Browsing by Browse by FOR 2008 "010299 Applied Mathematics not elsewhere classified"

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Thesis Masters Research
Applications of Hierarchical Statistical Models to Interpret Biological Data
(2009)
Penrose, Lindsay
;
Murison, Robert
This thesis looks at two biological applications of hierarchical models. The first application considers the non-linear decay of a contaminant in beef cattle, while the other quite different application considers the transmission probability of genetic markers in the mouse. In the first application, a contaminant was measured over time on 11 animals. Each animal was then a random effect in a small data set. Application of a non-linear random effects model was the only way to analyse these data. This required integrating over animal effects to obtain the marginal model for the small animal population. Lack of a closed form solution for the integration, necessitated use of LaPlace's approximation. A better analysis was demonstrated using MCMC, which provided confidence intervals for parameter estimates, and was not limited by the number of parameters in the model. The second application was seemingly very different. The data consisted of counts of genetic markers in mouse progeny, and so was binary with many elements in common to a time series. Of interest was change in transmission probability of markers from parents, as a function of marker location down each chromosome. An empirical (unconstrained) regression was found to potentially confound transmission effects with local error effects, and constrained the models sensitivity to very local effects. A better approach was to constrain transmission probabilities to specific states, which allowed nearby markers to have either abrupt changes in state, or no change. Hidden Markov Models confer this structure to data, having some similarity to step functions. The best method of fitting these models to data was shown to be the Baum-Welch algorithm, which allows model flexibility and expansion. It was shown that Hidden Markov Models provided a good fit to the mouse data. The ability to pool information within states provided a better estimate of transmission means and standard error, than is incurred by treating each locus in isolation and attaching a global error. In common to both hierarchical models, was the bringing of some sort of pre-existing information to models. For the random effects model, this was the constraint that random effects were normally distributed, and for the mouse data, that nearby markers may have the same transmission probability. This had the effect of adding to the information in the data, and so improved parameter estimation, and gave models stability. This latter benefit is particularly beneficial to small data sets. The bringing of pre-existing information to a model is implicitly a Bayesian approach to analysis.
2138
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Journal Article
Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with small/large diffusion coefficient
(Society for Industrial and Applied Mathematics, 2019)
Peng, Rui
;
Zhang, Guanghui
;
Zhou, Maolin
In this article, we are concerned with the following eigenvalue problem of a second order linear elliptic operator: -D∆∅ - 2α∇m(x) · ∇∅ + V(x)∅ = λ ∅ in Ω , complemented by a general boundary condition, including Dirichlet boundary condition and Robin boundary condition, ∂∅ ⁄ ∂n + β (x)∅ = 0 on ∂ Ω , where β ∈ C(∂ Ω) is allowed to be positive, sign-changing, or negative, and n(x) is the unit exterior normal to ∂ Ω at x. The domain Ω ⊂ ℝ^N is bounded and smooth, the constants D > 0 and α > 0 are, respectively, the diffusive and advection coefficients, and m ∈ C²(Ω), V ∈ C(Ω) are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient D → 0 or D → ∞ . Our results, together with those of [X. F. Chen and Y. Lou, Indiana Univ. Math. J., 61 (2012), pp. 45-80; A. Devinatz, R. Ellis, and A. Friedman, Indiana Univ. Math. J., 23 (1973/74), pp. 991-1011; and A. Friedman, Indiana U. Math. J., 22 (1973), pp. 1005-1015] where the Neumann boundary case (i.e., β = 0 on ∂ Ω ) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue.
785 1
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Open Access
Conference Publication
Comparison of Two Methods for Calculating 3-Dimensional Ball Spin, and its Application to Soccer Kicking
(International Society of Biomechanics in Sports (ISBS), 2013)
Shorter, Kathleen
;
Smith, Neal
The aim of this study was to select an appropriate computational method for determining both spin axis direction, and spin rate during soccer ball flight. Calculation methods of Cross-Product (CP) and Singular Value Decomposition (SVD) were compared on a stationary spinning ball under laboratory conditions, using data collected from 10 Vicon MX cameras tracking 5mm hemispherical ball markers at 500Hz. When the ball was spun at 371 ± 15 RPM, spin axis orientation appeared close to 'real' values, yet CP showed greater error in RPM estimation. Comparison of the methods during a kick showed no significant difference for spin rate calculation, yet CP underestimated the x and y spin axes, and overestimated spin around the z axis. It was proposed that SVD is used in future to estimate ball spin parameters, especially during kicking where marker occlusion may be more prevalent.
1906 4
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Journal Article
Effects of Large Degenerate Advection and Boundary Conditions on the Principal Eigenvalue and its Eigenfunction of A Linear Second-Order Elliptic Operator
(Indiana University, Department of Mathematics, 2018)
Peng, Rui
;
Zhou, Maolin
In this article, we study, as the coefficient s → ∞, the asymptotic behavior of the principal eigenvalue of the eigenvalue problem
−φ"(x)−2sm′(x)φ′(x)+c(x)φ(x)=λsφ(x), 0 < x < 1,
complemented by a general boundary condition. This problem is relevant to nonlinear propagation phenomena in reaction-diffusion equations. The main point is that the advection (or drift) term m allows natural degeneracy. For instance, m can be constant on [a, b] ⊂ [0, 1]. Depending on the behavior of m near the neighbourhood of the endpoints a and b, the limiting value could be the principal eigenvalue of
−φ"(x)+c(x)φ(x)=λφ(x), a < x < b,
coupled with Dirichlet or Newmann boundary condition at a and b. A complete understanding of the limiting behavior of the principal eigenvalue and its eigenfunction is obtained, and new fundamental effects of large degenerate advection and boundary conditions on the principal eigenvalue and the principal eigenfunction are revealed. In one space dimension, the results in the existing literature are substantially improved.
1529 6
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Conference Publication
Evaluation of a Multiple Calibration Procedure for Scapula Reconstruction
(International Society of Biomechanics, 2013)
Shorter, Kathleen
;
Smith, Neal
;
Lauder, Mike
The ability of an acromion cluster in conjunction with the calibrated anatomical systems technique (CAST) to reconstruct scapula landmarks with increasing arm elevation has been impaired due to poor congruence between the acromion and cluster. The aim of this investigation was to evaluate the accuracy of a multiple calibration procedure (mCAST) adapted from [1] to reconstruct scapula landmarks. Scapula kinematics from eight participants were recorded within the frontal plane, with both the CAST and mCAST methods evaluated using RMSE. RMSE associated with the mCAST method was lower than the CAST method, particularly at higher angles of elevation by up to 0.028m. Therefore, findings from this investigation support the application of mCAST as an alternative method to increase acromion cluster validity.
1756
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Journal Article
Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator
(American Mathematical Society, 2019-12)
Liu, Shuang
;
Lou, Yuan
;
Peng, Rui
;
Zhou, Maolin
We investigate the effect of frequency on the principal eigenvalue of a timeperiodic parabolic operator with Dirichlet, Robin or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the principal eigenvalue with respect to the frequency parameter are established. Our results prove a conjecture raised by Hutson, Michaikow and Poláčik.
783 7
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Journal Article
Multivariate limit of detection for non-linear sensor arrays
(Elsevier BV, 2020-06-15)
Alsaedi, Basim S O
;
McGraw, Christina M
;
Schaerf, Timothy M
;
Dillingham, Peter W
With the increased development of low-cost and miniature devices, sensors are increasingly being deployed as arrays of redundant sensors. However, little work has been done characterizing properties of these arrays. Here, we develop and test a Bayesian algorithm for estimating the limit of detection of sensor arrays. The algorithm is applicable for single sensors as well as sensor arrays, and works by reducing a vector in the signal domain to a univariate response in the measurand domain. We show that the new algorithm can reproduce results from a benchmark algorithm for single sensors, and then demonstrate the benefit of adding additional sensors to an array. Then, we provide guidelines that achieve numerical stability while minimising computational cost. Finally, we provide a real-world example using an array of ion-selective electrodes measuring carbonate in seawater. This application demonstrates how incorporation of a set of individual low-quality sensors into an array leads to a substantially reduced LOD that clearly meets the demands of the application.
1385 5
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Open Access
Conference Publication
On the noise-resolution duality, Heisenberg uncertainty and Shannon's information
(Cambridge University Press, 2015)
Gureyev, Timur
;
de Hoog, F R
;
Nesterets, Yakov
;
Paganin, D M
Several variations of the Heisenberg uncertainty inequality are derived on the basis of 'noise-resolution duality' recently proposed by us. The same approach leads to a related inequality that provides an upper limit for the information capacity of imaging systems in terms of the number of imaging quanta (particles) used in the experiment. These results are useful in the context of biomedical imaging constrained by the radiation dose delivered to the sample, or in imaging (eg., astronomical) problems under low light conditions.
992 2
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Journal Article
Refined estimates for the propagation speed of the transition solution to a free boundary problem with a nonlinearity of combustion type
(Academic Press, 2018-10-05)
Lei, Chengxia
;
Matsuzawa, Hiroshi
;
Peng, Rui
;
Zhou, Maolin
We are concerned with the nonlinear problem ut = uxx+f(u), where f is of combustion type, coupled with the Stefan-type free boundary h(t). According to [4,5], for some critical initial data, the transition solution u locally uniformly converges to θ, which is the ignition temperature off, and the free boundary satisfies h(t) =C√t+o(1)√t for some positive constant C and all large time t. In this paper, making use of two different approaches, we establish more accurate upper and lower bound estimates on h(t) for the transition solution, which suggest that the nonlinearity f can essentially influence the propagation speed.
1546 1
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Journal Article
The Stationary Maxwell-Dirac Equations
(Institute of Physics Publishing Ltd, 2003)
Radford, Christopher John
The Maxwell–Dirac equations are the equations for electronic matter, the 'classical' theory underlying QED. The system combines the Dirac equations with the Maxwell equations sourced by the Dirac current. A stationary Maxwell–Dirac system has ψ = e⁻[iEt],φ with φ independent of t. The system is said to be isolated if the dependent variables obey quite weak regularity and decay conditions. In this paper, we prove the following strong localization result for isolated, stationary Maxwell–Dirac systems,• there are no embedded eigenvalues in the essential spectrum, i.e. −m ≤ E ≤ m;• if |E| < m then the Dirac field decays exponentially as |x| → ∞;• if |E| = m then the system is 'asymptotically' static and decays exponentially if the total charge is non-zero.
823
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Open Access
Journal Article
The third homotopy group as a π₁-module
(Springer, 2015)
Baues, Hans-Joachim
;
Bleile, Beatrice
It is well-known how to compute the structure of the second homotopy group of a space, X, as a module over the fundamental group π₁X, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, π₃X as a module over π₁X. Moreover, we determine π₃X as an extension of π₁X-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces X=S¹Ue²Ue³ consisting of exactly one cell in each dimension ≤ 3.
2398