The Prouhet-Tarry-Escott problem

Given natural numbers n and k, with n>k, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of Z, say X={x_1,...,x_n} and Y={y_1,...,y_n}, such that x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. Many partial solutions to this problem were found in the late 19th century and early 20th century. When k=n-1, we call a solution X=(n-1)Y ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. This thesis focuses on the ideal case. We extend the framework of the problem to number fields, and prove generalizations of results from the literature. This information is used along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers. We also extend a computation from the literature and find new lower bounds for the constant C_n associated to ideal PTE solutions. Further, we present a new algorithm that determines whether an ideal PTE solution with a particular constant exists. This algorithm improves the upper bounds for C_n and in fact, completely determines the value of C_6. We also examine the connection between elliptic curves and ideal PTE solutions. We use quadratic twists of curves that appear in the literature to find ideal PTE solutions over number fields

Behaviour of traffic on a link with traffic light boundaries

This paper considers a single link with traffic light boundary conditions at both ends, and investigates the traffic evolution over time with various signal and system configurations. A hydrodynamic model and a modified stochastic domain wall theory are proposed to describe the local density variation. The Nagel-Schreckenberg model (NaSch), an agent based stochastic model, is used as a benchmark. The hydrodynamic model provides good approximations over short time scales. The domain wall model is found to reproduce the time evolution of local densities, in good agreement with the NaSch simulations for both short and long time scales. A systematic investigation of the impact of network parameters, including system sizes, cycle lengths, phase splits and signal offsets, on traffic flows suggests that the stationary flow is dominated by the boundary with the smaller split. Nevertheless, the signal offset plays an important role in determining the flow. Analytical expressions of the flow in relation to those parameters are obtained for the deterministic domain wall model and match the deterministic NaSch simulations. The analytic results agree qualitatively with the general stochastic models. When the cycle is sufficiently short, the stationary state is governed by effective inflow and outflow rates, and the density profile is approximately linear and independent of time

The Structural Basis for the Increased Immunogenicity of Two HIV-Reverse Transcriptase Peptide Variant/Class I Major Histocompatibility Complexes

Designing altered peptide ligands to generate specific immunological reactivity when bound to class I major histocompatibility complexes is important for both therapeutic and prophylactic reasons. We have previously shown that two altered peptides, derived from human immunodeficiency virus (HIV)-reverse transcriptase (RT) residues 309-317, are more immunogenic in vitro than the wild-type peptide. One peptide variant, I1Y, was able to stimulate RT-specific cytotoxic T cells from the blood of three HIV-infected individuals better than the wild-type RT peptide. Both I1Y and I1F peptide variants increase the cell surface half-life of the peptide-class I complex approximately 3-fold over that of the RT peptide but have different immunological activities. These peptides are candidates for the design of vaccines for HIV due to their increased immunogenicity. To understand the basis for the increased cell surface stability compared with wild-type peptide and to understand the differences in T cell recognition between I1Y and I1F, we determined the x-ray crystal structures of the two class I MHC-peptide complexes. These structures indicate that the increased cell surface half-life is due to pi-pi stacking interactions between Trp-167 of HLA-A2.1 and the aromatic P1 residues of I1F and I1Y. Comparison of the structures and modeling potential T cell receptor (TCR) interactions suggests that T cell interactions and immunogenicity are different between I1Y and I1F for two reasons. First, subtle changes in the steric and polar properties of the I1Y peptide affect TCR engagement. Second, water-mediated hydrogen bond interactions between the P1-Tyr and the P4-Glu peptide residues increase peptide side chain rigidity of residues critical for TCR engagement

Development and Validation of Computational Fluid Dynamics Models for Prediction of Heat Transfer and Thermal Microenvironments of Corals

We present Computational Fluid Dynamics (CFD) models of the coupled dynamics of water flow, heat transfer and irradiance in and around corals to predict temperatures experienced by corals. These models were validated against controlled laboratory experiments, under constant and transient irradiance, for hemispherical and branching corals. Our CFD models agree very well with experimental studies. A linear relationship between irradiance and coral surface warming was evident in both the simulation and experimental result agreeing with heat transfer theory. However, CFD models for the steady state simulation produced a better fit to the linear relationship than the experimental data, likely due to experimental error in the empirical measurements. The consistency of our modelling results with experimental observations demonstrates the applicability of CFD simulations, such as the models developed here, to coral bleaching studies. A study of the influence of coral skeletal porosity and skeletal bulk density on surface warming was also undertaken, demonstrating boundary layer behaviour, and interstitial flow magnitude and temperature profiles in coral cross sections. Our models compliment recent studies showing systematic changes in these parameters in some coral colonies and have utility in the prediction of coral bleaching

Estimated skeletal porosities.

*<p>Species not available. Realistic value for massive coral assumed.</p

Parameters used in the sensitivity analysis.

<p>Parameters used in the sensitivity analysis.</p

Conceptual representation for both models and the experiment apparatus used by Jimenez <i>et al.</i>

<p><i> Left:</i> massive/hemispherical coral. <i>Right:</i> branching coral.</p

Initial and boundary conditions for models.

<p>S: Slip, ZG: Zero Gradient, ZNG: Zero Normal Gradient, FV: Fixed Value.</p