For years, researchers have studied the movement of cells and mathematicians have attempted to model the movement of the cell using various methods. This work is an extension of the work done by Zheltukhin and Lui (2011), Mathematical Biosciences 229:30-40, who simulated the stress and displacement of a one-dimensional cell using a model based on viscoelastic theory. The report is divided into three main parts. The first part considers viscoelastic models with a first-order constitutive equation and uses the standard linear model as an example. The second part extends the results of the first to models with second-order constitutive equations. In this part, the two examples studied are Burger model and a Kelvin-Voigt element connected with a dashpot in series. In the third part, the effects of substrate with variable stiffness are explored. Here, the effective adhesion coefficient is changed from a constant to a spatially-dependent function. Numerical results are generated using two different functions for the adhesion coefficient. Results of this thesis show that stress on the cell varies greatly across each part of the cell depending on the constitute equation we use, while the position and velocity of the cell remain essentially unchanged from a large-scale point of view.

<p>The success of the current Internet relies to a large extent on a cooperation between the users and the network. The network signals its current state to the users by marking or dropping packets. The users then strive to maximize the sending rate without causing network congestion. To achieve this, the users implement a flow-control algorithm that controls the rate at which data packets are sent into the Internet. More specifically, the <i>Transmission Control Protocol (TCP)</i> is used by the users to adjust the sending rate in response to changing network conditions. TCP uses the observation of packet loss events and estimates of the round trip time (RTT) to adjust its sending rate. </p><p>In this thesis we investigate and propose stochastic models for TCP. The models are used to estimate network performance like throughput, link utilization, and packet loss rate. The first part of the thesis introduces the TCP protocol and contains an extensive TCP modeling survey that summarizes the most important TCP modeling work. Reviewed models are categorized as renewal theory models, fixed-point methods, fluid models, processor sharing models or control theoretic models. The merits of respective category is discussed and guidelines for which framework to use for future TCP modeling is given. </p><p>The second part of the thesis contains six papers on TCP modeling. Within the renewal theory framework we propose single source TCP-Tahoe and TCP-NewReno models. We investigate the performance of these protocols in both a DropTail and a RED queuing environment. The aspects of TCP performance that are inherently depending on the actual implementation of the flow-control algorithm are singled out from what depends on the queuing environment.</p><p>Using the fixed-point framework, we propose models that estimate packet loss rate and link utilization for a network with multiple TCP-Vegas, TCP-SACK and TCP-Reno on/off sources. The TCP-Vegas model is novel and is the first model capable of estimating the network's operating point for TCP-Vegas sources sending on/off traffic. All TCP and network models in the contributed research papers are validated via simulations with the network simulator <i>ns-2</i>. </p><p>This thesis serves both as an introduction to TCP and as an extensive orientation about state of the art stochastic TCP models.</p>

The success of the current Internet relies to a large extent on a cooperation between the users and the network. The network signals its current state to the users by marking or dropping packets. The users then strive to maximize the sending rate without causing network congestion. To achieve this, the users implement a flow-control algorithm that controls the rate at which data packets are sent into the Internet. More specifically, the Transmission Control Protocol (TCP) is used by the users to adjust the sending rate in response to changing network conditions. TCP uses the observation of packet loss events and estimates of the round trip time (RTT) to adjust its sending rate. In this thesis we investigate and propose stochastic models for TCP. The models are used to estimate network performance like throughput, link utilization, and packet loss rate. The first part of the thesis introduces the TCP protocol and contains an extensive TCP modeling survey that summarizes the most important TCP modeling work. Reviewed models are categorized as renewal theory models, fixed-point methods, fluid models, processor sharing models or control theoretic models. The merits of respective category is discussed and guidelines for which framework to use for future TCP modeling is given. The second part of the thesis contains six papers on TCP modeling. Within the renewal theory framework we propose single source TCP-Tahoe and TCP-NewReno models. We investigate the performance of these protocols in both a DropTail and a RED queuing environment. The aspects of TCP performance that are inherently depending on the actual implementation of the flow-control algorithm are singled out from what depends on the queuing environment. Using the fixed-point framework, we propose models that estimate packet loss rate and link utilization for a network with multiple TCP-Vegas, TCP-SACK and TCP-Reno on/off sources. The TCP-Vegas model is novel and is the first model capable of estimating the network's operating point for TCP-Vegas sources sending on/off traffic. All TCP and network models in the contributed research papers are validated via simulations with the network simulator ns-2. This thesis serves both as an introduction to TCP and as an extensive orientation about state of the art stochastic TCP models.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-19T14:01:17Z No. of bitstreams: 1 DAMIÃO JÚNIO GONÇALVES ARAÚJO - DISSERTAÇÃO PPGMAT 2008..pdf: 506877 bytes, checksum: 8b058ceadbb68cd8c6d372656749744e (MD5) / Made available in DSpace on 2018-07-19T14:01:17Z (GMT). No. of bitstreams: 1 DAMIÃO JÚNIO GONÇALVES ARAÚJO - DISSERTAÇÃO PPGMAT 2008..pdf: 506877 bytes, checksum: 8b058ceadbb68cd8c6d372656749744e (MD5) Previous issue date: 2008-04 / Capes / Neste trabalho são estudados dois sistemas de equações diferenciais parciais parabólicas sujeitas a condições iniciais e de contorno. O primeiro sistema tratado representa um modelo de solidificação envolvendo uma função campo de fase. O segundo problema tratado é uma simplificação de um modelo com duas funções campo de fase para solidificação de ligas. São estudados resultados sobre existência (via Método de Ponto Fixo), regularidade, continuidade em relação aos dados iniciais e ao termo forçante e unicidade de solução dos sistemas citados. / In this work we study two parabolic partial differential equations systems subject to initial and boundary conditions. The first system treated here represents a model for solidification with a phase field function. The second system is a simplification of a two-phase field model for alloy solidification. We study results concerning existence (by FixedPoint Method), regularityand uniquenessof solution for mentioned systems.

Le travail présenté dans ce mémoire nous a permis d’étudier d’un point de vue théorique et numérique le transfert de chaleur couplé par rayonnement et conduction à travers un milieu semi-transparent, gris et non diffusant dans une géométrie multidimensionnelle 2D. Ces deux modes de transfert de chaleur sont décrits par un couplage non linéaire de l’équation de la chaleur non linéaire (CT) et de l’équation du transfert radiatif (ETR). Nous avons présenté des résultats d’existence, d’unicité locale de la solution pour le système couplé avec des conditions aux limites de type Dirichlet homogènes en utilisant le théorème du point fixe de Banach. Par ailleurs, les travaux réalisés nous ont permis de mettre au point un code de calcul qui permet de simuler la température. Nous avons utilisé la quadrature S_N pour la discrétisation angulaire de l’ETR. La discrétisationde l'ETR dans la variable spatiale est effectuée par la méthode de Galerkin discontinue (DG) et en éléments finis pour l'équation de la chaleur non linéaire. Nous avons démontré la convergence du schémanumérique couplé en utilisant la méthode du point fixe discret. Le modèle discret, sous la forme d’équations différentielles ordinairesnon linéaires obtenu après une approximation nous a permis de fairel’analyse et la synthèse d’estimateurs d’état et de lois de commandepour la stabilisation. Grâce à la structure particulière du modèle età l’aide du DMVT. Nous avons proposé un observateur d’ordre réduit.D’autre part nous avons réussi à construire une matrice de gain quiassure la stabilité de l’observateur proposé. Une extension au filtrage $\mathcal{H}_{\infty}$ est également proposée. Une nouvelleinégalité matricielle (LMI) est donnée dans le cas d’une commandebasée observateur. Nous avons étendu à l’approche d’ordre réduit dans le cas de la commande basée observateur et nous avons montré la stabilité sous l’action de la rétroaction. De même une extension au filtrage $\mathcal{H}_{\infty}$ est également proposée. Tous les résultats sont validés par des simulations numériques. / This thesis investigates the theoretical and numerical analysis of coupled radiative conductive heat transfer in a semi-transparent, gray and non-scattering 2D medium. This two heat transfer modes are described by the radiative transfer equation (RTE) and the nonlinear heat equation (NHE). We proved the existence and uniqueness of the solution of coupled systems with homogeneous Dirichlet boundary conditions using the fixed-point theorem. Moreover, we developed a useful algorithm to simulate the temperature in the medium. We used the quadrature $S_{N}$ for the angular discretization of the RTE. The spatial discretization of RTE was made by the discontinuous Galerkin method (DG) and the finite element method for the non-linear heat equation. We have shown the convergence and the stability of the coupled numerical scheme using the discrete fixed point. The discrète model obtained after an approximation allowed us to do the analysis and synthesis of state estimators and feedback control design for stabilization of the system. Thanks to the special structure of the model and using the Differential Mean Value Theorem (DMVT), we proposed a reduced order observer and we construct a gain matrix, which ensures the exponential stability of the proposed observer and guarantees the boundedness of the estimate vector. An extension to $\mathcal{H}_{\infty}$ filtering is also provided. We have extended the reduced order approach in the case of the observer-based controller and we proved the exponential stability under the control feedback law. Similarly, an extension to $\mathcal{H}_{\infty}$ filtering is also provided. The obtained results were validated through several numerical simulations.