Numerical Analysis of Linear and Nonlinear Schrodinger Equations on Quantum Graphs

A wide variety of problems in quantum mechanics can be modeled with Schrodinger-type equations on quantum graphs. More specifically, graphs are useful for simplifying models of physical systems that feature nano-scaled branching structures. Since analytical solutions can only be found for a few trivial cases, it is necessary to consider how to accurately solve this type of problem numerically. While a wide variety of tools exist to solve these types of partial differential equations on lines, this is not well studied in the case of graphs - one critical difference between the two cases being that graphs require more complicated boundary conditions. This paper utilizes a new approach to include the boundary conditions in the discretized operator that preserves high levels of accuracy. Thus, a proper time evolution scheme must work in conjunction with a spatial operator that has incorporated boundary conditions and preserve the accuracy of our spatial component, and this is accomplished by implementing methods from differential algebraic equations. We study the numerical computation of linear and nonlinear states for Schrodinger equations on graphs, as well as numerically compute their linear stability properties. The latter result has implications for the future study of relative periodic orbits on graphs. All numerical components are being adapted into a MATLAB software package called QGLAB jointly with Roy Goodman on Github.Doctor of Philosoph

Stability of Periodic Waves in Nonlocal Dispersive Equations

In this work consisting of joint projects with my advisor, Dr. Mathew Johnson, we study the existence and stability of periodic waves in equations that possess nonlocal dispersion, i.e. equations in which the dispersion relation between the temporal frequency, omega, and wavenumber, k, of a plane wave is not polynomial in ik. In models that involve only classical derivative operators (known as local equations), the behavior of the system at a point is influenced solely by the behavior in an arbitrarily small neighborhood. In contrast, equations involving nonlocal operators incorporate long-range interactions as well. Such operators appear in numerous applications, including water wave theory and mathematical biology. Specifically, we establish the existence and nonlinear stability of a special class of periodic bound state solutions of the Fractional Nonlinear Schrodinger Equation, where the nonlocality of the fractional Laplacian presents formidable analytical challenges and elicits the development of functional-analytic tools to complement the absence of more-understood techniques commonly used to analyze local equations. Further, we use numerical methods to survey the existence and spectral stability of small- and large-amplitude periodic wavetrains in Bidirectional Whitham water wave models, which implement the exact (nonlocal) dispersion relation of the incompressible Euler equations and are thus expected to better capture high-frequency phenomena than the unidirectional Whitham and Korteweg-de Vries (KdV) equations

Stability of Periodic Waves in Nonlocal Dispersive Equations

In this work consisting of joint projects with my advisor, Dr. Mathew Johnson, we study the existence and stability of periodic waves in equations that possess nonlocal dispersion, i.e. equations in which the dispersion relation between the temporal frequency, omega, and wavenumber, k, of a plane wave is not polynomial in ik. In models that involve only classical derivative operators (known as local equations), the behavior of the system at a point is influenced solely by the behavior in an arbitrarily small neighborhood. In contrast, equations involving nonlocal operators incorporate long-range interactions as well. Such operators appear in numerous applications, including water wave theory and mathematical biology. Specifically, we establish the existence and nonlinear stability of a special class of periodic bound state solutions of the Fractional Nonlinear Schrodinger Equation, where the nonlocality of the fractional Laplacian presents formidable analytical challenges and elicits the development of functional-analytic tools to complement the absence of more-understood techniques commonly used to analyze local equations. Further, we use numerical methods to survey the existence and spectral stability of small- and large-amplitude periodic wavetrains in Bidirectional Whitham water wave models, which implement the exact (nonlocal) dispersion relation of the incompressible Euler equations and are thus expected to better capture high-frequency phenomena than the unidirectional Whitham and Korteweg-de Vries (KdV) equations

Meaning Behind the Metrics of Misery: Understanding Prevalence Estimates of Poor Mental Health in Two Samples of Older Rural Indonesians

Background: Late life is typically accompanied by unique physical and mental health challenges. Fewer older people are diagnosed with mood or anxiety-specific disorders than their younger counterparts. However, older people score more highly than younger people on symptom screens indicating high levels of clinically relevant depressive, anxiety, and nonspecific psychological distress symptoms which cause high morbidity, mortality, disability, and poor quality of life. The unique presentation of late life psychiatric syndromes, such as depression and anxiety, remain largely unaddressed in existing psychiatric nosology and measurement techniques, as do depictions of depression and anxiety across diverse cultural contexts. Very few studies exist investigating either the descriptive epidemiology of depression and anxiety among older adults living in low-middle income countries (LMIC) or the unique challenges of mental health measurement in LMIC contexts. This dissertation contributes to this developing evidence base by providing a critical analysis of point prevalence estimates of depression, anxiety, and nonspecific psychological distress (distress) symptoms in two samples of Indonesian rural older persons. Methods: We enumerated greater than or equal to 60-year-olds in 12 Indonesian rural villages as part of the Ageing in Rural Indonesia Study in 2015/16 (N=2526; sample 1). We re-enumerated two of the 12 villages surveyed in 2015 in 2017 (N=536; sample 2). Depressive and distress symptoms were each measured using three scales: PHQ-8/9, CES-D, GDS, and K6, DQ5 and SRQ-20 respectively. Anxiety symptoms were evaluated with the GAD-7. Classical Test Theory and Item Response Theory were used to investigate the psychometric properties of symptom screens. We also undertook mixed effects modelling and Moderated Nonlinear Factor Analysis to identify sources of variability in prevalence estimates. Results: Commonly used cut points of short symptom screens used to approximate diagnostic depressive disorders produced estimates that typically lacked comparability (e.g., sample 2 point-prevalence 3.2%-39.9%). Psychometric analysis further identified mental health scales with better (PHQ-8/9, GAD-7, K6, DQ5) and poorer (GDS, SRQ) construct validity. Sources of variability in point prevalence estimates of depression, anxiety and distress symptoms were identified, and related to study design, cognitive ability, marital status, financial means, level of social support, lifestyle, and health related status. Pervasive non-invariance was identified in participant responses to scale items related to gender, literacy, and ethnicity. However, when modelled, measurement non-invariance did not substantially modify means. Females, respondents with lower literacy levels, and Batak and Sundanese sample villages had significantly higher levels of depression, anxiety, and distress symptoms. Conclusion: The practice of using existing mental health symptom screens combined with commonly used cut points as proxies for depression and anxiety in older rural Indonesians and other diverse populations should be avoided. Rigorous psychometric and diagnostic validation evidence should be ascertained. In the interim, better performing symptom screening tools (i.e., PHQ-8/9, GAD-7, K6, DQ5) may be used as measures of continuous symptom severity. Future research should focus on evaluating the distinctive and overlapping features of mental ill-health in specific subpopulations of Indonesians

Convergence analysis of the NOJA algorithm using the ODE approach

10.1016/j.sigpro.2006.06.014Signal Processing86113490-3495SPRO

Lives and work of the Serbian scientists

This is the 10th volume of the edition entitled Lives and work of the Serbian scientists which is being published on a regular basis since the beginning of 1996. The first book of this edition was the result of dedicated and systematic work of the Committee for research into lives and work of the scientists in Serbia and scientists of Serbian origin, founded in 1992. Since the day of its foundation until the first book came out, the Committee has compiled a list of over 200 scientists from the fields of natural sciences, mathematics, medicine and technical sciences and who, with the results of their research work, have left invaluable and memorable mark on our world of science. This book contains life stories and working biographies of 13 new scientists from the fields of mathematics, mechanics, geology, bio-technology, medicine, mechanical engineering and civil engineering. The Edition contains now an impressive record of the biographies of 125 scientists, including the previously elaborated 112, whose endeavours will be permanently preserved. The Committee has, however, completed just a little over half of its planned course of action. It is laced with a difficult task of finding competent authors, who will write articles about the remaining scientists, and extending the already existing list by gradually adding new names to it. Book number ten of any edition may well be considered a very special publication. Remembering great contributions by the late academician Miloje R. Sarić (1925-2002), who initiated the founding of the Committee and was also the editor of the first eight books of the Edition set up by him, the Committee felt obligated to publish an article about the life and work of academician Sarić in this special issue. It is my pleasant duty to once again thank all of the Committee members for the efforts they invested in choosing competent authors and reviewers, and for their invaluable comments and suggestions which helped this book to maintain the quality of the previous volumes. I also wish to express my gratitude to the authors of individual articles, and reviewers, donators, technical team of the Serbian Academy of Sciences and Arts and the personnel from the printing company Poligraf.Биографије и библиографије / Српска академија наука и уметности : књ. 10. II одељење, Одбор за проучавање живота и рада научника српског порекла ; књ. 1