Although band theory is about a century old, it remains relevant today as a tool for the treatment of electrons in solids. The confluence of mathematical ideas like geometry and topology with band theory has proven to be a ripe avenue for research in the past few decades. The importance of Fermi surface geometry, especially in conjunction with electronic correlation, has been well recognized. One particular thread in this direction is probing the occurrence of non-trivial Fermi surface geometry, and its influence on macroscopic properties of materials. A notable example of exotic Fermi surface geometry arises from singular points of the dispersion, and these have been known since 1953. The investigation into these was reignited recently, culminating in the work presented in this thesis. In this dissertation, I investigate two broad categories of singular points in bands. At a singular point, either the dispersion or the Fermi surface fail to be smooth. This may cause distinct signatures in transport and spectroscopic properties when the singular point occurs close to the Fermi level. In the two dimensional setting, I classify using catastrophe theory, the point singularities arising from higher order saddles of the dispersion. These are the more exclusive cousins of the regular van Hove saddle that cause, among other things, a power law divergence in the density of states. The role of lattice symmetries in aiding or preventing the occurrence of these singularities is also carefully explored. In the case of three dimensional bands, I investigate the spectroscopic properties of the nodal point singularity, arising from a linear band crossing. In particular, I determine the distinct signature of nodal points in the analytic, momentum resolved, joint density of states (JDOS) and the numerically calculated resonant inelastic x-ray scattering (RIXS) spectrum, within the fast collision approximation that ignores core hole effects. The results presented here will be the stepping stone towards a careful future calculation, incorporating the potential edge singularity effects through core hole potential. Such a calculation may be directly comparable with ongoing experiments.

Longitudinal assessments are crucial in evaluating the disease state and trajectory in patients of neurodegenerative diseases. Neuropsychological outcomes measured over time often have a non-linear trajectory with autocorrelated residuals and skewed distributions. Due to these issues, statistical analysis and interpretation involving longitudinal cognitive outcomes can be a difficult and controversial task, thus hindering most convenient transformations (e.g. logarithmic) to avoid the assumption violations of common statistical modelling techniques. We propose the Adjusted Local Linear Trend (ALLT) model, an extended state space model in lieu of the commonly-used linear mixed-effects model (LMEM) in modeling longitudinal neuropsychological outcomes. Our contributed model has the capability to utilize information from the stochasticity of the data while accounting for subject-specific trajectories with the inclusion of covariates and unequally-spaced time intervals. The first step of model fitting involves a likelihood maximization step to estimate the unknown variances in the model before parsing these values into the Kalman Filter and Kalman Smoother recursive algorithms. Results from simulation studies showed that the ALLT model is able to attain lower bias, lower standard errors and high power, particularly in short longitudinal studies with equally-spaced time intervals, as compared to the LMEM. The ALLT model also outperforms the LMEM when data is missing completely at random (MCAR), missing at random (MAR) and, in certain cases, even in data with missing not at random (MNAR). In terms of model selection, likelihood-based inference is applicable for the ALLT model. Although a Chi-Square distribution with k degrees of freedom, where k is the number of parameter lost during estimation, was not the asymptotic distribution in the case of ALLT, we were able to derive an asymptotic distribution approximation of the likelihood ratio test statistics using the power transformation method for the utility of a Gaussian distribution to facilitate model selections for ALLT. In light of these findings, we believe that our proposed model will shed light into longitudinal data analysis not only in the neuropsychological data realm but also on a broader scale for statistical analysis of longitudinal data. / 2023-10-05T00:00:00Z

In the paper asymptotic expansions for second-order moments of integral functionals of a class of random functions are considered. The random functions are assumed to be $\epsilon$-correlated, i.e. the values are not correlated excluding a $\epsilon$-neighbourhood of each point. The asymptotic expansions are derived for $\epsilon \to 0$. With the help of a special weak assumption there are found easier expansions as in the case of general weakly correlated functions.

The paper considers approximations of time-continuous epsilon-correlated random processes by interpolation of time-discrete Moving-Average processes. These approximations are helpful for Monte-Carlo simulations of the response of systems containing random parameters described by epsilon-correlated processes. The paper focuses on the approximation of stationary epsilon-correlated processes with a prescribed correlation function. Numerical results are presented.

Empirical autocorrelation functions of returns of stochastic price processes show phenomena of correlation on small intervals of time, which decay to zero after a short time. The paper deals with the concept of weakly correlated random processes to describe a mathematical model which takes into account this behaviour of statistical data. Weakly correlated functions have been applied to model numerous problems of physics and engineering. The main idea is, that the values of the functions at two points are uncorrelated if the distance between the points exceeds a certain quantity epsilon > 0. In contrast to the white noise model, for distances smaller than epsilon a correlation between the values is permitted.

Title: Study of Diradicals by Explicitly Correlated Multireference Coupled Cluster Methods Author: Matej Švaňa Department: Department of Physical and Macromelecular Chemistry Supervisor: Mgr. Jiří Pittner, Dr. rer. nat., J. Heyrovský Institute of Physical Chemistry Abstract: Total energies of cyclopropane, trimethylene, and propylidene were calculated with conventional post-HF CCSD(T), BWCCSD(T), MkCCSD(T) methods and their explicitly correlated alternatives. Main aims of the the- sis were to compare the basis set convergence of total energies and relative energies between cyclopropane and trimethylene/propylidene, both at the conventional and the explicitly correlated levels. It was shown that use of explicit correlation accelerates the convergence of the total energy by one or- der of basis set quality, resulting in considerable savings in computational times. Also, the MkCCSD(T)-F12/QZ and the BWCCSD(T)-F12/QZ calcula- tions belong to the most sophisticated approaches employed for estimation of the relative energies of cyclopropane and trimethylene/propylidene to date. Keywords: explicitly correlated, coupled cluster, multi-reference, cyclopropane isomerisation, trimethylene, propylidene 1

Genetic association studies have successfully identified many genetic markers associated with complex human diseases and related quantitative traits. However, for most complex diseases and quantitative traits, all associated genetic markers identified to date only explain a small proportion of heritability. Thus, exploring the unexplained heritability in these traits will help us discover novel genetic determinants for these traits and better understand disease etiology and pathophysiology. Due to limited sample size, a single cohort study may not have sufficient power to identify novel genetic association with a small effect size, and meta-analysis approaches have been proposed and applied to combine results from multiple cohorts in large consortia, increasing the sample size and statistical power. Rare genetic variants and gene by environment interaction may both play a role in genetic association studies. In this dissertation, we develop statistical methods in meta-analysis, rare genetic variants analysis and gene by environment interaction analysis, conduct extensive simulation studies, and apply these methods in real data examples. First, we develop a method of moments estimator for the between-study covariance matrix in random effects model multivariate meta-analysis. Our estimator is the first such estimator in matrix form, and holds the invariance property to linear transformations. It has similar performance with existing methods in simulation studies and real data analysis. Next, we extend the Sequence Kernel Association Test (SKAT), a rare genetic variants analysis approach for unrelated individuals, to be applicable in family samples for quantitative traits. The extension is necessary, as the original test has inflated type I error when directly applied to related individuals, and selecting an unrelated subset from family samples reduces the sample size and power. Finally, we derive methods for rare genetic variants analysis in detecting gene by environment interaction on quantitative traits, in the context of univariate test on the interaction term parameter. We develop statistical tests in the settings of both burden test and SKAT, for both unrelated and related individuals. Our methods are relevant to genetic association studies, and we hope that they can facilitate research in this field and beyond.

In the exploration of chemical systems through quantum mechanics, accurate treatment of the electron wavefunction, and the related electron density, is fundamental to extracting information concerning properties of a system. This work examines challenges in achieving accurate chemical information through manipulation of the one-electron basis set.

This work is devoted to seeking methods for analysis of survival data with the Aalen model under special circumstances. We supposed, that all regression functions and all covariates of the observed individuals were nonnegative and we named this class of models monotone Aalen models. To find estimators of the unknown regres- sion functions we considered three maximum likelihood based approaches, namely the nonparametric maximum likelihood method, the Bayesian analysis using Beta processes as the priors for the unknown cumulative regression functions and the Bayesian analysis using a correlated prior approach, where the regression functions were supposed to be jump processes with a martingale structure.

In this thesis we study a variety of two- and three-dimensional (2D and 3D, respectively) nodal semimetals, subjected to local electronic interactions or disorder. Such systems constitute a minimal model for various real materials and capture a plethora of interesting physical phenomena therein. Our methodology includes an unbiased renormalization group analysis controlled by epsilon expansions about the appropriate lower critical dimension, mean-field analysis, as well as complementary numerical analyses. First, we focus on emergent symmetries at various infrared unstable quantum critical points, appearing in a renormalization group flow of interaction couplings. We investigate a 3D chiral Dirac semimetal, which in a noninteracting system enjoys a microscopic U(1)⊗SU(2) global symmetry. Though the chiral symmetry is absent in the interacting model, it gets restored (partially or fully) at various fixed points as emergent phenomena. Subsequently, we study a collection of 3D interacting effective spin-3/2 biquadratic Luttinger fermions, and demonstrate the emergence of full rotational symmetry between the distinct nematic sectors (namely Eg and T2g ) of the corresponding octahedral group. We then investigate the effects of electronic interactions at zero and finite temperature and chemical doping in a collection of (i) 2D Dirac and Luttinger fermions, constituting the linearly and quadratically dispersing low-energy excitations in monolayer and bilayer graphene, respectively, and (ii) 3D Luttinger fermions, describing a biquadratic touching of Kramers degenerate conduction and valence bands, relevant in the normal state of 227 pyrochlore iridates, and half-Heusler compounds, for example. These systems exhibit a plethora of competing broken symmetry phases (both magnetic and superconducting) when tuning the strength of interactions, temperature, and chemical doping. In this context we propose the selection rules, identifying the broken symmetry phases promoted by a given interaction channel, and the organizing principle, ordering these preselected phases along the temperature axis based on a generalized energy-entropy argument. Finally, we explore topological aspects of nodal Fermi liquids. We propose an experimentally feasible way to engineer higher-order topological phases via the application of uniaxial strain on a 3D Luttinger semimetal. Favoring a direction, strain explicitly breaks cubic symmetry. We show that the corresponding nematic orderings of Luttinger fermions result in a topological insulator or Dirac semimetal, depending on the sign (compressive or tensile, respectively) of the strain. We show that both of these phases host 1D hinge modes, localized along the edges parallel to the direction of strain, that are therefore second-order topological in nature. We then investigate the effects of disorder on such a second-order Dirac semimetal, and show its stability for weak enough disorder. At a critical disorder strength the system goes through a quantum phase transition into a diffusive metal phase and the toplogical hinge states melt into the bulk. The methodology presented in this thesis can be extended to a large family of correlated multiband systems, such as Weyl and nodal-loop semimetal.