameter negative binomial distribution and the extended quasi-likelihood. Wide analysis involving tests, data analysis and simulations indicate that the three parameter generalized negative binomial distribution does not improve in fit to count data over the simpler negative binomial distribution. Further data analysis and simulations using the extended quasi-likelihood indicate that the negative binomial variance function µ + cµ2 is preferable over a simpler variance function c3 µ2 for data with small mean and small over-dispersion. Otherwise c3 µ2 is a preferable variance function over the negative binomial variance function.