We consider the $k-varepsilon$ model in the theory of turbulence, where $k$ is the turbulent kinetic energy, $varepsilon$ is thedissipation rate of the turbulent energy, and $alpha,$ $eta,$ and $gamma$ are positive constants. In particular we examine the Barenblatt self-similar $k-varepsilon$ model, along with boundary conditions taken to ensure the symmetry and compactness of the support of solutions.Under the assumptions:$eta>alpha,$ $3alpha>2eta,$ and $gamma$>3/2,we show the existence of $mu$ for which there is a positive solutionto the system and corresponding boundary conditions by proving a seriesof lemmas. We also include graphs of solutions obtained by using XPPAUT 5.85.

This thesis is concerned with one of the most promising approaches to the numerical simulation of turbulent flows, the subgrid eddy viscosity models. We analyze both continuous and discontinuous finite element approximation of the new subgrid eddy viscosity model introduced in [43], [45], [44].First, we present a new subgrid eddy viscosity model introduced in a variationally consistent manner and acting only on the small scales of the fluid flow. We give complete convergence of themethod. We show convergence of the semi-discrete finite element approximation of the model and give error estimates of the velocity and pressure. In order to establish robustness of themethod with respect to Reynolds number, we consider the Oseen problem. We present the error is uniformly bounded with respect to the Reynolds number.Second, we establish the connection of the new eddy viscosity model with another stabilization technique, called VariationalMultiscale Method (VMM) of Hughes et.al. [35]. We then show the advantages of the method over VMM. The new approach defines mean by elliptic projection and this definition leads to nonzerofluctuations across element interfaces.Third, we provide a careful numerical assessment of a new VMM. We present how this model can be implemented in finite element procedures. We focus on herein error estimates of the model andcomparison to classical approaches. We then establish that the numerical experiments support the theoretical expectations.Finally, we present a discontinuous finite element approximation of subgrid eddy viscosity model. We derive semi-discrete and fullydiscrete error estimations for the velocity.

Biometric data can provide useful information about person's overall wellness. The focus of this dissertation is wellness monitoring and diagnostics based on behavioral and physiological traits. The research comprises of three studies: passive non-intrusive biometric monitoring, active monitoring using a wearable computer, and a diagnostics of early stages of Parkinson's disease. In the first study, a biometric analysis system for collecting voice and gait data from a target individual has been constructed. A central issue in that problem is filtering of data that is collected from non-target subjects. A novel approach to gait analysis using floor vibrations has been introduced. Naive Bayes model has been used for gait analysis, and the Gaussian Mixture Model has been implemented for voice analysis. It has been shown that the designed biometric system can provide sufficiently accurate data stream for health monitoring purposes.In the second study, a universal wellness monitoring algorithm based on a binary classification model has been developed. It has been tested on the data collected with a wearable body monitor SenseWearÂ®PRO and with the Support Vector Machines acting as an underlying binary classification model. The obtained results demonstrate that the wellness score produced by the algorithm can successfully discriminate anomalous data.The focus of the final part of this thesis is an ongoing project, which aims to develop an automated tool for diagnostics of early stages of Parkinson's disease. A spectral measure of balance impairment is introduced, and it is shown that that measure can separate the patients with Parkinson's disease from control subjects.

We explore three coupled networks. Each is an example of a network whose spatially coupled behavior is dratically different than the behavior of the uncoupled system. 1. An evolution equation such that the intrinsic dynamics of the system are those near a degenerate Hopf bifurcation is explored. The coupled system is bistable and solutions such as waves and persistent localized activity are found. 2. A trapping mechanism that causes long interspike intervals in a network of Hodgkin Huxley neurons coupled with excitatory synaptic coupling is unveiled. This trapping mechanism is formed through the interaction of the time scales present intrinsically and the time scale of the synaptic decay. 3. We construct a model to create the spatial patterns reported by subjects in an experiment when their eyes were stimulated electrically. Phase locked oscillators are used to create boundaries representing phosphenes. Asymmetric coupling causes the lines to move, as in the experiment. Stable stationary solutions and waves are found in a reduced model of evolution/ convolution type.

Solutions are shown to exist for a variety of differential equations. Both ordinary and partial differential equations are considered, with specified initial conditions, boundary conditions, or simultaneous initial and boundary conditions. A key feature of the these problems is a condition at infinity; it is demanded that solutions decay towards zero as the temporal variable becomes arbitrarily large. This feature removes from the problem a certain compactness property, which precludes the use of traditional methods which employ the Leray-Schauder topological degree. This difficulty is overcome by use of a much newer theory of topological degree, developed by Fitzpatrick, Pejsachowicz and Rabier in 1992, and later developed further by Pejsachowicz and Rabier in 1998. This degree theory requires several properties in lieu of compactness. It is shown that these properties are available in a wide range of problems, and that there is a practical way to verify this fact in specific cases. Specific examples are given.