Comparing methods to estimate the human lens power

Purpose: To compare accuracies of different methods for calculating human lens power when lens thickness is not available. Methods: Lens power was calculated by four methods. Three methods were used with previously published biometry and refraction data of 184 emmetropic and myopic eyes of 184 subjects (age range [18, 63] years, spherical equivalent range [–12.38, +0.75] D). These three methods consist of the Bennett method, which uses lens thickness, our modification of the Stenström method and the Bennett¬Rabbetts method, both of which do not require knowledge of lens thickness. These methods include c constants, which represent distances from lens surfaces to principal planes. Lens powers calculated with these methods were compared with those calculated using phakometry data available for a subgroup of 66 emmetropic eyes (66 subjects). Results: Lens powers obtained from the Bennett method corresponded well with those obtained by phakometry for emmetropic eyes, although individual differences up to 3.5D occurred. Lens powers obtained from the modified¬Stenström and Bennett¬Rabbetts methods deviated significantly from those obtained with either the Bennett method or phakometry. Customizing the c constants improved this agreement, but applying these constants to the entire group gave mean lens power differences of 0.71 ± 0.56D compared with the Bennett method. By further optimizing the c constants, the agreement with the Bennett method was within ± 1D for 95% of the eyes. Conclusion: With appropriate constants, the modified¬Stenström and Bennett¬Rabbetts methods provide a good approximation of the Bennett lens power in emmetropic and myopic eyes.

Mathematical modelling of the drying of sol gel microspheres

This thesis presents a mathematical model of the evaporation of colloidal sol droplets suspended within an atmosphere consisting of water vapour and air. The main purpose of this work is to investigate the causes of the morphologies arising within the powder collected from a spray dryer into which the precursor sol for Synroc™ is sprayed. The morphology is of significant importance for the application to storage of High Level Liquid Nuclear Waste. We begin by developing a model describing the evaporation of pure liquid droplets in order to establish a framework. This model is developed through the use of continuum mechanics and thermodynamic theory, and we focus on the specific case of pure water droplets. We establish a model considering a pure water vapour atmosphere, and then expand this model to account for the presence of an atmospheric gas such as air. We model colloidal particle-particle interactions and interactions between colloid and electrolyte using DLVO Theory and reaction kinetics, then incorporate these interactions into an expression for net interaction energy of a single particle with all other particles within the droplet. We account for the flow of material due to diffusion, advection, and interaction between species, and expand the pure liquid droplet models to account for the presence of these species. In addition, the process of colloidal agglomeration is modelled. To obtain solutions for our models, we develop a numerical algorithm based on the Control Volume method. To promote numerical stability, we formulate a new method of convergence acceleration. The results of a MATLAB™ code developed from this algorithm are compared with experimental data collected for the purposes of validation, and further analysis is done on the sensitivity of the solution to various controlling parameters.

A sequential Monte Carlo algorithm to incorporate model uncertainty in Bayesian sequential design

Here we present a sequential Monte Carlo (SMC) algorithm that can be used for any one-at-a-time Bayesian sequential design problem in the presence of model uncertainty where discrete data are encountered. Our focus is on adaptive design for model discrimination but the methodology is applicable if one has a different design objective such as parameter estimation or prediction. An SMC algorithm is run in parallel for each model and the algorithm relies on a convenient estimator of the evidence of each model which is essentially a function of importance sampling weights. Other methods for this task such as quadrature, often used in design, suffer from the curse of dimensionality. Approximating posterior model probabilities in this way allows us to use model discrimination utility functions derived from information theory that were previously difficult to compute except for conjugate models. A major benefit of the algorithm is that it requires very little problem specific tuning. We demonstrate the methodology on three applications, including discriminating between models for decline in motor neuron numbers in patients suffering from neurological diseases such as Motor Neuron disease.

Change Point Estimation in Monitoring Survival Time

Precise identification of the time when a change in a hospital outcome has occurred enables clinical experts to search for a potential special cause more effectively. In this paper, we develop change point estimation methods for survival time of a clinical procedure in the presence of patient mix in a Bayesian framework. We apply Bayesian hierarchical models to formulate the change point where there exists a step change in the mean survival time of patients who underwent cardiac surgery. The data are right censored since the monitoring is conducted over a limited follow-up period. We capture the effect of risk factors prior to the surgery using a Weibull accelerated failure time regression model. Markov Chain Monte Carlo is used to obtain posterior distributions of the change point parameters including location and magnitude of changes and also corresponding probabilistic intervals and inferences. The performance of the Bayesian estimator is investigated through simulations and the result shows that precise estimates can be obtained when they are used in conjunction with the risk-adjusted survival time CUSUM control charts for different magnitude scenarios. The proposed estimator shows a better performance where a longer follow-up period, censoring time, is applied. In comparison with the alternative built-in CUSUM estimator, more accurate and precise estimates are obtained by the Bayesian estimator. These superiorities are enhanced when probability quantification, flexibility and generalizability of the Bayesian change point detection model are also considered.

Tools and methods for capturing Twitter data during natural disasters

During the course of several natural disasters in recent years, Twitter has been found to play an important role as an additional medium for many–to–many crisis communication. Emergency services are successfully using Twitter to inform the public about current developments, and are increasingly also attempting to source first–hand situational information from Twitter feeds (such as relevant hashtags). The further study of the uses of Twitter during natural disasters relies on the development of flexible and reliable research infrastructure for tracking and analysing Twitter feeds at scale and in close to real time, however. This article outlines two approaches to the development of such infrastructure: one which builds on the readily available open source platform yourTwapperkeeper to provide a low–cost, simple, and basic solution; and, one which establishes a more powerful and flexible framework by drawing on highly scaleable, state–of–the–art technology.

Optimising container processes at multimodal seaport terminals : an integrated approach and application

A Multimodal Seaport Container Terminal (MSCT) is a complex system which requires careful planning and control in order to operate efficiently. It consists of a number of subsystems that require optimisation of the operations within them, as well as synchronisation of machines and containers between the various subsystems. Inefficiency in the terminal can delay ships from their scheduled timetables, as well as cause delays in delivering containers to their inland destinations, both of which can be very costly to their operators. The purpose of this PhD thesis is to use Operations Research methodologies to optimise and synchronise these subsystems as an integrated application. An initial model is developed for the overall MSCT; however, due to a large number of assumptions that had to be made, as well as other issues, it is found to be too inaccurate and infeasible for practical use. Instead, a method of developing models for each subsystem is proposed that then be integrated with each other. Mathematical models are developed for the Storage Area System (SAS) and Intra-terminal Transportation System (ITTS). The SAS deals with the movement and assignment of containers to stacks within the storage area, both when they arrive and when they are rehandled to retrieve containers below them. The ITTS deals with scheduling the movement of containers and machines between the storage areas and other sections of the terminal, such as the berth and road/rail terminals. Various constructive heuristics are explored and compared for these models to produce good initial solutions for large-sized problems, which are otherwise impractical to compute by exact methods. These initial solutions are further improved through the use of an innovative hyper-heuristic algorithm that integrates the SAS and ITTS solutions together and optimises them through meta-heuristic techniques. The method by which the two models can interact with each other as an integrated system will be discussed, as well as how this method can be extended to the other subsystems of the MSCT.

Machine vision toolbox for MATLAB, release 3 [Software]

The Toolbox, combined with MATLAB ® and a modern workstation computer, is a useful and convenient environment for investigation of machine vision algorithms. For modest image sizes the processing rate can be sufficiently ``real-time'' to allow for closed-loop control. Focus of attention methods such as dynamic windowing (not provided) can be used to increase the processing rate. With input from a firewire or web camera (support provided) and output to a robot (not provided) it would be possible to implement a visual servo system entirely in MATLAB. Provides many functions that are useful in machine vision and vision-based control. Useful for photometry, photogrammetry, colorimetry. It includes over 100 functions spanning operations such as image file reading and writing, acquisition, display, filtering, blob, point and line feature extraction, mathematical morphology, homographies, visual Jacobians, camera calibration and color space conversion.

Tests and Models of Non-compositional Concepts

The question of under what conditions conceptual representation is compositional remains debatable within cognitive science. This paper proposes a well developed mathematical apparatus for a probabilistic representation of concepts, drawing upon methods developed in quantum theory to propose a formal test that can determine whether a specific conceptual combination is compositional, or not. This test examines a joint probability distribution modeling the combination, asking whether or not it is factorizable. Empirical studies indicate that some combinations should be considered non-compositionally.

Meeting under the “Omei” Tree in the Torres Strait Islands : networks and funds of knowledge of mathematical ideas

This paper focuses on the turning point experiences that worked to transform the researcher during a preliminary consultation process to seek permission to conduct of a small pilot project on one Torres Strait Island. The project aimed to learn from parents how they support their children in their mathematics learning. Drawing on a community research design, a consultative meeting was held with one Torres Strait Islander community to discuss the possibility of piloting a small project that focused on working with parents and children to learn about early mathematics processes. Preliminary data indicated that parents use networks in their community. It highlighted the funds of knowledge of mathematics that exist in the community and which are used to teach their children. Such knowledges are situated within a community’s unique histories, culture and the voices of the people. “Omei” tree means the Tree of Wisdom in the Island community.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

Use of Bayesian MUNE to show differing rate of loss of motor units in subgroups of ALS

Objectives To evaluate differences among patients with different clinical features of ALS, we used our Bayesian method of motor unit number estimation (MUNE). Methods We performed serial MUNE studies on 42 subjects who fulfilled the diagnostic criteria for ALS during the course of their illness. Subjects were classified into three subgroups according to whether they had typical ALS (with upper and lower motor neurone signs) or had predominantly upper motor neurone weakness with only minor LMN signs, or predominantly lower motor neurone weakness with only minor UMN signs. In all subjects we calculated the half life of MUs, defined as the expected time for the number of MUs to halve, in one or more of the abductor digiti minimi (ADM), abductor pollicis brevis (APB) and extensor digitorum brevis (EDB) muscles. Results The mean half life of MUs was less in subjects who had typical ALS with both upper and lower motor neurone signs than in those with predominantly upper motor neurone weakness or predominantly lower motor neurone weakness. In 18 subjects we analysed the estimated size of the MUs and demonstrated the appearance of large MUs in subjects with upper or lower motor neurone predominant weakness. We found that the appearance of large MUs was correlated with the half life of MUs. Conclusions Patients with different clinical features of ALS have different rates of loss and different sizes of MUs. Significance: These findings could indicate differences in disease pathogenesis.

Modeling ion channel dynamics through reflected stochastic differential equations

Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks. © 2012 American Physical Society.

A new approach to automatically producing schedules for cane railways

The scheduling of locomotive movements on cane railways has proven to be a very complex task. Various optimisation methods have been used over the years to try and produce an optimised schedule that eliminates or minimises bin supply delays to harvesters and the factory, while minimising the number of locomotives, locomotive shifts and cane bins, and also the cane age. This paper reports on a new attempt to develop an automatic scheduler using a mathematical model solved using mixed integer programming and constraint programming approaches and blocking parallel job shop scheduling fundamentals. The model solution has been explored using conventional constraint programming search techniques and found to produce a reasonable schedule for small-scale problems with up to nine harvesters. While more effort is required to complete the development of the full model with metaheuristic search techniques, the work completed to date gives confidence that the metaheuristic techniques will provide near optimal solutions in reasonable time.

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].