Tuen Wai Ng

Meromorphic solutions of higher order Briot–Bouquet differential equations

For differential equations P ( y ( k ) , y )=0, where P is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.

A double epidemic model for the SARS propagation.

BackgroundAn epidemic of a Severe Acute Respiratory Syndrome (SARS) caused by a new coronavirus has spread from the Guangdong province to the rest of China and to the world, with a puzzling contagion behavior. It is important both for predicting the future of the present outbreak and for implementing effective prophylactic measures, to identify the causes of this behavior.ResultsIn this report, we show first that the standard Susceptible-Infected-Removed (SIR) model cannot account for the patterns observed in various regions where the disease spread. We develop a model involving two superimposed epidemics to study the recent spread of the SARS in Hong Kong and in the region. We explore the situation where these epidemics may be caused either by a virus and one or several mutants that changed its tropism, or by two unrelated viruses. This has important consequences for the future: the innocuous epidemic might still be there and generate, from time to time, variants that would have properties similar to those of SARS.ConclusionWe find that, in order to reconcile the existing data and the spread of the disease, it is convenient to suggest that a first milder outbreak protected against the SARS. Regions that had not seen the first epidemic, or that were affected simultaneously with the SARS suffered much more, with a very high percentage of persons affected. We also find regions where the data appear to be inconsistent, suggesting that they are incomplete or do not reflect an appropriate identification of SARS patients. Finally, we could, within the framework of the model, fix limits to the future development of the epidemic, allowing us to identify landmarks that may be useful to set up a monitoring system to follow the evolution of the epidemic. The model also suggests that there might exist a SARS precursor in a large reservoir, prompting for implementation of precautionary measures when the weather cools down.

A weighted Local Least Squares Imputation method for missing value estimation in microarray gene expression data

Many clustering techniques and classification methods for analysing microarray data require a complete dataset. However, very often gene expression datasets contain missing values due to various reasons. In this paper, we first propose to use vector angle as a measurement for the similarity between genes. We then propose the Weighted Local Least Square Imputation (WLLSI) method for missing values estimation. Numerical results on both synthetic data and real microarray data indicate that WLLSI method is more robust. The imputation methods are then applied to a breast cancer dataset and interesting results are obtained.

On Ritt's Factorization of Polynomials

Ritt has shown that any complex polynomial p can be written as the composition of polynomials p 1 ,…, p m , where each p j is prime in the sense that it cannot be written as a non-trivial composition of polynomials. The factors p j are not unique but the number m of them is, as is the set of the degrees of the p j . The paper extends Ritts theory and, in particular, a third invariant of the decomposition is introduced.

Random iteration of analytic maps

We consider analytic maps

Ritt's theory on the unit disk

f_j:D\to D

Residual Julia Sets of Meromorphic Functions

of a domain D into itself and ask when does the sequence

Permutable entire functions and their Julia sets

f_1\circ\dotsb\circ f_n

CpG/CpNpG motifs in the coding region are preferred sites for mutagenesis in the breast cancer susceptibility genes.

converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence

Smale's mean value conjecture for odd polynomials

\{w_1,w_2,\dotsc\}