This dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three separate papers. In Chapter 2 we consider gradient descent equations for energy functionals of the type [mathematical equation] where A is a second-order uniformly elliptic operator with smooth coefficients. We consider the gradient descent equation for S, where the gradient is an element of the Sobolev space H[superscipt beta], [beta is an element of](0, 1), with a metric that depends on A and a positive number [gamma] > sup |V₂₂|. The main result of Chapter 2 is a weak comparison principle for such a gradient flow. We extend our methods to the case where A is a fractional power of an elliptic operator, and we provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional. In Chapter 3 we investigate the differentiability of the minimal average energy associated to the functionals [mathematical equation] using numerical and perturbation methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the minimal average energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter [epsilon], and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series. In Chapter 4 we present a method for determining the stability of a class of stochastically forced ordinary differential equations, where the forcing term can be obtained by passing white noise through a filter of arbitrarily high degree. We use the Fokker-Planck equation to write a partial differential equation for the second moments, which we turn into an eigenvalue problem for a second-order differential operator. We develop ladder operators to determine analytic expressions for the eigenvalues and eigenfunctions of this differential operator, and thus determine the stability. / text

Fractional calculus has been used as a research tool in the fields of pharmacology, biology, chemistry, and other areas [3]. The main purpose of this thesis is to calculate Green's functions of fractional difference equations, and to model problems in pharmacokinetics. We claim that the discrete fractional calculus yields the best prediction performance compared to the continuous fractional calculus in the application of a one-compartmental model of drug concentration. In Chapter 1, the Gamma function and its properties are discussed to establish a theoretical basis. Additionally, the basics of discrete fractional calculus are discussed using particular examples for further calculations. In Chapter 2, we use these basic results in the analysis of a linear fractional difference equation. Existence of solutions to this difference equation is then established for both initial conditions (IVP) and two-point boundary conditions (BVP). In Chapter 3, Green's functions are introduced and discussed, along with examples. Instead of using Cauchy functions, the technique of finding Green's functions by a traditional method is demonstrated and used throughout this chapter. The solutions of the BVP play an important role in analysis and construction of the Green's functions. Then, Green's functions for the discrete calculus case are calculated using particular problems, such as boundary value problems, discrete boundary value problems (DBVP) and fractional boundary value problems (FBVP). Finally, we demonstrate how the Green's functions of the FBVP generalize the existence results of the Green's functions of DVBP. In Chapter 4, different compartmental pharmacokinetic models are discussed. This thesis limits discussion to the one-compartmental model. The Mathematica FindFit command and the statistical computational techniques of mean square error (MSE) and cross-validation are discussed. Each of the four models (continuous, continuous fractional, discrete and discrete fractional) is used to compute the MSE numerically with the given data of drug concentration. Then, the best fit and the best model are obtained by inspection of the resulting MSE. In the last Chapter, the results are summarized, conclusions are drawn, and directions for future work are stated.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The main objective of this dissertation is to establish the basic results of the Averaging Theory for Ordinary Differential Equations and used them in some problems of Classical Mechanics. Two mechanical problems are given here. The first one is about the existence and stability of periodic solutions in the mathematical pendulum with dissipation. The second one is about the existence of periodic solutions of regularized Hill lunar problem with comes from Celestial Mechanics / O principal objetivo desta dissertação é estabelecer resultados básicos da Teoria da Média para Equações Diferenciais Ordinárias e aplicá-los em alguns problemas de Mecânica Clássica. Estudaremos dois problemas de mecânica. O primeiro versa sobre existência e estabilidade de soluções periódicas no pêndulo com dissipação. O segundo consiste em estudar a existência de soluções periódicas do problema lunar de Hill regularizado proveniente da Mecânica Celeste. / Mestre em Matemática

Orientador: Laécio Carvalho de Barros / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T10:25:47Z (GMT). No. of bitstreams: 1 Menegotto_Juliana_M.pdf: 2478347 bytes, checksum: d968a14f6bf0c7261143ea6a2e25e99a (MD5) Previous issue date: 2011 / Resumo: Para estudar a concentração do fármaco no organismo utiliza-se modelos farmacocinéticos multicompartimentais que, via de regra, são dados por um sistema de equações diferenciais ordinárias (EDO). Neste trabalho propomos um modelo para descrever a dinâmica da concentração a partir de um sistema baseado em regras fuzzy. Para obter a curva da concentração, utilizamos o método de Takagi-Sugeno-Kang (TSK) e as curvas via TSK e EDO são comparadas. Quando o fármaco é administrado em doses múltiplas, o índice de acúmulo no organismo é avaliado através da razão entre as áreas sob a curva referente à dose dada - após atingir o estado estacionário - e a curva da primeira dose. Simulações são feitas em ambiente Matlab e os índices de acúmulo obtidos em ambas curvas são comparados / Abstract: To study the drug concentration in the organism we use multi-compartment pharmacokinetic models that normally are represented mathematically as a system of ordinary differential equations (ODE). In this work, we propose a model for describing the concentration dynamic from a Fuzzy Rule-Based System (FRBS). In order to obtain the concentration curve, we use the Takagi-Sugeno-Kang (TSK) method and the curves via TSK and ODE are compared. When the drug is administered in multiple doses, the drug accumulation index in the organism is computed from the ratio between the areas under the curve of the dose given - after reaching the steady-state - and the curve of the first dose. Simulations are made in Matlab environment and the accumulation index for both curves are compared / Mestrado / Biomatematica / Mestre em Matemática Aplicada

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-12-21T13:46:29Z No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-12-22T12:51:57Z (GMT) No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5) / Made available in DSpace on 2016-12-22T12:51:57Z (GMT). No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5) Previous issue date: 2016-05-12 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O uso de espuma para o controle da mobilidade é uma técnica potencial que melhora a eficiência na recuperação avançada de óleo. As propriedades da espuma são controladas pela dinâmica de criação e destruição seguindo os modelos mais usados de balanço de populaçãoemodelosdeequilíbriolocal,considerandoalgumashipótesescomodeslocamento unidimensional, método do fluxo fracionário. O surfactante como componente da fase liquida é responsável da criação de espuma. Em muitos artigos por simplicidade a concentração do surfactante é considerada constante. Neste trabalho não é considerado esta simplificação. O objetivo deste trabalho é desenvolver um modelo onde a concentração do surfactante é descrita por uma equação de balanço. O modelo é completado por equações de balanço de massa de água, gás e a concentração de bolhas de espuma. A geração e destruição de bolhas é descrita pela dinâmica do modelo cinético de primeira ordem. Para estudar matematicamenteomodelousamosferramentasdeequaçõesdiferenciaisordináriaseondas viajantes. Para estados de equilíbrio adequados mostramos a existência de ondas viajantes. Para o caso particular, desprezando a pressão capilar, a existência foi rigorosamente provada. Para o caso geral, uma investigação numérica foi realizada. / Theuseoffoamtocontrolthemobilityisapotentialtechniquethatimprovestheefficiency of the enhanced oil recovery. The properties of the foam are controlled by the dynamics of creation and destruction following the most used population balance models and models in local equilibrium. Under some assumptions, one-dimensional displacements, the fractional flow method. The surfactant as a component of the water phase is responsible for the foam generation e destruction. Some papers neglect this component for simplicity. In the present work the surfactant concentration is considered. Inthepresentworkthesurfactantphaseisconsideredinthemodelasseparatebalancelaw. The model is complete with mass balance equations of water, gas and the concentration of bubbles foam. The bubble generation and destruction is described by dynamic of the first order kinetic model. The mathematically study was based on ordinary differential equation tools and traveling waves analysis. For reasonable equilibrium conditions we study the existence of the traveling wave solution. For the particular case neglecting the capillary pressure, the existence was proved rigorously. For the general case numerical investigation was performed.

This thesis presents methods for efficient numerical approximation of linear and non-linear systems of highly oscillatory ordinary differential equations. Phenomena of high oscillation is considered a major computational problem occurring in Fourier analysis, computational harmonic analysis, quantum mechanics, electrodynamics and fluid dynamics. Classical methods based on Gaussian quadrature fail to approximate oscillatory integrals. In this work we introduce numerical methods which share the remarkable feature that the accuracy of approximation improves as the frequency of oscillation increases. Asymptotically, our methods depend on inverse powers of the frequency of oscillation, turning the major computational problem into an advantage. Evolving ideas from the stationary phase method, we first apply the asymptotic method to solve highly oscillatory linear systems of differential equations. The asymptotic method provides a background for our next, the Filon-type method, which is highly accurate and requires computation of moments. We also introduce two novel methods. The first method, we call it the FM method, is a combination of Magnus approach and the Filon-type method, to solve matrix exponential. The second method, we call it the WRF method, a combination of the Filon-type method and the waveform relaxation methods, for solving highly oscillatory non-linear systems. Finally, completing the theory, we show that the Filon-type method can be replaced by a less accurate but moment free Levin-type method.

Diplomová práce se zabývá problematikou obyčejných stochastických diferenciálních rovnic. Po souhrnu teorie stochastických procesů, zejména tzv. Brownova pohybu je zaveden stochastický Itôův integrál, diferenciál a tzv. Itôova formule. Poté je definováno řešení počáteční úlohy stochastické diferenciální rovnice a uvedena věta o existenci a jednoznačnosti řešení. Pro případ lineární rovnice je odvozen tvar řešení a rovnice pro jeho střední hodnotu a rozptzyl. Závěr tvoří rozbor vybraných rovnic.

Spatiotemporal Predictive Learning (SPL) is an essential research topic involving many practical and real-world applications, e.g., motion detection, video generation, precipitation forecasting, and traffic flow prediction. The problems and challenges of this field come from numerous data characteristics in both time and space domains, and they vary depending on the specific task. For instance, spatial analysis refers to the study of spatial features, such as spatial location, latitude, elevation, longitude, the shape of objects, and other patterns. From the time domain perspective, the temporal analysis generally illustrates the time steps and time intervals of data points in the sequence, also known as interval recording or time sampling. Typically, there are two types of time sampling in temporal analysis: regular time sampling (i.e., the time interval is assumed to be fixed) and the irregular time sampling (i.e., the time interval is considered arbitrary) related closely to the continuous-time prediction task when data are in continuous space. Therefore, an efficient spatiotemporal predictive method has to model spatial features properly at the given time sampling types. In this thesis, by taking advantage of Machine Learning (ML) and Deep Learning (DL) methods, which have achieved promising performance in many complicated computational tasks, we propose three DL-based models used for Spatiotemporal Sequence Prediction (SSP) with several types of time sampling. First, we design the Trajectory Gated Recurrent Unit Attention (TrajGRU-Attention) with novel attention mechanisms, namely Motion-based Attention (MA), to improve the performance of the standard Convolutional Recurrent Neural Networks (ConvRNNs) in the SSP tasks. In particular, the TrajGRU-Attention model can alleviate the impact of the vanishing gradient, which leads to the blurry effect in the long-term predictions and handle both regularly sampled and irregularly sampled time series. Consequently, this model can work effectively with different scenarios of spatiotemporal sequential data, especially in the case of time series with missing time steps. Second, by taking the idea of Neural Ordinary Differential Equations (NODEs), we propose Trajectory Gated Recurrent Unit integrating Ordinary Differential Equation techniques (TrajGRU-ODE) as a continuous time-series model. With Ordinary Differential Equation (ODE) techniques and the TrajGRU neural network, this model can perform continuous-time spatiotemporal prediction tasks and generate resulting output with high accuracy. Compared to TrajGRU-Attention, TrajGRU-ODE benefits from the development of efficient and accurate ODE solvers. Ultimately, we attempt to combine those two models to create TrajGRU-Attention-ODE. NODEs are still in their early stage of research, and recent ODE-based models were designed for many relatively simple tasks. In this thesis, we will train the models with several video datasets to verify the ability of the proposed models in practical applications. To evaluate the performance of the proposed models, we select four available spatiotemporal datasets based on the complexity level, including the MovingMNIST, MovingMNIST++, and two real-life datasets: the weather radar HKO-7 and KTH Action. With each dataset, we train, validate, and test with distinct types of time sampling to justify the prediction ability of our models. In summary, the experimental results on the four datasets indicate the proposed models can generate predictions properly with high accuracy and sharpness. Significantly, the proposed models outperform state-of-the-art ODE-based approaches under SSP tasks with different circumstances of interval recording.

Immunotherapy, by enhancing the endogenous anti-tumor immune responses, is showing promising results for the treatment of numerous cancers refractory to conventional therapies. However, its effectiveness for advanced castration-resistant prostate cancer remains unsatisfactory and new therapeutic strategies need to be developed. To this end, mathematical modeling provides a quantitative framework for testing in silico the efficacy of new treatments and combination therapies, as well as understanding unknown biological mechanisms. In this dissertation we present two mathematical models of prostate cancer immunotherapy defined as systems of ordinary differential equations. The first work, introduced in Chapter 2, provides a mathematical model of prostate cancer immunotherapy which has been calibrated using data from pre-clinical experiments in mice. This model describes the evolution of prostate cancer, key components of the immune system, and seven treatments. Numerous combination therapies were evaluated considering both the degree of tumor inhibition and the predicted synergistic effects, integrated into a decision tree. Our simulations predicted cancer vaccine combined with immune checkpoint blockade as the most effective dual-drug combination immunotherapy for subjects treated with androgen-deprivation therapy that developed resistance. Overall, this model serves as a computational framework to support drug development, by generating hypotheses that can be tested experimentally in pre-clinical models. The Chapter 3 is devoted to the description of a human prostate cancer mathematical model. The potential effect of immunotherapies on castration-resistant form has been analyzed. In particular, the model includes the dendritic vaccine sipuleucel-T, the only currently available immunotherapy option for advanced prostate cancer, and the ipilimumab, a drug targeting the cytotoxic T-lymphocyte antigen 4 , exposed on the CTLs membrane, currently under Phase II clinical trial. From a mathematical analysis of a simplified model, it seems likely that, under continuous administration of ipilimumab, the system lies in a bistable situation where both the no-tumor equilibrium and the high-tumor equilibrium are attractive. The schedule of periodic treatments could then determine the outcome, and mathematical models could help in deciding an optimal schedule.

<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)