Department of Mathematics and Statistics

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https://hdl.handle.net/10315/901

This collection contains pre-prints and post-prints of journal articles written by selected York University Mathematics faculty.

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Open Access
Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type
(Journal of Differential Equations (Elsevier Science), 2002) Zhu, Huaiping; Rousseau, Christiane
In this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside C . families of planar vector fields. In some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3. By blowing up the family of vector fields, we obtain all the limit periodic sets. We calculate two different types of Dulac maps in the blown-up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point. The finite cyclicity theorems are derived by a generalized derivation–division method introduced by Roussarie.
Open Access
Bifurcation Analysis of a Predator-Prey System With Nonmonotonic Function Response
(SIAM Journal on Applied Mathematics, 2002) Zhu, Huaiping; Campbell, Sue Ann; Wolkowicz, Gail S. K.
We consider a predator-prey system with nonmonotonic functional response: p(x) = mx /(ax2 +bx+1) . By allowing b to be negative (b > −2√a), p(x) is concave up for small values of x > 0 as it is for the sigmoidal functional response. We show that in this case there exists a Bogdanov–Takens bifurcation point of codimension 3, which acts as an organizing center for the system. We study the Hopf and homoclinic bifurcations and saddle-node bifurcation of limit cycles of the system. We also describe the bifurcation sequences in each subregion of parameter space as the death rate of the predator is varied. In contrast with the case b ≥ 0, we prove that when −2√a < b < 0, a limit cycle can coexist with a homoclinic loop. The bifurcation sequences involving Hopf bifurcations, homoclinic bifurcations, as well as the saddle-node bifurcations of limit cycles are determined using information from the complete study of the Bogdanov–Takens bifurcation point of codimension 3 and the geometry of the system. Examples of the predicted bifurcation curves are also generated numerically using XPPAUT. Our work extends the results in [F. Rothe and D. S. Shafer, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), pp. 313–347] and [S. Ruan and D. Xiao, SIAM J. Appl. Math., 61 (2001), pp. 1445–1472].
Open Access
The Courant-Herrmann conjecture
(Journal of Applied Mathematics and Mechanics, 2003) Gladwell, Graham M. L.; Zhu, Hongmei
The Courant‐Herrmann Conjecture (CHC) concerns the sign properties of combinations of the Dirichlet eigenfunctions of elliptic pde's, the most important of which is the Helmholtz equation for $D \in \mathbb{R}^N$. If the eigenvalues are ordered increasingly, CHC states that the nodal set of a combination of the first eigenfunctions, divides into no more than sign domains in which has one sign. The conjecture is classically known to hold for , we conjecture that it is true for rectangular boxes in $\mathbb{R}^N (N\geq2)$, but show that it is false in general.
Open Access
Metapopulation Dynamics with Migration and Local Competition
(American Mathematical Society, 2003) Feng, Zhilan; Yi, Yingfei; Zhu, Huaiping
Many patch-based metapopulation models assume that the local population within each patch is at its equilibrium and independent of changes in patch occupancy. We study a metapoulation model which explicitly incorporates the local population dynamics of two competing species. Singular perturbation method is used to separate the fast dynamics of the local competition and the slow process of patch colonization and extinction. Our results show that the coupled system leads to much more complex outcomes than simple patch models that do not include explicit local dynamics.
Open Access
PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem
(Journal of Differential Equations (Elsevier Science), 2004) Rousseau, Christiane; Zhu, Huaiping
This paper is part of the program launched in (J. Differential Equations 110(1) (1994) 86) to prove the finiteness part of Hilbert’s 16th problem for quadratic system, which consists in proving that 121 graphics have finite cyclicity among quadratic systems. We show that any pp-graphic through a multiplicity 3 nilpotent singularity of elliptic type which does not surround a center has finite cyclicity. Such graphics may have additional saddles and/or saddle-nodes. Altogether we show the finite cyclicity of 15 graphics of (J. Differential Equations 110(1) (1994) 86). In particular we prove the finite cyclicity of a pp-graphic with an elliptic nilpotent singular point together with a hyperbolic saddle with hyperbolicity a1 which appears in generic 3-parameter families of vector fields and hence belongs to the zoo of Kotova and Stanzo (Concerning the Hilbert 16th problem, American Mathematical Society Translation Series 2, Vol. 165, American Mathematical Society, Providence, RI, 1995, pp. 155–201).
Open Access
Removal of phase artifacts from fMRI data using a Stockwell transform filter improves brain activity detection
(Magnetic Resonance in Medicine, 2004) Goodyear, Bradley G.; Zhu, Hongmei; Brown, Robert A.; Mitchell, Joseph
A novel and automated technique is described for removing fMRI image artifacts resulting from motion outside of the imaging field of view. The technique is based on the Stockwell transform (ST), a mathematical operation that provides the frequency content at each time point within a time-varying signal. Using this technique, 1D Fourier transforms (FTs) are performed on raw image data to obtain phase profiles. The time series of phase magnitude for each and every point in the phase profile is then subjected to the ST to obtain a time-frequency spectrum. The temporal location of an artifact is identified based on the magnitude of a frequency component relative to the median magnitude of that frequency’s occurrence over all time points. After each artifact frequency is removed by replacing its magnitude with the median magnitude, an inverse ST is applied to regain the MR signal. Brain activity detection within fMRI datasets is improved by ignificantly reducing image artifacts that overlap anatomical regions of interest. The major advantage of ST-filtering is that artifact frequencies may be removed within a arrow time-window, while preserving the frequency information at all other time points.
Open Access
Fast and slow dynamics of malaria and the S-gene frequency.
(Journal of Dynamics and Differential Equations (Springer), 2004) Feng, Zhilan; Yi, Yingfei; Zhu, Huaiping
A mathematical model incorporating both malaria epidemics and human population genetics of the sickle-cell gene is studied. Singular perturbation techniques are used to separate the dynamics of the model into two time-scales with a faster time-scale for the epidemics and a slower time-scale for the change in gene frequencies. A complete analysis of the dynamics on the slow manifold is conducted, which provides insights into how malaria epidemics may have an impact on the maintenance of the sickle-cell gene in a population where malaria is prevalent.
Open Access
Asymptotic Properties of Weighted Likelihood Estimators
(Journal of Statistical Planning and Inference, 2004) Wang, Xiaogang; Eeden, Constance van; Zidek, James V.
The relevance weighted likelihood method was introduced by Hu and Zidek (Technical Report No. 161, Department of Statistics, The University of British Columbia, Vancouver, BC, Canada, 1995) to formally embrace a variety of statistical procedures for trading bias for precision. Their approach combines all relevant information through a weighted version of the likelihood function. The present paper is concerned with the asymptotic properties of a class of maximum weighted likelihood estimators that contains those considered by Hu and Zidek (Technical Report No. 161, Department of Statistics, The University of British Columbia, Vancouver, BC, Canada, 1995, in: Ahmed, S.E. Reid, N. (Eds.), Empirical Bayes and Likelihood Inference, Springer, New York, 2001, p. 211). Our results complement those of Hu (Can. J. Stat. 25 (1997) 45). In particular, we invoke a di>erent asymptotic paradigm than that in Hu (Can. J. Stat. 25 (1997) 45). Moreover, our adaptive weights are allowed to depend on the data.
Open Access
Distributed Vector Processing of A New Local Multiscale: Fourier Transform for Medical Imaging Applications
(IEEE Transactions on Medical Imaging, 2005) Brown, Robert; Zhu, Hongmei; Mitchell, Joseph
The recently developed S-transform (ST) combines features of the Fourier andWavelet transforms; it reveals frequency variation over both space and time. It is a potentially powerful tool that can be applied to medical image processing including texture analysis and noise filtering. However, calculation of the ST is computationally intensive, making conventional implementations too slow for many medical applications. This problem was addressed by combining parallel and vector computations to provide a 25-fold reduction in computation time. This approach could help accelerate many medical image processing algorithms.
Open Access
A Mathematical Model for Assessing Control Strategies Against West Nile Virus
(Bulletin of Mathematical Biology, Society for Mathematical Biology, 2005) Bowman, C.; Gumel, Abba; van den Driessche, P.; Wu, J.; Zhu, Huaiping
Since its incursion into North America in 1999, West Nile virus (WNV) has spread rapidly across the continent resulting in numerous human infections and deaths. Owing to the absence of an effective diagnostic test and therapeutic treatment against WNV, public health officials have focussed on the use of preventive measures in an attempt to halt the spread of WNV in humans. The aim of this paper is to use mathematical modelling and analysis to assess two main anti-WNV preventive strategies, namely: mosquito reduction strategies and personal protection. We propose a single-season ordinary differential equation model for the transmission dynamics of WNV in a mosquito–bird–human community, with birds as reservoir hosts and culicine mosquitoes as vectors. The model exhibits two equilibria; namely the disease-free equilibrium and a unique endemic equilibrium. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if a certain threshold quantity (R0 ), which depends solely on parameters associated with the mosquito–bird cycle, is less than unity. The public health implication of this is that WNV can be eradicated from the mosquito–bird cycle (and, consequently, from the human population) if the adopted mosquito reduction strategy (or strategies) can make R0 < 1. On the other hand, it is shown, using a novel and robust technique that is based on the theory of monotone dynamical systems coupled with a regular perturbation argument and a Liapunov function, that if R0 > 1, then the unique endemic equilibrium is globally stable for small WNV-induced avian mortality. Thus, in this case, WNV persists in the mosquito–bird population.
Open Access
Derivation of Mixture Distribution and Weighted Likelihood as minimizers of KL-divergence subject to constraints
(Annals - Institute of Statistical Mathematics, 2005) Wang, Xiaogang; Zidek, James V.
In this article, mixture distributions and weighted likelihoods are derived within an information-theoretic framework and shown to be closely related. This surprising relationship obtains in spite of the arithmetic form of the former and the geometric form of the latter. Mixture distributions are shown to be optima that minimize the entropy loss under certain constraints. The same framework implies the weighted likelihood when the distributions in the mixture are unknown and information from independent samples generated by them have to be used instead. Thus the likelihood weights trade bias for precision and yield inferential procedures such as estimates that can be more reliable than their classical counterparts.
Open Access
From the PP-graphics to the finiteness part of Hilbert's 16th problem for quadratic systems
(World Scientific Publishing, 2005) Zhu, Huaiping
This is part of the effort, the program launched by Dumortier, Roussarie and Rousseau, in proving the finiteness part of Hilbert’s 16th problem. In this paper, we highlight the ideas of proving the finite cyclicity of pp-graphics in quadratic systems.
Open Access
Nonparametric tests for differential gene expression and interaction effects in multi-factorial microarray experiments
(BioMed Central, 2005-07-21) Gao, Xin; Song, Peter XK
Background Numerous nonparametric approaches have been proposed in literature to detect differential gene expression in the setting of two user-defined groups. However, there is a lack of nonparametric procedures to analyze microarray data with multiple factors attributing to the gene expression. Furthermore, incorporating interaction effects in the analysis of microarray data has long been of great interest to biological scientists, little of which has been investigated in the nonparametric framework. Results In this paper, we propose a set of nonparametric tests to detect treatment effects, clinical covariate effects, and interaction effects for multifactorial microarray data. When the distribution of expression data is skewed or heavy-tailed, the rank tests are substantially more powerful than the competing parametric F tests. On the other hand, in the case of light or medium-tailed distributions, the rank tests appear to be marginally less powerful than the parametric competitors. Conclusion The proposed rank tests enable us to detect differential gene expression and establish interaction effects for microarray data with various non-normally distributed expression measurements across genome. In the presence of outliers, they are advantageous alternative approaches to the existing parametric F tests due to the robustness feature.
Open Access
A Derivative Free Optimization Algorithm based on Conditional Moments
(Journal of Mathematical Analysis and Applications, 2006) Wang, Xiaogang; Liang, Dong; Feng, Xingdong; Ye, Lu
In this paper we propose a derivative-free optimization algorithm based on conditional moments for finding the maximizer of an objective function. The proposed algorithm does not require calculation or approximation of any order derivative of the objective function. The step size in iteration is determined adaptively according to the local geometrical feature of the objective function and a pre-specified quantity representing the desired precision. The theoretical properties including convergence of the method are presented. Numerical experiments comparing with the Newton, Quasi-Newton and trust region methods are given to illustrate the effectiveness of the algorithm.
Open Access
Clustering Large Software Systems at Multiple Layers
(Information and Software Technology, 2006) Andreopoulos, Bill; An, Aijun; Tzerpos, Vassilios; Wang, Xiaogang
Software clustering algorithms presented in the literature rarely incorporate in the clustering process dynamic information, such as the number of function invocations during run time. Moreover, the structure of a software system is often multi-layered, while existing clustering algorithms often create flat system decompositions. This paper presents a software clustering algorithm called MULICsoft that incorporates in the clustering process both static and dynamic information. MULICsoft produces layered clusters with the core elements of each cluster assigned to the top layer. We present experimental results of applying MULICsoft to a large open-source system. Comparison with existing software clustering algorithms indicates that MULICsoft is able to produce decompositions that are close to those created by system experts.
Open Access
Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response
(SIAM Journal on Applied Mathematics, 2006) Xiao, Dongmei; Zhu, Huaiping
In this paper, we develop a criterion to calculate the multiplicity of a multiple focus for general predator prey systems. As applications of this criterion, we calculate the most multiplicity of a multiple focus in a predator prey system with nonmonotonic functional response p(x) = x/(ax^2+bx+1) studied by Zhu, Campbell and Wolkowicz [26] and prove that the degenerate Hopf bifurcation is of codimension two. Furthermore, we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles, the inner one is unstable and outer one is stable. Numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus were also given in support of the criterion.
Open Access
Linear grouping using orthogonal regression
(Computational Statistics and Data Analysis, 2006) Van Aelst, Stefan; Wang, Xiaogang; Zamar, Ruben H.; Zhu, Rong
A new method to detect different linear structures in a data set, called Linear Grouping Algorithm (LGA), is proposed. LGA is useful for investigating potential linear patterns in data sets, that is, subsets that follow different linear relationships. LGA combines ideas from principal components, clustering methods and resampling algorithms. It can detect several different linear relations at once. Methods to determine the number of groups in the data are proposed. Diagnostic tools to investigate the results obtained from LGA are introduced. It is shown how LGA can be extended to detect groups characterized by lower dimensional hyperplanes as well. Some applications illustrate the usefulness of LGA in practice.
Open Access
An Iterative Non-parametric Clustering Algorithm Based on Local Shrinking
(Computational Statistics and Data Analysis, 2006) Wang, Xiaogang; Qiu, Weiliang; Zamar, Ruben H.
In this paper, we propose a new non-parametric clustering method based on local shrinking. Each data point is transformed in such a way that it moves a specific distance toward a cluster center. The direction and the associated size of each movement are determined by the median of its K-nearest neighbors. This process is repeated until a pre-defined convergence criterion is satisfied. The optimal value of the K is decided by optimizing index functions that measure the strengths of clusters. The number of clusters and the final partition are determined automatically without any input parameter except the stopping rule for convergence. Our performance studies have shown that this algorithm converges fast and achieves high accuracy.
Open Access
The Loop Quantities and Bifurcations of Homoclinic Loops
(Journal of Differential Equations (Elsevier Science), 2007-01-31T19:36:32Z) Han, Maoan ; Zhu, Huaiping
The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.
Open Access
Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation
(Elsevier, Applied Mathematics and Computation, 2011-11-01) Abrarov, S. M.; Quine, B. M.
We show that a Fourier expansion of the exponential multiplier yields an exponential series that can compute high-accuracy values of the complex error function in a rapid algorithm. Numerical error analysis and computational test reveal that with essentially higher accuracy it is as fast as FFT-based Weideman’s algorithm at a regular size of the input array and considerably faster at an extended size of the input array. As this exponential series approximation is based only on elementary functions, the algorithm can be implemented utilizing freely available functions from the standard libraries of most programming languages. Due to its simplicity, rapidness, high-accuracy and coverage of the entire complex plane, the algorithm is efficient and practically convenient in numerical methods related to the spectral line broadening and other applications requiring error-function evaluation over extended input arrays.