Department of Mathematics and Statistics
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This collection contains pre-prints and post-prints of journal articles written by selected York University Mathematics faculty.
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Item Open Access Cost and social distancing dynamics in a mathematical model of COVID-19 with application to Ontario, Canada(The Royal Society Publishing, 2021-02-24) Moyles, Iain; Heffernan, Jane; Kong, JudeA mathematical model of COVID-19 is presented where the decision to increase or decrease social distancing is modelled dynamically as a function of the measured active and total cases as well as the perceived cost of isolating. Along with the cost of isolation, we define an overburden healthcare cost and a total cost. We explore these costs by adjusting parameters that could change with policy decisions. We observe that two disease prevention practices, namely increasing isolation activity and increasing incentive to isolate do not always lead to optimal health outcomes. We demonstrate that this is due to the fatigue and cost of isolation. We further demonstrate that an increase in the number of lock-downs, each of shorter duration can lead to minimal costs. Our results are compared with case data in Ontario, Canada from March to August 2020 and details of expanding the results to other regions are presented.Item Restricted An improved approximation for hydraulic conductivity for pipes of triangular cross-section by asymptotic means(Springer, 2021-01-28) Keane, Laura; Moyles, Iain; Hall, CameronIn this paper, we explore single-phase flow in pores with triangular cross-sections at the pore-scale level. We use analytic and asymptotic methods to calculate the hydraulic conductivity in triangular pores, a typical geometry used in network models of porous media flow. We present an analytical formula for hydraulic conductivity based on Poiseuille flow that can be used in network models contrasting the typical geometric approach leading to many different estimations of the hydraulic conductivity. We consider perturbations to an equilateral triangle by changing the length of one of the triangle sides. We look at both small and large triangles in order to capture triangles that are near and far from equilateral. In each case, the calculations are compared with numerical solutions and the corresponding network approximations. We show that the analytical solution reduces to a quantitatively justifiable formula and agrees well with the numerical solutions in both the near and far from equilateral cases.Item Restricted Quasi-steady uptake and bacterial community assembly in a mathematical model of soil-phosphorus mobility(Elsevier, 2021-01-21) Moyles, Iain; Fowler, Andrew; Donohue, JohnWe mathematically model the uptake of phosphorus by a soil community consisting of a plant and two bacterial groups: copiotrophs and oligotrophs. Four equilibrium states emerge, one for each of the species monopolising the resource and dominating the community and one with coexistence of all species. We show that the dynamics are controlled by the ratio of chemical adsorption to bacterial death permitting either oscillatory states or quasi-steady uptake. We show how a steady state can emerge which has soil and plant nutrient content unresponsive to increased fertilization. However, the additional fertilization supports the copiotrophs leading to community reassembly. Our results demonstrate the importance of time-series measurements in nutrient uptake experiments.Item Open Access Cost-effectiveness of a potential Zika vaccine candidate: a case study for Colombia(Biomed Central, 2018-07-03) Shoukat, Affan; Vilches, Thomas; Moghadas, SeyedBackground: A number of Zika vaccine platforms are currently being investigated, some of which have entered clinical trials. We sought to evaluate the cost-effectiveness of a potential Zika vaccine candidate under the WHO Vaccine Target Product Profile for outbreak response, prioritizing women of reproductive age to prevent microcephaly and other neurological disorders. Methods: Using an agent-based simulation model of ZIKV transmission dynamics in a Colombian population setting, we conducted cost-effectiveness analysis with and without pre-existing herd immunity. The model was parameterized with estimates associated with ZIKV infection, risks of microcephaly in different trimesters, direct medical costs, and vaccination costs. We assumed that a single dose of vaccine provides a protection efficacy in the range 60% to 90% against infection. Cost-effectiveness analysis was conducted from a government perspective. Results: Under a favorable scenario when the reproduction number is relatively low (R0 = 2.2) and the relative transmissibility of asymptomatic infection is 10% compared with symptomatic infection, a vaccine is cost-saving (with negative incremental cost-effective ratio; ICER) for vaccination costs up to US$6 per individual without herd immunity, and up to US$4 per individual with 8% herd immunity. For positive ICER values, vaccination is highly cost-effective for vaccination costs up to US$10 (US$7) in the respective scenarios with the willingness-to-pay of US$6610 per disability-adjusted life-year, corresponding to the average per capita GDP of Colombia between 2013 and 2017. Our results indicate that the effect of other control measures targeted to reduce ZIKV transmission decreases the range of vaccination costs for cost-effectiveness due to reduced returns of vaccine-induced herd immunity. In all scenarios investigated, the median reduction of microcephaly exceeded 64% with vaccination. Conclusions: Our study suggests that a Zika vaccine with protection efficacy as low as 60% could significantly reduce the incidence of microcephaly. From a government perspective, Zika vaccination is highly cost-effective, and even cost-saving in Colombia if vaccination costs per individual is sufficiently low. Efficacy data from clinical trials and number of vaccine doses will be important requirements in future studies to refine our estimates, and conduct similar studies in other at-risk populations. Keywords: Zika, Microcephaly, Vaccination, Agent-based modeling, Cost-effectivenessItem Open Access The rational number u2 for the two-term Machin-like formula for pi computed by iteration(2017-06-02) Abrarov, S. M.; Quine, B. M.Item Open Access Master-Slave Algorithm for Highly Accurate and Rapid Computation of the Voigt/Complex Error Function(Journal of Mathematics Research, 2014-06) Abrarov, S. M.; Quine, B. M.We obtain a rational approximation of the Voigt/complex error function by Fourier expansion of the exponential function ${e^{ - {{\left( {t - 2\sigma } \right)}^2}}}$ and present master-slave algorithm for its efficient computation. The error analysis shows that at $y > {10^{ - 5}}$ the computed values match with highly accurate references up to the last decimal digits. The common problem that occurs at $y \to 0$ is effectively resolved by main and supplementary approximations running computation flow in a master-slave mode. Since the proposed approximation is rational function, it can be implemented in a rapid algorithm.Item Open Access On the Fourier expansion method for highly accurate computation of the Voigt/complex error function in a rapid algorithm(2012-06-21) Abrarov, S. M.; Quine, B. M.In our recent publication [1] we presented an exponential series approximation suitable for highly accurate computation of the complex error function in a rapid algorithm. In this Short Communication we describe how a simplified representation of the proposed complex error function approximation makes possible further algorithmic optimization resulting in a considerable computational acceleration without compromise on accuracy.Item 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.Item 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 XKBackground 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.Item Open Access The Courant-Herrmann conjecture(Journal of Applied Mathematics and Mechanics, 2003) Gladwell, Graham M. L.; Zhu, HongmeiThe 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.Item 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, JosephA 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.Item 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, JosephThe 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.Item 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, LuIn 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.Item 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.Item Open Access Clustering Large Software Systems at Multiple Layers(Information and Software Technology, 2006) Andreopoulos, Bill; An, Aijun; Tzerpos, Vassilios; Wang, XiaogangSoftware 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.Item Open Access Linear grouping using orthogonal regression(Computational Statistics and Data Analysis, 2006) Van Aelst, Stefan; Wang, Xiaogang; Zamar, Ruben H.; Zhu, RongA 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.Item 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.Item 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.Item 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, ChristianeIn 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.Item 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].