<p>We study bifurcations of periodic travelling waves in granular dimer chains from the anti-continuum limit, when the mass ratio between the light and heavy beads tends to zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter and the periodic waves with the wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic travelling waves of granular monomer chains exist.</p> / Master of Science (MSc)

In this dissertation we introduce two variable selection procedures for multivariate responses. Our procedures are based on sufficient dimension reduction concepts and are model-free. In the first procedure we consider the dual marginal coordinate hypotheses, where the role of the predictor and the response is not important. Motivated by canonical correlation analysis (CCA), we propose a CCA-based test for the dual marginal coordinate hypotheses, and devise a joint backward selection algorithm for dual model-free variable selection. The second procedure is based on ordinary least squares (OLS). We derive and study the asymptotic properties of the OLS-based test under the normality assumption of the predictors as well as an asymmetry assumption. When these assumptions are violated, the asymptotic test with elliptical trimming and clustering is still valid with desirable numerical performances. A backward selection algorithm for the predictor is also provided for the OLS-based test. The performances of the proposed tests and the variable selection procedures are evaluated through synthetic examples and a real data analysis. / Statistics

[EN] Mathematical models based on deterministic differential equations do not take into account the inherent uncertainty of the physical phenomenon (in a wide sense) under study. In addition, inaccuracies in the collected data often arise due to errors in the measurements. It thus becomes necessary to treat the input parameters of the model as random quantities, in the form of random variables or stochastic processes. This gives rise to the study of random ordinary and partial differential equations. The computation of the probability density function of the stochastic solution is important for uncertainty quantification of the model output. Although such computation is a difficult objective in general, certain stochastic expansions for the model coefficients allow faithful representations for the stochastic solution, which permits approximating its density function. In this regard, Karhunen-Loève and generalized polynomial chaos expansions become powerful tools for the density approximation. Also, methods based on discretizations from finite difference numerical schemes permit approximating the stochastic solution, therefore its probability density function. The main part of this dissertation aims at approximating the probability density function of important mathematical models with uncertainties in their formulation. Specifically, in this thesis we study, in the stochastic sense, the following models that arise in different scientific areas: in Physics, the model for the damped pendulum; in Biology and Epidemiology, the models for logistic growth and Bertalanffy, as well as epidemiological models; and in Thermodynamics, the heat partial differential equation. We rely on Karhunen-Loève and generalized polynomial chaos expansions and on finite difference schemes for the density approximation of the solution. These techniques are only applicable when we have a forward model in which the input parameters have certain probability distributions already set. When the model coefficients are estimated from collected data, we have an inverse problem. The Bayesian inference approach allows estimating the probability distribution of the model parameters from their prior probability distribution and the likelihood of the data. Uncertainty quantification for the model output is then carried out using the posterior predictive distribution. In this regard, the last part of the thesis shows the estimation of the distributions of the model parameters from experimental data on bacteria growth. To do so, a hybrid method that combines Bayesian parameter estimation and generalized polynomial chaos expansions is used. / [ES] Los modelos matemáticos basados en ecuaciones diferenciales deterministas no tienen en cuenta la incertidumbre inherente del fenómeno físico (en un sentido amplio) bajo estudio. Además, a menudo se producen inexactitudes en los datos recopilados debido a errores en las mediciones. Por lo tanto, es necesario tratar los parámetros de entrada del modelo como cantidades aleatorias, en forma de variables aleatorias o procesos estocásticos. Esto da lugar al estudio de las ecuaciones diferenciales aleatorias. El cálculo de la función de densidad de probabilidad de la solución estocástica es importante en la cuantificación de la incertidumbre de la respuesta del modelo. Aunque dicho cálculo es un objetivo difícil en general, ciertas expansiones estocásticas para los coeficientes del modelo dan lugar a representaciones fieles de la solución estocástica, lo que permite aproximar su función de densidad. En este sentido, las expansiones de Karhunen-Loève y de caos polinomial generalizado constituyen herramientas para dicha aproximación de la densidad. Además, los métodos basados en discretizaciones de esquemas numéricos de diferencias finitas permiten aproximar la solución estocástica, por lo tanto, su función de densidad de probabilidad. La parte principal de esta disertación tiene como objetivo aproximar la función de densidad de probabilidad de modelos matemáticos importantes con incertidumbre en su formulación. Concretamente, en esta memoria se estudian, en un sentido estocástico, los siguientes modelos que aparecen en diferentes áreas científicas: en Física, el modelo del péndulo amortiguado; en Biología y Epidemiología, los modelos de crecimiento logístico y de Bertalanffy, así como modelos de tipo epidemiológico; y en Termodinámica, la ecuación en derivadas parciales del calor. Utilizamos expansiones de Karhunen-Loève y de caos polinomial generalizado y esquemas de diferencias finitas para la aproximación de la densidad de la solución. Estas técnicas solo son aplicables cuando tenemos un modelo directo en el que los parámetros de entrada ya tienen determinadas distribuciones de probabilidad establecidas. Cuando los coeficientes del modelo se estiman a partir de los datos recopilados, tenemos un problema inverso. El enfoque de inferencia Bayesiana permite estimar la distribución de probabilidad de los parámetros del modelo a partir de su distribución de probabilidad previa y la verosimilitud de los datos. La cuantificación de la incertidumbre para la respuesta del modelo se lleva a cabo utilizando la distribución predictiva a posteriori. En este sentido, la última parte de la tesis muestra la estimación de las distribuciones de los parámetros del modelo a partir de datos experimentales sobre el crecimiento de bacterias. Para hacerlo, se utiliza un método híbrido que combina la estimación de parámetros Bayesianos y los desarrollos de caos polinomial generalizado. / [CA] Els models matemàtics basats en equacions diferencials deterministes no tenen en compte la incertesa inherent al fenomen físic (en un sentit ampli) sota estudi. A més a més, sovint es produeixen inexactituds en les dades recollides a causa d'errors de mesurament. Es fa així necessari tractar els paràmetres d'entrada del model com a quantitats aleatòries, en forma de variables aleatòries o processos estocàstics. Açò dóna lloc a l'estudi de les equacions diferencials aleatòries. El càlcul de la funció de densitat de probabilitat de la solució estocàstica és important per a quantificar la incertesa de la sortida del model. Tot i que, en general, aquest càlcul és un objectiu difícil d'assolir, certes expansions estocàstiques dels coeficients del model donen lloc a representacions fidels de la solució estocàstica, el que permet aproximar la seua funció de densitat. En aquest sentit, les expansions de Karhunen-Loève i de caos polinomial generalitzat esdevenen eines per a l'esmentada aproximació de la densitat. A més a més, els mètodes basats en discretitzacions mitjançant esquemes numèrics de diferències finites permeten aproximar la solució estocàstica, per tant la seua funció de densitat de probabilitat. La part principal d'aquesta dissertació té com a objectiu aproximar la funció de densitat de probabilitat d'importants models matemàtics amb incerteses en la seua formulació. Concretament, en aquesta memòria s'estudien, en un sentit estocàstic, els següents models que apareixen en diferents àrees científiques: en Física, el model del pèndol amortit; en Biologia i Epidemiologia, els models de creixement logístic i de Bertalanffy, així com models de tipus epidemiològic; i en Termodinàmica, l'equació en derivades parcials de la calor. Per a l'aproximació de la densitat de la solució, ens basem en expansions de Karhunen-Loève i de caos polinomial generalitzat i en esquemes de diferències finites. Aquestes tècniques només són aplicables quan tenim un model cap avant en què els paràmetres d'entrada tenen ja determinades distribucions de probabilitat. Quan els coeficients del model s'estimen a partir de les dades recollides, tenim un problema invers. L'enfocament de la inferència Bayesiana permet estimar la distribució de probabilitat dels paràmetres del model a partir de la seua distribució de probabilitat prèvia i la versemblança de les dades. La quantificació de la incertesa per a la resposta del model es fa mitjançant la distribució predictiva a posteriori. En aquest sentit, l'última part de la tesi mostra l'estimació de les distribucions dels paràmetres del model a partir de dades experimentals sobre el creixement de bacteris. Per a fer-ho, s'utilitza un mètode híbrid que combina l'estimació de paràmetres Bayesiana i els desenvolupaments de caos polinomial generalitzat. / This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P. / Calatayud Gregori, J. (2020). Computational methods for random differential equations: probability density function and estimation of the parameters [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138396 / Premios Extraordinarios de tesis doctorales

<p dir="ltr">Deep neural networks (DNNs) have demonstrated outstanding performance in numerous tasks such as image recognition and natural language processing. However, in dynamic systems modeling, the tasks of estimating and uncovering the potentially nonlinear structure of systems represented by ordinary differential equations (ODEs) pose a significant challenge. In this dissertation, we employ DNNs to enable precise and efficient parameter estimation of dynamic systems. In addition, we introduce a highly flexible neural ODE model to capture both nonlinear and sparse dependent relations among multiple functional processes. Nonetheless, DNNs are susceptible to overfitting and often struggle to accurately assess predictive uncertainty despite their widespread success across various AI domains. The challenge of defining meaningful priors for DNN weights and characterizing predictive uncertainty persists. In this dissertation, we present a novel neural adaptive empirical Bayes framework with a new class of prior distributions to address weight uncertainty.</p><p dir="ltr">In the first part, we propose a precise and efficient approach utilizing DNNs for estimation and inference of ODEs given noisy data. The DNNs are employed directly as a nonparametric proxy for the true solution of the ODEs, eliminating the need for numerical integration and resulting in significant computational time savings. We develop a gradient descent algorithm to estimate both the DNNs solution and the parameters of the ODEs by optimizing a fidelity-penalized likelihood loss function. This ensures that the derivatives of the DNNs estimator conform to the system of ODEs. Our method is particularly effective in scenarios where only a set of variables transformed from the system components by a given function are observed. We establish the convergence rate of the DNNs estimator and demonstrate that the derivatives of the DNNs solution asymptotically satisfy the ODEs determined by the inferred parameters. Simulations and real data analysis of COVID-19 daily cases are conducted to show the superior performance of our method in terms of accuracy of parameter estimates and system recovery, and computational speed.</p><p dir="ltr">In the second part, we present a novel sparse neural ODE model to characterize flexible relations among multiple functional processes. This model represents the latent states of the functions using a set of ODEs and models the dynamic changes of these states utilizing a DNN with a specially designed architecture and sparsity-inducing regularization. Our new model is able to capture both nonlinear and sparse dependent relations among multivariate functions. We develop an efficient optimization algorithm to estimate the unknown weights for the DNN under the sparsity constraint. Furthermore, we establish both algorithmic convergence and selection consistency, providing theoretical guarantees for the proposed method. We illustrate the efficacy of the method through simulation studies and a gene regulatory network example.</p><p dir="ltr">In the third part, we introduce a class of implicit generative priors to facilitate Bayesian modeling and inference. These priors are derived through a nonlinear transformation of a known low-dimensional distribution, allowing us to handle complex data distributions and capture the underlying manifold structure effectively. Our framework combines variational inference with a gradient ascent algorithm, which serves to select the hyperparameters and approximate the posterior distribution. Theoretical justification is established through both the posterior and classification consistency. We demonstrate the practical applications of our framework through extensive simulation examples and real-world datasets. Our experimental results highlight the superiority of our proposed framework over existing methods, such as sparse variational Bayesian and generative models, in terms of prediction accuracy and uncertainty quantification.</p>

Infectious diseases caused by bacteria, fungi, viruses and parasites have affected humans historically. Infectious diseases remain a major cause of premature death and a public health concern globally with increased mortality and significant economic burden. Unvaccinated individuals, people with suppressed and compromised immune systems are at higher risk of suffering from infectious diseases. In spite of significant advancements in infectious diseases research, the control or treatment process faces challenges. The mucosal immune system plays a crucial role in safeguarding the body from harmful pathogens, while being constantly exposed to the environment. To develop treatment options for infectious diseases, it is vital to understand the immune responses that occur during infection. The two infectious diseases presented here are: i) Helicobacter pylori infection and ii) human immunodeficiency (HIV) and human papillomavirus (HPV) co-infection. H pylori, is a bacterium that colonizes the stomach and causes gastric cancer in 1-2% but is beneficial for protection against allergies and gastroesophageal diseases. An estimated 85% of H pylori colonized individuals show no detrimental effects. HIV is a virus that causes AIDS, one of the deadliest and most persistent epidemics. HIV-infected patients are at an increased risk of co-infection with HPV, and report an increased incidence of oral cancer. The goal of this thesis is to elucidate the host immune responses in infectious diseases via the use of computational and mathematical models. First, the thesis reviews the need for computational and mathematical models to study the immune responses in the course of infectious diseases. Second, it presents a novel sensitivity analysis method that identifies important parameters in a hybrid (agent-based/equation-based) model of H. pylori infection. Third, it introduces a novel model representing the HIV/HPV coinfection and compares the simulation results with a clinical study. Fourth, it discusses the need of advanced modeling technologies to achieve a personalized systems wide approach and the challenges that can be encountered in the process. Taken together, the work in this dissertation presents modeling approaches that could lead to the identification of host immune factors in infectious diseases in a predictive and more resource-efficient manner. / Doctor of Philosophy / Infectious diseases caused by bacteria, fungi, viruses and parasites have affected humans historically. Infectious diseases remain a major cause of premature death and a public health concern globally with increased mortality and significant economic burden. These infections can occur either via air, travel to at-risk places, direct person-to-person contact with an infected individual or through water or fecal route. Unvaccinated individuals, individuals with suppressed and compromised immune system such as that in HIV carriers are at higher risk of getting infectious diseases. In spite of significant advancements in infectious diseases research, the control and treatment of these diseases faces numerous challenges. The mucosal immune system plays a crucial role in safeguarding the body from harmful pathogens, while being exposed to the environment, mainly food antigens. To develop treatment options for infectious diseases, it is vital to understand the immune responses that occur during infection. In this work, we focus on gut immune system that acts like an ecosystem comprising of trillions of interacting cells and molecules, including membars of the microbiome. The goal of this dissertation is to develop computational models that can simulate host immune responses in two infectious diseases- i) Helicobacter pylori infection and ii) human immunodeficiency virus (HIV)-human papilloma virus (HPV) co-infection. Firstly, it reviews the various mathematical techniques and systems biology based methods. Second, it introduces a "hybrid" model that combines different mathematical and statistical approaches to study H. pylori infection. Third, it highlights the development of a novel HIV/HPV coinfection model and compares the results from a clinical trial study. Fourth, it discusses the challenges that can be encountered in adapting machine learning based computational technologies. Taken together, the work in this dissertation presents modeling approaches that could lead to the identification of host immune factors in infectious diseases in a predictive and more resourceful way.

The Forest Vegetation Simulator (FVS) is a growth and yield modeling system widely-used for predicting stand and tree-level attributes for management and planning applications in North American forests. The accuracy of FVS predictions for a range of tree and stand level attributes depends a great deal on the performance of the diameter increment model and its predictions of change in diameter at breast height (DBH) over time. To address the challenge of predicting growth in highly variable and geographically expansive forest systems, FVS was designed to include an internal calibration algorithm that makes use of growth observations, when available, from permanent inventory plots. The basic idea is that observed growth rates on a collection of remeasured trees are used to adjust or "calibrate" FVS diameter growth predictions. Therefore, DBH modeling was the focus of this investigation. Five methods were proposed for local calibration of individual tree DBH growth predictions and compared to two sets of results generated without calibration. Data from the US Forest Service's Forest Inventory and Analysis (FIA) program were used to test the methods for eleven widely-distributed forest tree species in Virginia. Two calibration approaches were based on median prediction errors from locally-observed DBH increments spanning a five year average time interval. Two were based on simple linear regression models fitted to the locally-observed prediction errors, and one method employed a mixed effects regression model with a random intercept term estimated from locally-observed DBH increments. Data witholding, specifically a leave-one-out cross-validation was used to compare results of the methods tested. Results showed that any of the calibration approaches tested in general led to improved accuracy of DBH growth predictions, with either of the median-based methods or regression based methods performing better than the random-effects-based approach. Equivalence testing showed that median or regression-based local calibration methods met error tolerances within ± 12% of observed DBH increments for all species with the random effects approach meeting a larger tolerance of ± 17%. These results showed improvement over uncalibrated models, which failed to meet tolerances as high as ± 30% for some species in a newly-fitted DBH growth model for Virginia, and as high as ± 170% for an existing model fitted to data from a much larger region of the Southeastern United States. Local calibration of regional DBH increment models provides an effective means of substantially reducing prediction errors when a relatively small set of observations are available from local sources such as permanent forest inventory plots, or the FIA database. / MS / The Forest Vegetation Simulator (FVS) is a growth and yield model widely-used for predicting stand dynamics, management and decision support in North American forests. Diameter increment is a major component in modeling tree growth. The system of integrated analytical tools in FVS is primarily based on the performance of the diameter increment model and the subsequent use of predicted in diameter at breast height (DBH) over time in forecasting tree attributes. To address the challenge of predicting growth in highly variable and geographically expansive forest systems, FVS was designed to include an internal calibration algorithm that makes use of growth observations, when available, from permanent inventory plots. The basic idea was that observed growth rates on a small set of remeasured trees are used to adjust or “calibrate” FVS growth predictions. The FVS internal calibration was the subject being investigated here. Five alternative methods were proposed attributed to a specific site or stand of interest and compared to two sets of results, which were based on median prediction errors, generated without calibration. Results illustrated that median-based methods or regression based methods performed better than the random-effects-based approach using independently observed growth data from Forest Service FIA re-measurements in Virginia. Local calibration of regional DBH increment models provides an effective means of substantially reducing prediction errors. The results of this study should also provide information to evaluate the efficiency of FVS calibration alternatives and a possible method for future implementation.

In this thesis we construct a new model of the immune response to the growth of breast cancer cells and investigate the impact of certain drug therapies on the cancer. We use delay differential equations to model the interaction of breast cancer cells with the immune system. The new model is constructed by combining two previous models. The first model accounts for different cell cycles and includes terms to evaluate drug treatments, but ignores quiescent tumor cells. The second model includes quiescent cells, but ignores the immune response and drug treatments. The new model is obtained by combining and modifying these two models to account for quiescent cells, immune cells and includes drug intervention terms. This new model is used to evaluate the effects of pulsed applications of the drug Paclitaxel for models with and without quiescent cells. We use sensitivity equation methods to analyze the sensitivity of the model with respect to the initial number of immune cytotoxic T-cells. Numerical experiments are conducted to compare the model predictions to observed behavior. / Master of Science

In this dissertation, I examine the Sonoran Desert trilogy by Kathleen Alcalá through the lens of post-oppositional theory as developed by AnaLouise Keating. Moving beyond the use of post-oppositional theory to analyze non-fiction works, I apply this theory instead to the fiction of Kathleen Alcalá—whose work appears in such anthologies as The Norton Anthology of Latino Literature. Alcalá, though well published, is underrepresented in contemporary literary criticism, as can be seen by the only eight entries under her name in the MLA International Bibliography. Therefore, I have chosen her most significant fiction work, her trilogy about the Sonoran Desert, as the perfect text upon which to map post-oppositional theory. Through analysis of her three novels, I show that her work is an ideal example of post-oppositionality in action and that her characters act as post-oppositional revolutionaries and prophets within the pages of the text. The first chapter outlines the parameters of the project. In Chapter 2, I argue that post-oppositionality can be seen in Alcalá through gender bending, looking at the characters of Membrillo and Manzana, Corey, and Rosalinda. In Chapter 3, I argue that the characters of Estela, La Señorita, and Magdalena are enacting post-oppositionality through their transcendence of traditional women's roles in sexuality. In Chapter 4, I argue that the female characters of the novels act as revolutionaries through their political and social agency—reaching out to other characters through such work as educating and writing. In Chapters 5 and 6, I feature my interviews with Alcalá and Keating, who were generous enough to speak with me over Zoom during lockdown. Finally, in the conclusion chapter, Chapter 7, I examine how post-oppositionality in the novels prepares the reader for post-oppositional action in reality. Throughout all of these chapters, I rely on other theories and historiographies such as gender theory (Judith Butler, Foucault, West and Zimmerman, etc.), the history of women's sexuality, and the roles of women in nineteenth century Mexico (looking especially at the works of Nancy LaGreca and Anna Macías).

The main purpose of this study is to empirically assess the relationship between inflation and economic growth in Ethiopia using quarterly dataset from 1992Q1 to 2010Q4. In doing so, an interesting policy issue arises. What is the threshold level of inflation for the Ethiopian economy? Based on the Engle-Granger and Johansen co-integration tests it is found out that there is a positive long-run relationship between inflation and economic growth. The error correction models show that in cases of short-run disequilibrium, the inflation model adjusts itself to its long-run path correcting roughly 40% of the imbalance in each quarter. In addition, based on the conditional least square technique, the estimated threshold model suggests 10% as the optimal level of inflation that facilitates growth. An inflation level higher or lower than the threshold level of inflation affects the economic growth negatively and hence fiscal and monetary policy coordination is vital to keep inflation at the threshold. / Economics / M. Com. (Economics)

Interval analysis is an essential tool in the construction of validated numerical solutions of Initial Value Problems (IVP) for Ordinary (ODE) and Partial (PDE) Differential Equations. A validated solution typically consists of guaranteed lower and upper bounds for the exact solution or set of exact solutions in the case of uncertain data, i.e. it is an interval function (enclosure) containing all solutions of the problem. IVP for ODE: The central point of discussion is the wrapping effect. A new concept of wrapping function is introduced and applied in studying this effect. It is proved that the wrapping function is the limit of the enclosures produced by any method of certain type (propagate and wrap type). Then, the wrapping effect can be quantified as the difference between the wrapping function and the optimal interval enclosure of the solution set (or some norm of it). The problems with no wrapping effect are characterized as problems for which the wrapping function equals the optimal interval enclosure. A sufficient condition for no wrapping effect is that there exist a linear transformation, preserving the intervals, which reduces the right-hand side of the system of ODE to a quasi-isotone function. This condition is also necessary for linear problems and "near" necessary in the general case. Hyperbolic PDE: The Initial Value Problem with periodic boundary conditions for the wave equation is considered. It is proved that under certain conditions the problem is an operator equation with an operator of monotone type. Using the established monotone properties, an interval (validated) method for numerical solution of the problem is proposed. The solution is obtained step by step in the time dimension as a Fourier series of the space variable and a polynomial of the time variable. The numerical implementation involves computations in Fourier and Taylor functoids. Propagation of discontinuo~swaves is a serious problem when a Fourier series is used (Gibbs phenomenon, etc.). We propose the combined use of periodic splines and Fourier series for representing discontinuous functions and a method for propagating discontinuous waves. The numerical implementation involves computations in a Fourier hyper functoid. / Mathematical Sciences / D. Phil. (Mathematics)