" A novel Markov Random Field (MRF) based method for the mosaicing of 3D ultrasound volumes is presented in this dissertation. The motivation for this work is the production of training volumes for an affordable ultrasound simulator, which offers a low-cost/portable training solution for new users of diagnostic ultrasound, by providing the scanning experience essential for developing the necessary psycho-motor skills. It also has the potential for introducing ultrasound instruction into medical education curriculums. The interest in ultrasound training stems in part from the widespread adoption of point-of-care scanners, i.e. low cost portable ultrasound scanning systems in the medical community. This work develops a novel approach for producing 3D composite image volumes and validates the approach using clinically acquired fetal images from the obstetrics department at the University of Massachusetts Medical School (UMMS). Results using the Visible Human Female dataset as well as an abdominal trauma phantom are also presented. The process is broken down into five distinct steps, which include individual 3D volume acquisition, rigid registration, calculation of a mosaicing function, group-wise non-rigid registration, and finally blending. Each of these steps, common in medical image processing, has been investigated in the context of ultrasound mosaicing and has resulted in improved algorithms. Rigid and non-rigid registration methods are analyzed in a probabilistic framework and their sensitivity to ultrasound shadowing artifacts is studied. The group-wise non-rigid registration problem is initially formulated as a maximum likelihood estimation, where the joint probability density function is comprised of the partially overlapping ultrasound image volumes. This expression is simplified using a block-matching methodology and the resulting discrete registration energy is shown to be equivalent to a Markov Random Field. Graph based methods common in computer vision are then used for optimization, resulting in a set of transformations that bring the overlapping volumes into alignment. This optimization is parallelized using a fusion approach, where the registration problem is divided into 8 independent sub-problems whose solutions are fused together at the end of each iteration. This method provided a speedup factor of 3.91 over the single threaded approach with no noticeable reduction in accuracy during our simulations. Furthermore, the registration problem is simplified by introducing a mosaicing function, which partitions the composite volume into regions filled with data from unique partially overlapping source volumes. This mosaicing functions attempts to minimize intensity and gradient differences between adjacent sources in the composite volume. Experimental results to demonstrate the performance of the group-wise registration algorithm are also presented. This algorithm is initially tested on deformed abdominal image volumes generated using a finite element model of the Visible Human Female to show the accuracy of its calculated displacement fields. In addition, the algorithm is evaluated using real ultrasound data from an abdominal phantom. Finally, composite obstetrics image volumes are constructed using clinical scans of pregnant subjects, where fetal movement makes registration/mosaicing especially difficult. Our solution to blending, which is the final step of the mosaicing process, is also discussed. The trainee will have a better experience if the volume boundaries are visually seamless, and this usually requires some blending prior to stitching. Also, regions of the volume where no data was collected during scanning should have an ultrasound-like appearance before being displayed in the simulator. This ensures the trainee's visual experience isn't degraded by unrealistic images. A discrete Poisson approach has been adapted to accomplish these tasks. Following this, we will describe how a 4D fetal heart image volume can be constructed from swept 2D ultrasound. A 4D probe, such as the Philips X6-1 xMATRIX Array, would make this task simpler as it can acquire 3D ultrasound volumes of the fetal heart in real-time; However, probes such as these aren't widespread yet. Once the theory has been introduced, we will describe the clinical component of this dissertation. For the purpose of acquiring actual clinical ultrasound data, from which training datasets were produced, 11 pregnant subjects were scanned by experienced sonographers at the UMMS following an approved IRB protocol. First, we will discuss the software/hardware configuration that was used to conduct these scans, which included some custom mechanical design. With the data collected using this arrangement we generated seamless 3D fetal mosaics, that is, the training datasets, loaded them into our ultrasound training simulator, and then subsequently had them evaluated by the sonographers at the UMMS for accuracy. These mosaics were constructed from the raw scan data using the techniques previously introduced. Specific training objectives were established based on the input from our collaborators in the obstetrics sonography group. Important fetal measurements are reviewed, which form the basis for training in obstetrics ultrasound. Finally clinical images demonstrating the sonographer making fetal measurements in practice, which were acquired directly by the Philips iU22 ultrasound machine from one of our 11 subjects, are compared with screenshots of corresponding images produced by our simulator. "

This thesis investigates stabilisation of the SIMPLE-family discretisations for incompressible flow and their discrete adjoint counterparts. The SIMPLE method is presented from typical \prediction-correction" point of view, but also using a pressure Schur complement approach, which leads to a wider class of schemes. A novel semicoupled implicit solver with velocity coupling is proposed to improve stability. Skewness correction methods are applied to enhance solver accuracy on non-orthogonal grids. An algebraic multi grid linear solver from the HYPRE library is linked to flow and discrete adjoint solvers to further stabilise the computation and improve the convergence rate. With the improved implementation, both of flow and discrete adjoint solvers can be applied to a wide range of 2D and 3D test cases. Results show that the semi-coupled implicit solver is more robust compared to the standard SIMPLE solver. A shape optimisation of a S-bend air flow duct from a VW Golf vehicle is studied using a CAD-based parametrisation for two Reynolds numbers. The optimised shapes and their flows are analysed to con rm the physical nature of the improvement. A first application of the new stabilised discrete adjoint method to a reverse osmosis (RO) membrane channel flow is presented. A CFD model of the RO membrane process with a membrane boundary condition is added. Two objective functions, pressure drop and permeate flux, are evaluated for various spacer geometries such as open channel, cavity, submerged and zigzag spacer arrangements. The flow and the surface sensitivity of these two objective functions is computed and analysed for these geometries. An optimisation with a node-base parametrisation approach is carried out for the zigzag con guration channel flow in order to reduce the pressure drop. Results indicate that the pressure loss can be reduced by 24% with a slight reduction in permeate flux by 0.43%.

Neste trabalho foi estudado o espectro das ondas de MHD ideal em um plasma cilíndrico. Considerando o plasma inomogêneo, o espectro apresenta regiões discretas e contínuas: onda lenta e onda de Alfvén. Os automodos das regiões discretas são as soluções da equação de Hain-Lüst e, nas regiões contínuas, as autofunções apresentam singularidades. Foram determinadas expressões analíticas em termos da função de Bessel que os autovalores da onda de Alfvén apresentam o comportamento discreto. Os modos globais discretos de Alfvén foram calculados usando um equilíbrio da configuração de tokamak, beta < 1, e estes modos apresentam uma pequena contribuição da onda compressional. Foi feita uma análise da propagação da energia da onda por meio do vetor de Poynting e este conhecimento tem a sua importância no problema de aquecimento do plasma por ondas de Alfvén. Foi feita, também, uma análise da polarização das ondas. Foi estudado o espectro da onda devido ao efeito de beta, beta > 1. Esta condição é mais aplicável em problemas de plasma espacial, por exemplo, na fotosfera solar. Foram obtidos, pela primeira vez, os modos discretos de onda lenta, previstos teoricamente. As suas soluções (autofunções e autovalores) possuem também a característica global, aqui denominadas de modos globais discretos da onda lenta. Outro estudo do problema espectral foi realizado para urna configuração de equilíbrio para RFP (\"Reversed Field Pinch\"). Determinou-se, além dos automodos Sturmianos da onda de Alfvén, os automodos anti-Sturmianos da onda lenta. / The spectrum of the ideal MHD waves in cylindrical plasmas has been studied. Assuming non homogeneous plasma, the spectrum presents the discrete and continuum (slow and Alfvén waves) regions. The eigenfunctions of the discrete regions are the solutions of the Hain-Lüst equation. In the continua, the solutions of this equation have singularities. It has been determined analytical expressions for the discrete eigenfunctions. These happen to be the Bessel\'s function and its eigenvalues agree with the numerical calculations. The discrete modes (global modes) of Alfvén waves have been calculated by numerically using the equilibrium configuration of tokamaks, beta < 1, and it was observed that these modes present a small contribution from the compressional waves. An analysis of the energy propagation was done using the Poynting vector. This has its importance in the problem of plasma heating by Alfvén waves. Its polarization was also studied. The effect of beta, through values greater than 1, was also studied. This condition is more aplicable to the spacial plasma problems, for example, in the solar photosphere\'s plasma. It has been calculated, for the first time, the discrete modes of slow waves, proposed theoretically. Its solutions (eigenfunctions and eigenvalues) have also the characteristics of the global modes, called global discrete slow waves. The study of spectral problems for the Reversed Field Pinch configurations was also determined here for both Sturmian eigenmodes for the Alfvén waves and Anti-Sturmian eigenmodes for the slow waves.

Background: Tooth loss has a negative impact on patients’ general health and wellbeing. Dental prostheses can restore oral function, aesthetics and improve oral health related quality of life. Preferences for dental prostheses cannot be fully captured using existing clinical studies and questionnaires. Discrete choice experiment (DCE) is a novel method in health economics to elicit people’s preference for treatments and it allows the researcher to integrate all aspects relevant to treatment into evaluation and measurement of interrelationship between factors. The aim of this PhD thesis is to use a mixed method of DCE and qualitative interviews to analyse dentists and patient’s preferences for dental prosthesis choices in replacing missing teeth. Methods: Discrete choice experiment questionnaires were developed, describing dental prosthdontic treatments in multi-dimensions, including outcome, process and economic factors. Survey and analysis using the questionnaires were conducted with dentists and patients in Edinburgh. Qualitative interviews with Edinburgh dentists and patients were carried out to derive factors to aid the DCE questionnaire design and provide in-depth understanding of DCE results. Systematic reviews were performed to summarise existing evidence on prosthesis evaluation in traditional quantitative studies and perception of prostheses in qualitative interviews. The current application of DCEs in dentistry was also systematically reviewed. Results: Treatment longevity was identified as the most important factor for dentists and patients’ treatment decisions of anterior missing tooth replacements, followed by appearance and chewing function. Dentists put more value on fixation/comfort and treatment procedure than patients. Patients cared about cost of treatment whereas dentists were relatively insensitive. Gender, age and treatment experience significantly influenced patients’ preference for treatment characteristics. Dental implant supported crown was preferred by dentists, whereas patients gave higher utility to traditional prosthodontic treatments. The monetary benefit of fixed dental prostheses ranged from £1856 -£3848 for patients, far exceeding their willingness-to-pay (WTP), which was £120 - £240. Dentists were willing to pay £600-£3000, more than the perceived benefit £503 to £1649. Qualitative study identified the above factors and provided interpretation of DCE results. Problems in the dental care system related to referral and training for dental implant treatments were raised. Discussion: This thesis is the first DCE application in dentistry evaluating and comparing dentists and patients preferences for missing tooth replacements. Dentists and patients’ preferences were elicited qualitatively and qualitatively integrating multidimensional factors. Patients’ preference for treatments, monetary benefit and WTP were demonstrated to be different from dentists’. Treatment benefits exceeded patients WTP for fixed dental prostheses.

Quantile regression (QR) (Koenker and Bassett, 1978), as a comprehensive extension to standard mean regression, has been steadily promoted from both theoretical and applied aspects. Bayesian quantile regression (BQR), which deals with unknown parameter estimation and model uncertainty, is a newly proposed tool of QR. This thesis aims to make some novel contributions to the following three issues related to QR. First, whereas QR for continuous responses has received much attention in literatures, QR for discrete responses has received far less attention. Second, conventional QR methods often show that QR curves crossing lead to invalid distributions for the response. In particular, given a set of covariates, it may turn out, for example, that the predicted 95th percentile of the response is smaller than the 90th percentile for some values of the covariates. Third, mean-based clustering methods are widely developed, but need improvements to deal with clustering extreme-type, heavy tailed-type or outliers problems. This thesis focuses on methods developed over these three challenges: modelling quantile regression with discrete responses, ensuring non-crossing quantile curves for any given sample and modelling tails for collinear data with outliers. The main contributions are listed as below: * The first challenge is studied in Chapter 2, in which a general method for Bayesian inference of regression models beyond the mean with discrete responses is developed. In particular, this method is developed for both Bayesian quantile regression and Bayesian expectile regression. This method provides a direct Bayesian approach to these regression models with a simple and intuitive interpretation of the regression results. The posterior distribution under this approach is shown to not only be coherent to the response variable, irrespective of its true distribution, but also proper in relation to improper priors for unknown model parameters. * Chapter 3 investigates a new kernel-weighted likelihood smoothing quantile regression method. The likelihood is based on a normal scale-mixture representation of an asymmetric Laplace distribution (ALD). This approach benefits of the same good design adaptation just as the local quantile regression (Spokoiny et al., 2014) does and ensures non-crossing quantile curves for any given sample. * In Chapter 4, we introduce an asymmetric Laplace distribution to model the response variable using profile regression, a Bayesian non-parametric model for clustering responses and covariates simultaneously. This development allows us to model more accurately for clusters which are asymmetric and predict more accurately for extreme values of the response variable and/or outliers. In addition to the three major aforementioned challenges, this thesis also addresses other important issues such as smoothing extreme quantile curves and avoiding insensitive to heteroscedastic errors as well as outliers in the response variable. The performances of all the three developments are evaluated via both simulation studies and real data analysis.

L'objectif de ce travail de recherche est le développement d'un outil numérique capable de simuler le comportement d'infrastructures sensibles soumises à des charges dynamiques extrêmes sous l'effet d'aléas naturels ou humains tels que les impacts aériens. L'étude propose pour ce faire une modélisation 3D par éléments discrets, capable de décrire des états de destructions avancés en obtenant des macro-fissures et des fragments réalistes grâce à la nature discontinue du modèle.Dans un premier temps, on a étudié de manière exhaustive l'influence des paramètres de création du maillage sur les caractéristiques dudit maillage et sur le comportement macroscopique du béton. Ensuite, on a transformé le modèle de charge dynamique en une modélisation plus réaliste de l'énergie de rupture dynamique en contrôlant l'augmentation de distance limite maximum à l'interaction. Par ailleurs, on a défini une condition de ratio entre la taille des éléments discrets et celle des éléments finis pour s'assurer d'un contact correct entre eux.La procédure d'identification des paramètres du modèle est réalisée en simulation numérique avec des essais en laboratoire: Compression et traction quasi-statiques, essai tri-axial à haut confinement, écaillage dynamique. Enfin, la fiabilité de l'approche est vérifiée sur trois essais différents d'impact violent:1) Perforation et pénétration de projectiles à tête ogivale dans des cibles cylindriques confinées (CEA-Gramat)2) Essais d'impact tranchant de projectile avec une géométrie homothétique particulière sur des dalles en béton (Erzar) / The objective of this work is development of a numerical tool capable to simulate sensitive infrastructures subjected to severe dynamic loadings due to natural or manmade hazards, such as aircrafts impacts. This study proposes a 3D discrete element method able to predict advance damage states obtaining realistic macro-cracks and materials fragments due to its discontinue nature.We thoroughly studied the influence of mesh creation parameters on the mesh characteristics and on the macroscopic concrete behaviour. Then, we modified the constitutive model for dynamic loading with a more realistic modelling of the dynamic fracture energy by controlling the increase of the maximum distance limit at the interaction scale. In addition, we defined a condition for ratio between the size of finite and discrete elements for proper contact between them.The identification procedure for the parameters of the constitutive model is analysed with simulation on laboratory test: quasi-static compression and tension, high confinement triaxial and dynamic spalling. Finally, the reliability of our approach is verified on three different types of hard impact test: 1) perforation and penetration of ogive-nosed projectiles on confined cylindrical targets (CEA-Gramat); 2) edge-on impact tests of projectiles with a particular homothetic geometry on concrete tiles (Erzar); 3) drop-weight impact on a reinforced concrete beam (University of Toronto).

The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or a 5 x 4 puzzle, respectively. We consider the possibilities when we delete a color to make the game a 3 × 3 puzzle and when we add a color, making the game a 5 × 5 puzzle. Finally, we determine if solution two is a permutation of solution one.

abstract: In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique. This thesis explores one possible construction (originally due to Hunt) in depth and uses it to produce arithmetic lattices, non-arithmetic lattices, and thin subgroups in SU(2,1). / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

The hypercube Qn is the graph whose vertices are the ordered n-tuples of zeros and ones, where two vertices are adjacent iff they differ in exactly one coordinate. A partial edge coloring f of a graph G is a mapping from a subset of edges of G to a set of colors; it is called proper if no pair of adjacent edges share the same color. A (possibly partial and unproper) coloring f is avoidable if there exists a proper coloring g such that no edge has the same color under f and g. An unavoidable coloring h is called minimal if it would be avoidable by letting any colored edge turn noncolored. We construct a computer program to find all minimal unavoidable edge colorings of Q3 using up to 3 colors, and draw some conclusions for general Qn.

The Bernoulli distribution is a basic, well-studied distribution in probability. In this thesis, we will consider repeated Bernoulli trials in order to study runs of identical outcomes. More formally, for t ∈ N, we let Xt ∼ Bernoulli(p), where p is the probability of success, q = 1 − p is the probability of failure, and all Xt are independent. Then Xt gives the outcome of the tth trial, which is 1 for success or 0 for failure. For n, m ∈ N, we define Tn to be the number of trials needed to first observe n consecutive successes (where the nth success occurs on trial XTn ). Likewise, we define Tn,m to be the number of trials needed to first observe either n consecutive successes or m consecutive failures. We shall primarily focus our attention on calculating E[Tn] and E[Tn,m]. Starting with the simple cases of E[T2] and E[T2,2], we will use a variety of techniques, such as counting arguments and Markov chains, in order to derive the expectations. When possible, we shall also provide closed-form expressions for the probability mass function, cumulative distribution function, variance, and other values of interest. Eventually we will work our way to general formulas for E[Tn] and E[Tn,m]. We will also derive formulas for conditional averages, and discuss how famous results from probability such as Wald’s Identity apply to our problem. Numerical examples will also be given in order to supplement the discussion and clarify the results.