Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possible definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive understanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. 1

Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possi- ble definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive un- derstanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. Method for Hurst index estimation for fractional Gaussian noise and it's application on logarithmic returns of shares of selected produ- cers of beer are included in this work. 1

Low-income students and blacks make up nearly half of public school students, and on nearly every indicator of educational access, particularly technology, these students have less access than white affluent students (Darling-Hammond, Zielezinski, and Goldman, 2014). The National Center for Education Statistics (2005) reported that teacher quality and missed opportunities to learn accounted for 93% of African Americans, and 87% of Hispanics performing below proficiency in mathematics. Students that do not master mathematics standards by the end of compulsory education are less likely to complete general mathematics courses in upper secondary school and beyond successfully (Levpušček, Zupančič, & Sočan, 2013). Interactive whiteboards (IWBs) can support student engagement, interest and possibly increased achievement in mathematics if used effectively. The purposes of this study were to (a) examine the perspectives of secondary mathematics teachers with regard to the use of IWBs for teaching, (b) determine how secondary mathematics teachers in one school district use the IWB to guide students toward mathematical proficiency, and (c) consider how secondary mathematics teachers’ perspectives in one school district were influenced by 1st order and 2nd order barriers to technology integration. The following factors were considered when examining the context needed to better understand the complexities using IWBs effectively in mathematics: (a) Niess et al. (2009) Mathematics Teachers’ TPACK Development Model, (b) Miller and Glover (2005) stages of IWB use, and (c) Ertmer (1999) first-order and second-order barriers to technology integration. The data revealed that at each stage of IWB use (a) supported didactic, (b)interactive, and (c) enhanced interactivity, teachers faced a unique combination of first-order and second-order barriers to IWB integration that affected how IWBs were used for teaching mathematics. The results of the data suggest that as barriers are resolved at each stage of IWB use, the likelihood mathematics teachers will effectively use IWBs to teach mathematics will increase. Suggestions including administrator support and modifying professional development practices are included to provide educators and policy makers the practical knowledge needed to inform sustainable plans for integrating IWBs effectively.

Voltage stability is a challenging problem in the design and operation of terrestrial and shipboard power systems. DC links can be integrated in the AC systems to increase the transmission capacity or to enhance the distribution performance. However, DC links introduce voltage stability issues related to the reactive power shortage due to power converters. Multi-infeed DC systems make this existing phenomenon more complicated. In addition, shipboard power systems have unique characteristics, and some concepts and methodologies developed for terrestrial power systems need to be investigated and modified before they are extended for shipboard power systems. One goal of this work was to develop a systematic method for voltage stability assessment of hybrid AC/DC systems, independent of system configuration. The static and dynamic approaches have been used as complementary methods to address different aspects in voltage stability. The other goal was to develop or to apply voltage stability indicators for voltage stability assessment. Two classical indicators (the minimum eigenvalue and loading margin) and an improvement (the 2nd order performance indicator) have been jointly used for the prediction of voltage stability, providing information on the system state and proximity to and mechanism of instability. The eliminated variable method has been introduced to calculate the partial derivatives of AC/DC systems for modal analysis. The previously mentioned methodologies and the associated indicators have been implemented for the application of integrated shipboard power system including DC zonal arrangement. The procedure of voltage stability assessment has been performed for three test systems, the WSCC 3-machine 9-bus system, the benchmark integrated shipboard power system, and the modified I RTS-96. The static simulation results illustrate the critical location and the contributing factors to the voltage instability, and screen the critical contingencies for dynamic simulation. The results obtained from various static methods have been compared. The dynamic simulation results demonstrate the response of dynamic characteristics of system components, and benchmark the static simulation results.

Finite volume methods employing second-order gradient reconstruction schemes are often utilized to computationally solve the governing equations of fluid mechanics and transport. These schemes, while not as dissipative as first-order schemes, frequently produce oscillatory solutions in regions of discontinuities and/or unsatisfactory levels of dissipation in smooth regions of the variable field. Limiters are often employed to reduce the inherent variable over- and under-shoot; however, they can significantly increase the numerical dissipation of a solution, eroding a scheme’s performance in smooth regions. A novel gradient reconstruction scheme, which shows significant improvement over traditional second-order schemes, is presented in this work. Two implementations of this Optimization-based Gradient REconstruction (OGRE) scheme are examined: minimizing an objective function based on the mismatch between local reconstructions at midpoints or selected quadrature points between cell stencil neighbors. Regardless of the implementation employed, the resulting gradient calculation is a compact, implicit method that can be used with unstructured meshes by employing an arbitrary computational stencil. An adjustable weighting parameter is included in the objective function that allows the scheme to be tuned towards either greater accuracy or greater stability. To address over- and undershoot of the variable field near discontinuities, non-local, non-monotonic (NLNM) and local, non-monotonic (LNM) limiters have also been developed, which operate by enforcing cell minima and maxima on dependent variable values projected to cell faces. The former determines minimum and maximum values for a cell through recursive reference to the minimum and maximum values of its upwind neighbors. The latter determines these bounding values through examination of the extrema of values of the dependent variable projected from the face-neighbor cell into the original cell. Steady state test cases on structured and unstructured grids are presented, exhibiting the low-dissipative nature of the scheme. Results are primarily compared to those produced by existing limited and unlimited second-order upwind (SOU) and first-order upwind (FOU). Solution accuracy, convergence rate and computational costs are examined.

The focus on specific principal leadership behaviors that positively impact student achievement has become more and more pronounced since the inception of the No Child Left Behind Act of 2001. Recently, researchers have begun to focus on a more dramatic type of change as a method for improving student achievement in schools. Marzano, Waters, and McNulty (2005) conducted a meta-analysis of more than 5,000 studies and identified seven leadership behaviors that related to improved student achievement and were viewed as second-order in nature. In many cases, second-order change was needed (a) to accomplish the student achievement improvements necessary to attain Adequate Yearly Progress (AYP) and (b) to ensure that all students would read on grade level by 2014. For this study, 66 principals from schools with fewer than 60% of students who qualified for free and reduced-price lunches from five urban Florida school districts completed an online survey, Principal Actions Survey (PAS), created to determine which of the seven leadership behaviors successful principals utilized in their schools. Principals were specifically asked to comment on those actions that they felt impacted student achievement and achievement of AYP. Principals consistently responded that they used the seven leadership behaviors, but the results from this study indicated very few statistically significant relationships or predictive relationships. The 66 principal responses were also compared to responses on the PAS of principals from urban Florida elementary schools with more than 60% of students who qualified for free and reducedprice lunches (La Cava, 2009). These comparisons indicated that principals of schools iv with a higher level of poverty reported utilization of the seven leadership behaviors on a more frequent basis or with a higher success rate than principals at schools with lower poverty levels.

The purpose of this mixed-method, descriptive study was to determine the teachers' perceptions who were employed in the target school district from 2003 to 2011, regarding school district second-order change leadership decisions, events, and challenges, and the extent to which equity and access to excellence for all students were achieved. Also investigated was the relationship, if any, that existed in achieving equity and access to excellence based on school district second-order change leadership from 2003 to 2011. Teacher perception data were analyzed from a survey presented to teachers in over 16 schools who had been consecutively employed in the target school district from 2003 to 2011. The findings of this research suggest that teachers' perceptions of specific school district leadership decisions, events, and challenges contributed to improving opportunities for students who historically were not provided equitable opportunities for academic achievement and post high school career advancement. Beginning in 2003, the target school district underwent a leadership transition period in which a new superintendent established history-making goals and objectives for the school district. The findings suggest that based on teachers' perceptions, school district efforts provided for greater access to technology, high quality instruction, specific programs of study such as the implementation of magnet programs, and college preparation courses. The greater access provided the opportunity for equity and access to excellence for all students, especially those who historically lacked access and investment with respect to their demographics (race, gender, socioeconomic status, and ethnicity). There were limitations iv to this study. Objectivity may be questioned since the participants were employees of the school district. It was assumed that participants in the study responded accurately and honestly to the questions asked in the interviews and survey. Future research is recommended that would include a larger and more diverse sample. Further recommendations include separate studies to examine the differences between student achievement as a result of school district leadership efforts to attain access to equity and excellence based on college readiness assessment exam scores such as the SAT and/or the ACT, and college or technical school entrance and completion, with regard to student subgroups such as race, ethnicity, and family income.

Modern theories of authoritarianism have stressed the importance of threat to the expression of authoritarian attitudes and intolerance. Arguably, authoritarian tendencies may have increased during COVID-19 pandemic, a major threat to life and security. One issue arising when comparing mean scores is that of measurement invariance. Meaningful comparisons are only possible, if latent constructs are similar between groups and/or across time. This prerequisite is rarely ever tested in research on authoritarianism. In this study, we aim to analyze the short scale for authoritarianism KSA-3 by investigating its measurement invariance on two levels (three first-order and one second-order factors) and latent mean changes using two German representative samples (N = 4,905). Specifically, we look at differences before and during the pandemic (2017 vs. 2020). While measurement invariance holds across both levels in all conditions, we find a decrease in latent means in 2020, contrary to expectations and established theories. Moreover, latent means differ with regard to gender, education, and east–west Germany. We conclude that analyses of latent means and measurement invariance instead of mean comparisons with composites should become the standard. Future studies should focus on threat as a moderator between authoritarianism and intolerance, and on possible interactions with context variables.

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This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to interface problems of related partial differential equations. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problems. We introduce an energy norm stronger than the one used in [111]. Then we derive an estimate for the IFE interpolation error with this energy norm using patches of interface elements. We prove both the continuity and coercivity of the bilinear form in a partially penalized IFE (PPIFE) method. These properties allow us to derive an error bound for the PPIFE solution in the energy norm under the standard piecewise $H^2$ regularity assumption instead of the more stringent $H^3$ regularity used in [111]. As an important consequence, this new estimation further enables us to show the optimal convergence in the $L^2$ norm which could not be done by the analysis presented in [111]. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. The first application is for the time-harmonic wave interface problem that involves the Helmholtz equation with a discontinuous coefficient. We design PPIFE and DGIFE schemes including the higher degree IFEs for Helmholtz interface problems. We present an error analysis for the symmetric linear/bilinear PPIFE methods. Under the standard piecewise $H^2$ regularity assumption for the exact solution, following Schatz's arguments, we derive optimal error bounds for the PPIFE solutions in both an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. {In the second group of applications, we focus on the error analysis for IFE methods developed for solving typical time-dependent interface problems associated with the second order elliptic operator with a discontinuous coefficient.} For hyperbolic interface problems, which are typical wave propagation interface problems, we reanalyze the fully-discrete PPIFE method in [143]. We derive the optimal error bounds for this PPIFE method for both an energy norm and the $L^2$ norm under the standard piecewise $H^2$ regularity assumption in the space variable of the exact solution. Simulations for standing and travelling waves are presented to corroborate the results of the error analysis. For parabolic interface problems, which are typical diffusion interface problems, we reanalyze the PPIFE methods in [113]. We prove that these PPIFE methods have the optimal convergence not only in an energy norm but also in the usual $L^2$ norm under the standard piecewise $H^2$ regularity. / Doctor of Philosophy / This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to a few types of interface problems. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problem. We can show that the IFE methods for the elliptic interface problems converge optimally when the exact solution has lower regularity than that in the previous publications. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. For interface problems of the Helmholtz equation which models time-Harmonic wave propagations, we design IFE schemes, including higher degree schemes, and derive error estimates for a lower degree scheme. For interface problems of the second order hyperbolic equation which models time dependent wave propagations, we derive better error estimates for the IFE methods and provides numerical simulations for both the standing and traveling waves. For interface problems of the parabolic equation which models the time dependent diffusion, we also derive better error estimates for the IFE methods.