The last five years have shown us the profound impact that SARS-CoV-2 pandemic has had on human kind and made us aware of the dangers that emerging pathogens can present. The goal of this dissertation is to use mathematical models in connection with data to uncover mechanistic interactions governing viral infections. To acquire a holistic understanding of the impact of viral infections, it is necessary to develop mathematical techniques and models that bridge knowledge on multiple biological scales. This dissertation explores the relationship between within-host virus dynamics, the environment and the between-host viral transmission. We will validate the models against data from SARS-CoV-2 infections, and data from infections with an emerging pathogen, the Usutu virus. Our models of SARS-CoV-2 infection looked at the relationship between infectious virus and viral RNA in the body and in the environment. Using golden hamster data and within-host mathematical models, we determined that infectious virus shedding early in infection correlates with transmission events, shedding of infectious virus diminishes late in the infection, and high viral RNA levels late in the infection are a poor indicator of transmission. We further showed that viral infectiousness increases in a density dependent manner with viral RNA and that their relative ratio is time-dependent. Such information is useful for designing interventions. Our models of Usutu virus infection looked at differences between different virus strains during bird infections. Within-host models applied to data showed heterogeneity in viral strain dynamics, and correlated high basic reproductive number with short infected cell lifespan (indicative of immune responses) and correlated low basic reproductive number with low viral peaks and longer lasting viremia (due to lower infection rates and high infected cell lifespan). We expanded the models to investigate multiscale dynamics connecting within-host scale, bird-to-vector transmission scale, and vector-borne epidemiological scale. One important direction of this dissertation is the investigation of uncertainty in parameter estimation and overall model identifiability. We conducted identifiability studies (using several theoretical tools) in the multiscale models of Usutu virus infection and in several within-host influenza models. Model identifiability is critical to the reproducibility of modeling results in any biological systems. In this dissertation, we will show how insights from such analyses inform both modeling practices and experimental design. / Doctor of Philosophy / The last five years have shown us the profound impact that SARS-CoV-2 pandemic has had on human kind and made us aware of the dangers that emerging pathogens can present. Within-host mathematical models are tools that can be used to study the dynamics of virus infections. These models help us gain an understanding of biological quantities of interest, relationships between biological processes in a quantitative and qualitative ways, and disease outcome. However, to acquire a holistic understanding of the impact of viral infections, it is necessary to develop mathematical tools and models that bridge knowledge on multiple biological scales. This dissertation explores the relationship between virus infection characteristics over time in a single host and larger biological scales including virus' release into the environment and spread of virus between hosts. Biological and public health insights about SARS-CoV-2 and Usutu virus were gained through these modeling efforts.

<p> The work in this thesis is to develop a tool for calculating the parameters corresponding to certain theoretical model of dielectric relaxation phenomena and then doing the curve fitting using the result after fetching the data from the user. To our best knowledge, this the first such tool to calculate the parameters corresponding to certain theoretical model of dielectric relaxation phenomena while the user only need to provide the experimental data. The parameters are calculated by using a nonlinear least square algorithm implemented in Matlab and a nonlinear function minimizer available in Matlab. The way to do the curve fitting is not by the traditional way such as cubic spline but by calculating the simulated data using the chosen model and the calculated result for the parameters. </p> <p> The available mathematical models include all of popular theoretical models, the Cole-Davidson (DC), the Kohlrausch-Williams-Watts (KWW), the Havriliak-Negami (HN) and the model proposed by R. Hilfer (FD). </p> <p> There are two ways to calculate the parameters for each model as mentioned before. The result returned by this system may not be unique. Especially if the frequency range of data is not wide enough, the result would most likely be non-unique. Since the iterative method is used in the system, it is suggested that the user provides the initial values for the system with his best knowledge or background for the data and the tested sample related to dielectric relaxation process. </p> <p> It is normal if there is a part having worse fitting than the other parts. One of reasons could be the mathematical model's defect, which the model does not work for that part. For the further information, please contact me by email at zouhaijun at yahoo.com. </p> / Thesis / Master of Science (MSc)

The aim of this thesis is to develop discrete fractional models of tumor growth for a given data and to estimate parameters of these models in order to have better data tting. We use discrete nabla fractional calculus because we believe the discrete counterpart of this mathematical theory will give us a better and more accurate outcome. This thesis consists of ve chapters. In the rst chapter, we give the history of the fractional calculus, and we present some basic de nitions and properties that are used in this theory. We de ne nabla fractional exponential and then nabla fractional trigonometric functions. In the second chapter, we concentrate on completely monotonic functions on R, and we introduce completely monotonic functions on discrete domain. The third chapter presents discrete Laplace N-transform table which is a great tool to nd solutions of -th order nabla fractional di erence equations. Furthermore, we nd the solution of nonhomogeneous up to rst order nabla fractional di erence equation using N-transform. In the fourth chapter, rst we give the de nition of Casoration for the set of solutions up to n-th order nabla fractional equation. Then, we state and prove some basic theorems about linear independence of the set of solutions. We focus on the solutions of up to second order nabla fractional di erence equation. We examine these solutions case by case namely, for the real and distinct characteristic roots, real and same, and complex ones. The fth chapter emphasizes the aim of this thesis. First, we give a vi brief introduction to parameter estimation with Gomperts and Logistic curves. In addition, we recall a statistical method called cross-validation for prediction. We state continuous, discrete, continuous fractional and discrete fractional forms of Gompertz and Logistic curves. We use the tumor growth data for twenty-eight mice for the comparison. These control mice were inoculated with tumors but did not receive any succeeding treatment. We claim that the discrete fractional type of sigmoidal curves have the best data tting results when they are compared to the other types of models.

Title: Analysis of Point Clouds Representing Surfaces of Engineering Practice Author: Petra Surynková Department: Department of Mathematics Education Supervisor: Mgr. Šárka Voráčová, Ph.D., Faculty of Transportation Sciences, Czech Technical University in Prague Abstract: The doctoral dissertation Analysis of Point Clouds Representing Surfaces of Engineering Practice addresses the development and application of methods of digital reconstruction of surfaces of engineering and construction practice from point clouds. The main outcome of the dissertation is a presentation of new procedures and methods that contribute to each of the stages of the reconstruction process from the input point clouds. The work is mainly focused on the analysis of input clouds that describe special types of surfaces. Several completely new algorithms and improvements of existing algorithms that contribute to individual steps of surface reconstruction are presented. New procedures are based on geometrical characteristics of the reconstructed object. An important result of the dissertation is an analysis of not only synthetically generated point clouds but above all an analysis of real point clouds that have been obtained from measurements of real objects. The significant contribution of the dissertation is also an...

<div>Many practical applications benefit from the reconstruction of a smooth multivariate function from discrete data for purposes such as reducing file size or improving analytic and visualization performance. Among the different reconstruction methods, tensor product B-spline has a number of advantageous properties over alternative data representation. However, the problem of constructing a best-fit B-spline approximation effectively contains many roadblocks. Within the many free parameters in the B-spline model, the choice of the knot vectors, which defines the separation of each piecewise polynomial patch in a B-spline construction, has a major influence on the resulting reconstruction quality. Yet existing knot placement methods are still ineffective, computationally expensive, or impose limitations on the dataset format or the B-spline order. Moving beyond the 1D cases (curves) and onto higher dimensional datasets (surfaces, volumes, hypervolumes) introduces additional computational challenges as well. Further complications also arise in the case of undersampled data points where the approximation problem can become ill-posed and existing regularization proves unsatisfactory.</div><div><br></div><div>This dissertation is concerned with improving the efficiency and accuracy of the construction of a B-spline approximation on discrete data. Specifically, we present a novel B-splines knot placement approach for accurate reconstruction of discretely sampled data, first in 1D, then extended to higher dimensions for both structured and unstructured formats. Our knot placement methods take into account the feature or complexity of the input data by estimating its high-order derivatives such that the resulting approximation is highly accurate with a low number of control points. We demonstrate our method on various 1D to 3D structured and unstructured datasets, including synthetic, simulation, and captured data. We compare our method with state-of-the-art knot placement methods and show that our approach achieves higher accuracy while requiring fewer B-spline control points. We discuss a regression approach to the selection of the number of knots for multivariate data given a target error threshold. In the case of the reconstruction of irregularly sampled data, where the linear system often becomes ill-posed, we propose a locally varying regularization scheme to address cases for which a straightforward regularization fails to produce a satisfactory reconstruction.</div>

This work focuses on issues relating to the perfusion analysis. The aim of this work is to perform experimental measurements of the phantom and then evaluate the perfusion curves. This curves are used to himation of perfusion hemodynamic parameters, which indicates important informatik about monitoring area. All processes associated with the designation and evaluation are performed in a program named Matlab. The output of work is a system that provides the reader into the problem of perfusion analysis and allows him to understand and know what is the meaning od analysis, what demands are placed on the evaluation and what is the result of this perfusion analysis.

The 1st and 2nd order Lee–Carter methods were compared with the Hyndman–Ullah method in regards to goodness of fit and forecasting ability of mortality rates. Swedish population data was used from the Human Mortality Database. The robust estimation property of the Hyndman–Ullah method was also tested with inclusion of the Spanish flu and a hypothetical scenario of the COVID-19 pandemic. After having presented the three methods and making several comparisons between the methods, it is concluded that the Hyndman–Ullah method is overall superior among the three methods with the implementation of the chosen dataset. Its robust estimation of mortality shocks could also be confirmed.

Biomolecular interactions, including protein-protein, protein-DNA, and protein-ligand interactions, are of special importance in all biological systems. These interactions may occer during the loading of biomolecules to interfaces, the translocation of biomolecules through transmembrane protein pores, and the movement of biomolecules in a crowded intracellular environment. The molecular interaction of a protein with its binding partners is crucial in fundamental biological processes such as electron transfer, intracellular signal transmission and regulation, neuroprotective mechanisms, and regulation of DNA topology. In this dissertation, a customized surface plasmon resonance (SPR) has been optimized and new theoretical and label free experimental methods with related analytical calculations have been developed for the analysis of biomolecular interactions. Human neuroglobin (hNgb) and cytochrome c from equine heart (Cyt c) proteins have been used to optimize the customized SPR instrument. The obtained Kd value (~13 µM), from SPR results, for Cyt c-hNgb molecular interactions is in general agreement with a previously published result. The SPR results also confirmed no significant impact of the internal disulfide bridge between Cys 46 and Cys 55 on hNgb binding to Cyt c. Using SPR, E. coli topoisomerase I enzyme turnover during plasmid DNA relaxation was found to be enhanced in the presence of Mg2+. In addition, a new theoretical approach of analyzing biphasic SPR data has been introduced based on analytical solutions of the biphasic rate equations. In order to develop a new label free method to quantitatively study protein-protein interactions, quartz nanopipettes were chemically modified. The derived Kd (~20 µM) value for the Cyt c-hNgb complex formations matched very well with SPR measurements (Kd ~16 µM). The finite element numerical simulation results were similar to the nanopipette experimental results. These results demonstrate that nanopipettes can potentially be used as a new class of a label-free analytical method to quantitatively characterize protein-protein interactions in attoliter sensing volumes, based on a charge sensing mechanism. Moreover, the molecule-based selective nature of hydrophobic and nanometer sized carbon nanotube (CNT) pores was observed. This result might be helpful to understand the selective nature of cellular transport through transmembrane protein pores.

The Silicon Carbide Bipolar Transistor is a device that is barely brought into real application so far. It features very low conduction losses and a high power density. The application is in some points different and unusual in comparison to the mainstream power semiconductors as IGBTs or MOSFETs. The Silicon Carbide Bipolar Transistor, the SiC-BJT, is a current driven device and the effort in driving is uncommonly high. As an outcome of the present work it can be said that it is more like a shift of requirements from the power semiconductor power unit to the driver stage. With consideration of all system losses, including driving losses, the final unoptimized COSIVU prototype inverter system gained an increase of efficiency of 40-60% in comparison to the IGBT-based reference system dependent on the applied load points. In terms of reliability and possible failure modes, the SiC-BJT behaves differently from the mainstream devices. One result of the project is that the chips itself are quite robust but the packaging needs some improvements. Thermal impedance spectroscopy is a method for detecting possible deterioration in the cooling path of a device. A method for temperature estimation of the SiC-BJT during on-state will be presented in this work. The electronic hardware for thermal impedance spectroscopy has been developed to do the measurements in a non-laboratory setup in the inverter in real application. Furthermore, the hardware implementation was realized on a very small space for integration into an in-wheel motor inverter system. / Der Siliciumkarbid Bipolartransistor ist ein leistungselektronisches Bauelement, was bis heute kaum über Labor- und Forschungsprojekte hinaus anwendungsnah zum Einsatz kam. Er verfügt über sehr geringe Durchlassverluste und eine hohe Leistungsdichte. Seine Verwendung und Anwendung ist in mancher Hinsicht anders und unüblich im Vergleich zu den etablierten leistungselektronischen Bauelementen wie IGBT und MOSFET. Der Siliciumkarbid Bipolartransistor, also der SiC-BJT, ist ein stromgesteuertes Bauteil, weswegen der Aufwand für die Treiber sehr hoch ist. Die praktische Arbeit im Rahmen des Forschungsprojektes „COSIVU“ mit den SiC-BJTs in Verbindung mit dem fertigen integrierten Invertersystem hat unter anderem gezeigt, dass es mehr eine Verschiebung der Anforderungen von der Leistungselektronik hin zu den Treibern für die Leistungselektronik ist. Unter Betrachtung der Verluste des gesamten Systems, einschließlich der Motor-, Treiber- und Steuerverluste, hat das fertige Prototyp-Invertersystem, welches durchaus noch Potential zur Optimierung besaß, eine deutliche Verbesserung des Wirkungsgrades erreicht. Gegenüber dem auf IGBT basierenden Referenz-Invertersystem, hat das COSIVU Invertersystem eine Verbesserung des Wirkungsgrades um 40-60 % erreicht. Eine Erkenntnis aus dem Forschungsprojekt in Bezug auf Zuverlässigkeit und mögliche Fehler und Defekte ist, dass der Chip selbst zwar ziemlich robust ist, aber dass die Gehäuse-, Aufbau- und Verbindungstechnik angepasst und verbessert werden sollte. Thermische Impedanzspektroskopie ist eine Methode um Verschlechterungen im Kühlpfad eines leistungselektronischen Halbleiters zu erkennen, was ein Kriterium für die Alterung des Bauteils ist. Eine Methode zur Bestimmung der Sperrschichttemperatur von SiC-BJTs während des normalen Durchlassbetriebes wird in dieser Arbeit vorgestellt. Die Platine für die thermische Impedanzspektroskopie wurde entwickelt, um die Messung in einem laborfernen Aufbau in einer echten Inverteranwendung durchzuführen. Zudem wurden die Platinenaufbauten auf sehr kleiner Fläche realisiert. Die Integration musste nämlich sehr kompakt gestaltet werden, da es sich um ein „in-wheel“ Motor-Inverter-System handelt, was zum größten Teil innerhalb eines Fahrzeugrades untergebracht ist.