CECM Home > Events > CECM Annual Summer Meeting 2007 > Program > Poster Abstracts

### Poster Abstracts

###### A Delay Differential Equation Model of HIV Testing and Treatment

*Azadeh Alimadad, Krisztina Vasarhelyi and Alexander R. Rutherford*

We develop a compartmental model for evaluating HIV testing and treatment scenarios. A system of coupled delay differential equations is used to represent three stages of HIV infection, plus the fourth stage of symptomatic AIDS. The model is implemented using a system dynamics approach and numerical analysis is used to study the stability of solutions. The model is used to evaluate an early treatment scenario. A steady decrease in new HIV infections is observed. However, the number of AIDS patients in treatment shows a strong latency effect.

###### The Height of the 3,234,846,615th Cyclotomic Polynomial is Big (2,888,582,082,500,892,851)

*Andrew Arnold and Michael Monagan*

In this poster we will discuss some interesting properties of cyclotomic polynomials, and introduce a fast algorithm for constructing these polynomials. We also will present our findings, including the largest cyclotomic polynomial coefficients ever computed.

###### Solving Linear Systems of Equations Over Cyclotomic Fields

*Liang Chen and Michael Monagan*

Let A ∈ Q^{n×n}[z] be a matrix of polynomials and b ∈ Q^{n}[z] be a vector of polynomials.
Let m(z) = Φ_{k}[z] be the k^{th} cyclotomic polynomial.
We want to find the solution vector x ∈ Q^{n}[z] such that the equation Ax ≡ b mod m(z) holds.
One may obtain x using Gaussian elimination, however, it is inefficient because of the large rational
numbers that appear in the coefficients of the polynomials in the matrix during the elimination.
In this thesis, we present two modular algorithms
namely, Chinese remaindering and linear p-adic lifting.
We have implemented both algorithms in Maple and have determined the time complexity of both algorithms.
We present timing comparison
tables on two sets of data, firstly, systems with random generated coefficients and secondly real
systems given to us by Vahid Dabbaghian which arise from computational group theory.
The results show that both of our algorithms are much faster than Gaussian elimination.

###### On the Density of Integers Bi-representable as the Sum of two Cubes

*Michael Coons and Paul Vrbik*

Denote by ν(x) the number of positive integers less than or equal to x that are representable in at least two ways as the sum of two positive cubes. We address the following question: find the least ϑ > 0 so that ν(x) ≪ x^{ϑ+ε}. The best known result is ϑ = 4/9. We provide computational evidence for the conjectured asymptotic bound of ϑ = 2/5.

###### Inferring a duplication and speciation history from a gene tree

*Jean-Philippe Doyon*

We consider two algorithmical questions related to the evolution of gene families. First, given a gene tree G, can the evolutionary history of this gene family be explained with only speciation and duplication events. We show that this can be answered in linear time, and that G induces a single species tree consistent with this history. We then present a heuristic for the following problem: if G can not be explained without gene losses, what is the minimum number of losses involved in an history of the gene family. We evaluate our algorithms on two gene families datasets: plant and yeast.

###### Zeros of truncated Taylor series of the symmetric Riemann Zeta function

*Greg Fee*

The Riemann Zeta function is defined by: Zeta(x) = sum(k^(-x),k=1..infinity). We may remove the pole and the trivial zeros at negative even integers by using Riemann's symmetric Zeta function. First we compute approximate truncated Taylor series of the above function expanded about the symmetry point, x=1/2 . Next we find the zeros of these Taylor polynomials. Then we fit a curve to the extraneous zeros.

###### On Kloosterman sums divisible by 3

*Kseniya Garaschuk and Petr Lisonek*

By counting the coset leaders for cosets of weight 3 of the Melas code we find a new proof for the characterization of Kloosterman sums divisible by 3 over F_{2m} where m is odd. New results due to Charpin, Helleseth and Zinoviev then provide a connection to a characterization of all elements a in F_{2m} such that Tr(a^{1/3}) = 0. Moreover, we prove a generalization to the case Tr(a^{1/(2k-1)}) = 0. We present an application to constructing caps in PG(n, 2) with many free pairs of points.

###### The Lander-Waterman Genome Sequencing Model

*Huiyi Lu*

We describe an exploration, using Maple, of the Lander-Waterman genome sequencing model in the framework of the sequencing of viral metagenomes.

###### Classification of Random Walks on Triangular Lattices

*Jamie Lutley and Rebecca Nie*

The equilateral triangular lattice has a rotational symmetry in addition to the reflectional symmetry similar to the traditional rectangular lattice. The wedges that can naturally be formed in the triangular lattice are subtended by angles of π/3, 2π/3, 4π/3, and 5π/3. The classification of walks on the two convex regions is quite similar, both in the form of what has been proven by general criteria and the outlier cases which have been proven individually. The form of these outliers suggests additional general criteria may exist. On the other hand, the classification on the larger regions are of more intricate nature, but a provisional classification based on decomposition is established, while this classification suggests direction for further research.

###### Sparse Polynomial Arithmetic using Heaps of Pointers

*Roman Pearce*

We present algorithms to multiply and divide sparse polynomials that construct terms of the result one at a time using very little memory. Their asymptotic and real world performance is the same or better than the best algorithms commonly used today.

###### Anomalous Scalings in Spatiotemporal Chaos

*Ka Fai Poon*

Anomalous scaling of amplitudes and time scales is found that contradicts the predictions of a leading-order multiple-scale analysis of the Nikolaevskii equation. By adding a single Burgers-like term at the next order, it is able to capture corrections to scaling. We show evidence that these two sets of equations exhibit different dynamics.

###### A Cellular Automata Model of the Spread of HIV in a Community of Injection Drug Users

*Natasha Richardson and Vahid Dabbaghian-Abdoly*

Within a health promotion and harm reduction framework, modelling the spread of Human Immunodeficiency Virus (HIV) among Injection Drug Users (IDU) is a priority issue. In this poster we present a Cellular Automata model which describes the dynamics of HIV based on hypothetical social interactions between members of an IDU community. We show the results of the implementations of this model on the computer algebra system Maple.

###### Demo of SAGE

*William Stein*

The goal of the SAGE project is to create a viable free open source alternative to Mathematica, Magma, Matlab, and Maple. See http://www.sagemath.org/.