Scientific machine learning (SciML) is a new branch of AI research at the edge of scientific computing (Sci) and machine learning (ML). It deals with efficient amalgamation of data-driven algorithms along with scientific computing to discover the dynamics of the time-evolving process. The output of such algorithms is represented in the form of a governing equation(s) (e.g., ordinary differential equation(s), ODE(s)), which one can solve then for any time point and, thus, obtain a rigorous prediction.  In this thesis, we present a methodology on how to incorporate the SciML approach in the context of clinical trials to predict IPF disease progression in the form of governing equation. Our proposed methodology also quantifies the uncertainties associated with the model by fitting 95\% high density interval (HDI) for the ODE parameters and 95\% posterior prediction interval for posterior predicted samples. We have also investigated the possibility of predicting later outcomes by using the observations collected at early phase of the study. We were successful in combining ML techniques, statistical methodologies and scientific computing tools such as bootstrap sampling, cubic spline interpolation, Bayesian inference and sparse identification of nonlinear dynamics (SINDy) to discover the dynamics behind the efficacy outcome as well as in quantifying the uncertainty of the parameters of the governing equation in the form of 95 \% HDI intervals. We compared the resulting model with the existed disease progression model described by the Weibull function. Based on the mean squared error (MSE) criterion between our ODE approximated values and population means of respective datasets, we achieved the least possible MSE of 0.133,0.089,0.213 and 0.057. After comparing these MSE values with the MSE values obtained after using Weibull function, for the third dataset and pooled dataset, our ODE model performed better in reducing error than the Weibull baseline model by 7.5\% and 8.1\%, respectively. Whereas for the first and second datasets, the Weibull model performed better in reducing errors by 1.5\% and 1.2\%, respectively. Comparing the overall performance in terms of MSE, our proposed model approximates the population means better in all the cases except for the first and second datasets, assuming the latter case's error margin is very small. Also, in terms of interpretation, our dynamical system model contains the mechanistic elements that can explain the decay/acceleration rate of the efficacy endpoint, which is missing in the Weibull model. However, our approach had a limitation in predicting final outcomes using a model derived from  24, 36, 48 weeks observations with good accuracy where as on the contrast, the Weibull model do not possess the predicting capability. However, the extrapolated trend based on 60 weeks of data was found to be close to population mean and the ODE model built on 72 weeks of data. Finally we highlight potential questions for the future work.

Diploma thesis on Machining hardened steels is focused on the analysis of longitudinal turning of hardened steel 14 109 by using a tool from PKNB in terms of measuring the force load using a dynamometer and then evaluating the surface quality. The theoretical part deals with the issue of turning technology, cutting materials and heat treatment of steel. In the practical part, the influence of cutting conditions on the resulting values was assessed during the experiment. Twelve samples with different combinations of cutting conditions were tested, then was selected a sample that met the criterion of combining minimum cutting forces values and surface quality. The experiment shows that force load values can be achieved twice less by combining cutting conditions with an appropriate combination.

The theme of the work is design of the administrative building with supporting concrete structure. The design must include changes of concrete cubic capacity and their additional effects on the stress of the structure. Minimization of the impact of additional stress is necessary to solve with the appropriate design of structural systems and materials.

Strength and strain analysis of high pressure compound vessel, which is used to produce superhard materials (for example synthetic diamond). This work was elaborated to compare the stresses and strains calculated by Prof. Jan Vrbka making use of the FEM program ‘Prokop’17 years ago with those gained with the contemporary FEM Ansys program. The vessel is loaded by internal pressure of size 6 GPa. The elastic-plastic material be-haviour is taken into account. Real value of friction between rings and non-uniform temperature field is included into the calculation. The process of assembling the compound vessel is simulated step by step.

In the past years the so-called body-centered cubic grid (bcc) has been examined and proved to be superior over Cartesian lattices for certain applications. Our work deals with parallel thinning on these bcc grids. We introduce conditions which are sufficient for retaining topology and suggest additional conditions to influence the shape of the resulting skeleton. We further developed an algorithm to extract curve skeletons out of 3d objects in parallel which we also present here. We show in our results that the developed thinning approach on bcc grids is extremely efficient.

Flexible beam structures are popular in civil and mechanical engineering. Many of these structures undergo large displacements and finite rotations, but with small deformations. Their dynamic behaviors are usually investigated using finite beam elements. A well known method to derive such beam elements is the corotational approach. This method has been extensively used in nonlinear static analysis. However, its application in nonlinear dynamics is rather limited. The purpose of this thesis is to investigate the nonlinear dynamic behavior of flexible beam structures using the corotational method. For the 2D case, a new dynamic corotational beam formulation is presented. The idea is to adopt the same corotational kinetic description in static and dynamic parts. The main novelty is to use cubic interpolations to derive both inertia terms and internal terms in order to capture correctly all inertia effects. This new formulation is compared with two classic formulations using constant Timoshenko and constant lumped mass matrices. This work is presented in the first appended journal paper. For the 3D case, update procedures of finite rotations, which are central issues in development of nonlinear beam elements in dynamic analysis, are discussed. Three classic and one new formulations of beam elements based on the three different parameterizations of the finite rotations are presented. In these formulations, the corotational method is used to develop expressions of the internal forces and the tangent stiffness matrices, while the dynamic terms are formulated into a total Lagrangian context. Many aspects of the four formulations are investigated. First, theoretical derivations as well as practical implementations are given in details. The similarities and differences between the formulations are pointed out. Second, numerical accuracy and computational efficiency of these four formulations are compared. Regarding efficiency, the choice of the predictor at each time step and the possibility to simplify the tangent inertia matrix are carefully investigated. This work is presented in the second appended journal paper. To make this thesis self-contained, two chapters concerning the parametrization of the finite rotations and the derivation of the 3D corotational beam element in statics are added. / QC 20120521

This thesis aims to study a connection between indecomposable elements in the cubic fields and the Jacobi-Perron algorithm (JPA). JPA is a multidimensional generalization of the usual continued fractions algorithm. We work in the family of Ennola's cubic fields and we examine how the indecomposable elements are related to elements originating from this algorithm and whether some of these elements generate all indecomposable elements in the fields. We formulate conjectures on how to determine which elements will generate the indecomposable elements. We also prove some necessary conditions that have to hold for elements originating from this algorithm to generate indecomposable elements. 1

Damage to the central nervous system (CNS) leads to the generation of an immune response which culminates with the encapsulation of the damaged area. The encapsulation, known as a glial scar, essentially breaks neural signal pathways and blocks signal transmissions to and from the CNS. The effect is the loss of motor and sensory control for the damaged individual. One method that has been used successfully to treat this problem is the use of a brain-machine interface (BMI) which can intercept signals from the brain and use these signals to control a machine. Although there are many types of BMI devices, implantable devices show the greatest promise with the ability to target specific areas of the CNS, with reduced noise levels and faster signal interception, and the fact that they can also be used to send signals to neurons. The largest problem that has plagued this type of BMI device is that the materials that have been used for their construction are not chemically resilient, elicit a negative biological response, or have difficulty functioning for extended periods of time in the harsh body environment. Many of these implantable devices experience catastrophic failure within weeks to months because of these negative factors. New materials must be examined to advance the future utilization of BMI devices to assist people with CNS damage or disease. We have proposed that two semiconductor materials, cubic silicon carbide (3C-SiC) and nanocrystalline diamond (NCD), which should provide solutions to the material biocompatibility problems experienced by implantable BMI devices. We have shown in this study that these two materials show chemical resilience to neuronal cellular processes, and we show evidence which indicates that these materials possess good biocompatibility with neural cell lines that, in the worst case, is comparable to celltreated polystyrene and, in most cases, even surpasses polystyrene. We have utilized 3C-SiC within an electrode device and activated the action potential of differentiated PC12 cells. This work details our initial efforts to modify the surfaces of these materials in order to improve cellular interaction and biocompatibility, and we examine our current and future work on improving our implantable BMI devices.

<p> This thesis reports a systematic study of geometrically frustrated f.c.c. double perovskites with both monoclinic (P2₁/n) La₂LiMO₆ and cubic (Fm3m) Ba₂YMO₆ symmetries, where M=Mo (S=1/2), Re (S=1) and Ru (S=3/2). The roles of both the spin quantum number, i.e. quantum spin fluctuations, and the local site symmetry, i.e. orbital ordering, on the determination of the ground magnetic state were studied. All the compounds were prepared by solid state reaction and the structural information and magnetic properties of the compounds were collected using diffraction techniques (X-ray and neutron), de susceptibility, heat capacity, muon spin relaxation (μSR) and solid state NMR. </p> <p> The S=3/2 materials, La₂LiRuO₆ and Ba₂YRuO₆, while highly frustrated with frustration indices f ~ 16 and 8, respectively, both show antiferromagnetic (AF) long range ordering at 24K and 37K, respectively. The Neel temperature of the latter compound was determined for the first time by the heat capacity and neutron diffraction. This compound shows an unusual AF transition as two broader peaks were observed in the susceptibility while La₂LiRuO₆ shows a typical AF behavior. There is about 1 % of Y/Ru site mixing observed by 89Y MAS NMR in Ba₂YRuO₆. </p> <p> For the S=1 materials, monoclinic La₂LiReO₆ shows collective singlet like behavior as zero magnetization was observed in the ZFC susceptibility and a static and diluted spin system was indicated by μSR. On the other hand, the cubic phase, Ba₂YReO₆ surprisingly shows a spin glass behavior confirmed by μSR while no Y/Re site mixing was observed by MAS NMR. It is also a surprising observation that this compound retains cubic symmetry down to 3K where it would have a structure transition subject to the Jahn-Teller theorem. </p> <p> Finally, the S=1/2 compounds, La₂LiMoO₆ and Ba₂YMoO₆, show quite different magnetic behavior. Monoclinic La₂LiMoO₆ shows the presence of at least short range order achieved at 18K according to the heat capacity and μSR measurements. Ba₂YMoO₆ retains cubic symmetry down to 3K and no Jahn-Teller distortion was observed at the limit of the resolution of neutron diffraction. This compound surprisingly appears to remain paramagnetic down to 2K, yet evidence for a collective singlet state was observed by a paramagnetic Knight shift measurement in NMR. This is consistent with an existing theoretical prediction. </p> <p> An extended study on other S=1/2 Mo analogues, Ba₂LuMoO₆ and Ba₂ScMoO₆ is also presented. Both compounds show cubic structure confirmed by X-ray diffraction and paramagnetic behavior down to 2K in the susceptibility. </p> / Thesis / Doctor of Philosophy (PhD)

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