An Analysis of the Theory of Functions of One Real Variable

Few undergraduates are aware that the Riemann integral taught in introductory calculus courses has only limited application-essentially this integral can be used only to integrate continuous functions over intervals. The necessity to integrate a broader class of functions over a wider range of sets that arises in many applications motivates the theory of abstract integration and functional analysis. The founder of this theory was the French mathematician Henri Lebesgue, who in 1902 defined the Lebesgue measure of subsets of the real line. The purpose of this project is to elucidate the theory of abstract measure spaces and of important spaces of functions (a critical example of which are Banach spaces), and extend the application of this theory. Developing the tools for doing so has been the focus of my advisor Professor Dmitry Khavinson and me over the past three years. The primary goal of the thesis is to make this highly formal and abstract material accessible to an undergraduate having only a year of coursework in advanced calculus. These concepts are typically introduced at the graduate level, but the ideas require only a familiarity with the analytic style of proof learned as an undergraduate. It would be advantageous to expose advanced undergraduates to this material since these ideas form the foundation for how mathematical research is done at the professional level. The addition of interesting and practical examples (which are scarce in the standard graduate texts) will help to make the concepts more familiar and down-to-earth. The motivation for a new theory of integration came from the Riemann integral\u27s apparent inability to operate on functions that fail to be continuous. For example, the Riemann integral of the function that assigns the value 1 to rational numbers and 0 to irrational numbers can be evaluated over the interval [0, 1] with equally valid justification to be 0 or 1. This is because the definition of the Riemann integral depends on partitioning the domain of the function to be integrated, and finding the maximum and minimum values of the function over each partition. The Lebesgue integral, on the other hand, partitions the range of the function to be integrated and then considers the length of the Jason Reed and Dmitry Khavinson pre-image of each partition as well as the maximum and minimum values of the function of the partition. The utility of this change of perspective arises when we refine what is meant by length in the aforementioned pre-image. The Riemann integral requires that the domain consist of intervals of real numbers (where length makes sense), while the Lebesgue integral can be used with a much broader class of sets. Lebesgue modified the notion of length by defining the measure of a set E to be the smallest possible total length of all collections of intervals that cover E. Using this ingenious method, Lebesgue constructed a theory of integration which forms the most useful example of all general integration theories. The theory has important applications in many areas of science and engineering as well as probability and statistics. Our approach to the subject has emphasized theory developed in H.L. Royden\u27s classic text, Real Analysis. My project has included analysis of each concept in the text, and I have developed for each major subject a collection of problems solved and applications of major theorems that were explored. The result has been comprehension of many of the foundational ideas in the field. We have used a number of supplemental texts to gain depth of understanding where Royden\u27s text provides only a survey, such as the Riesz Representation theorem, and to extend important ideas, such as the consideration of complex-valued (in addition to real-valued) measures. The synthesis has been a comprehensive paper which describes the theoretical directions the research has taken, the major results and theorems with proof, and applications and examples which are worked out in detail. The final record of my research will be divided into the following six sections: Lebesgue measure, Lebesgue integral, relationship between differentiation and Lebesgue integration, Banach space theory, abstract measure theory, and general integration theory. The analysis encompasses discussion of the main ideas (what it means for a set function to be a measure, how an integral can be defined in a coherent way with respect to a measure, when the derivative of an integral of a function is the function itself, different ideas about what it means for a sequence of functions to converge to a function, what are the properties of Banach spaces and why they are useful, etc.), as well as important ideas and theorems that interrelate these concepts (i.e., when we can interchange the limit of a sequence of functions and the integral, how we can represent a bounded linear functional, the structure of certain spaces of integrable functions)

Optimizing the fine lock performance of the Hubble Space Telescope fine guidance sensors

This paper summarizes the on-orbit performance to date of the three Hubble Space Telescope Fine Guidance Sensors (FGS's) in Fine Lock mode, with respect to acquisition success rate, ability to maintain lock, and star brightness range. The process of optimizing Fine Lock performance, including the reasoning underlying the adjustment of uplink parameters, and the effects of optimization are described. The Fine Lock optimization process has combined theoretical and experimental approaches. Computer models of the FGS have improved understanding of the effects of uplink parameters and fine error averaging on the ability of the FGS to acquire stars and maintain lock. Empirical data have determined the variation of the interferometric error characteristics (so-called 's-curves') between FGS's and over each FGS field of view, identified binary stars, and quantified the systematic error in Coarse Track (the mode preceding Fine Lock). On the basis of these empirical data, the values of the uplink parameters can be selected more precisely. Since launch, optimization efforts have improved FGS Fine Lock performance, particularly acquisition, which now enjoys a nearly 100 percent success rate. More recent work has been directed towards improving FGS tolerance of two conditions that exceed its original design requirements. First, large amplitude spacecraft jitter is induced by solar panel vibrations following day/night transitions. This jitter is generally much greater than the FGS's were designed to track, and while the tracking ability of the FGS's has been shown to exceed design requirements, losses of Fine Lock after day/night transitions are frequent. Computer simulations have demonstrated a potential improvement in Fine Lock tracking of vehicle jitter near terminator crossings. Second, telescope spherical aberration degrades the interferometric error signal in Fine Lock, but use of the FGS two-thirds aperture stop restores the transfer function with a corresponding loss of throughput. This loss requires the minimum brightness of acquired stars to be about one magnitude brighter than originally planned

Software Project Management: Scenarios of Failure and Potential Solutions

This paper provides the project manager (PM) with three (3) scenarios for examining project failure. For each scenario a decision tree is provided. By asking questions about the task, the PM can determine points of failure, and potential remedies. This is intended to be a learning tool and should affect how a PM prepares for his/her next project

Longitudinal Monitoring of SARS-CoV-2 IgM and IgG Seropositivity to Detect COVID-19.

BackgroundSevere acute respiratory syndrome coronavirus 2 (SARS-CoV-2), is a novel beta-coronavirus that has recently emerged as the cause of the 2019 coronavirus pandemic (COVID-19). Polymerase chain reaction (PCR) based tests are optimal and recommended for the diagnosis of an acute SARS-CoV-2 infection. Serology tests for viral antibodies provide an important tool to diagnose previous exposure to the virus. Here we evaluate the analytical performance parameters of the Diazyme SARS-CoV-2 IgM/IgG serology assays and describe the kinetics of IgM and IgG seroconversion observed in patients with PCR-confirmed COVID-19 who were admitted to our hospital.MethodsWe validated the performance of the Diazyme assay in 235 presumed SARS-CoV-2 negative subjects to determine specificity. Subsequently, we evaluated the SARS-CoV-2 IgM and IgG seroconversion of 54 PCR-confirmed COVID-19 patients and determined sensitivity of the assay at three different timeframes.ResultSensitivity and specificity for detecting seropositivity at ≥15 days following a positive SARS-CoV-2 PCR result, was 100.0% and 98.7% when assaying for the panel of IgM and IgG. The median time to seropositivity observed for a reactive IgM and IgG result from the date of a positive PCR was 5 days (IQR: 2.75-9 days) and 4 days (IQR: 2.75-6.75 days), respectively.ConclusionsOur data demonstrate that the Diazyme IgM/IgG assays are suited for the purpose of detecting SARS-CoV-2 IgG and IgM in patients with suspected SARS-CoV-2 infections. For the first time, we report longitudinal data showing the evolution of seroconversion for both IgG and IgM in a cohort of acutely ill patients in the United States. We also demonstrate a low false positive rate in patients who were presumed to be disease free

On Information Theory, Spectral Geometry and Quantum Gravity

We show that there exists a deep link between the two disciplines of information theory and spectral geometry. This allows us to obtain new results on a well known quantum gravity motivated natural ultraviolet cutoff which describes an upper bound on the spatial density of information. Concretely, we show that, together with an infrared cutoff, this natural ultraviolet cutoff beautifully reduces the path integral of quantum field theory on curved space to a finite number of ordinary integrations. We then show, in particular, that the subsequent removal of the infrared cutoff is safe.Comment: 4 page

Thin-film fixed-bed reactor (TFFBR) for solar photocatalytic inactivation of aquaculture pathogen Aeromonas hydrophila

Background: Outbreaks of infectious diseases by microbial pathogens can cause substantial losses of stock in aquaculture systems. There are several ways to eliminate these pathogens including the use of antibiotics, biocides and conventional disinfectants, but these leave undesirable chemical residues. Conversely, using sunlight for disinfection has the advantage of leaving no chemical residue and is particularly suited to countries with sunny climates. Titanium dioxide (TiO2) is a photocatalyst that increases the effectiveness of solar disinfection. In recent years, several different types of solar photocatalytic reactors coated with TiO2 have been developed for waste water and drinking water treatment. In this study a thin-film fixed-bed reactor (TFFBR), designed as a sloping flat plate reactor coated with P25 DEGUSSA TiO2, was used. Results: The level of inactivation of the aquaculture pathogen Aeromonas hydrophila ATCC 35654 was determined after travelling across the TFFBR under various natural sunlight conditions (300- 1200 W m-2), at 3 different flow rates (4.8, 8.4 and 16.8 L h-1). Bacterial numbers were determined by conventional plate counting using selective agar media, cultured (i) under conventional aerobic conditions to detect healthy cells and (ii) under conditions designed to neutralise reactive oxygen species (agar medium supplemented with the peroxide scavenger sodium pyruvate at 0.05% w/v, incubated under anaerobic conditions), to detect both healthy and sub-lethally injured (oxygen-sensitive) cells. The results clearly demonstrate that high sunlight intensities (≥600 W m-2) and low flow rates (4.8 L h-1) provided optimum conditions for inactivation of A. hydrophila ATCC 3564, with greater overall inactivation and fewer sub-lethally injured cells than at low sunlight intensities or high flow rates. Low sunlight intensities resulted in reduced overall inactivation and greater sub-lethal injury at all flow rates.Conclusions: This is the first demonstration of the effectiveness of the TFFBR in the inactivation of Aeromonas hydrophila at high sunlight intensities, providing proof-of-concept for the application of solar photocatalysis in aquaculture systems

Limiting absorption principle for the dissipative Helmholtz equation

Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when trapped trajectories meet the set where the imaginary part of the potential is non-zero. We also give the resolvent estimates in Besov spaces

Dietetics program directors in the United States support teaching vegetarian and vegan nutrition and half connect vegetarian and vegan diets to environmental impact

Registered Dietitian Nutritionists (RDNs) are clinicians trained in the application of food, nutrition, and dietetics. Vegetarians and vegans have a lower risk of many nutrition-related chronic diseases that are epidemic while vegetarian and vegan diets are associated with reduced environmental impact. Despite this strong diet-disease and diet-environment connection, it is not known if dietetics students are taught the principles of vegetarian and vegan nutrition. The overarching goal of our study was to investigate curricular practices in accredited dietetics training programs in the United States (U.S.) including (1) the prevalence and perceived importance of vegetarian and vegan nutrition instruction and (2) if program directors connect vegetarian and vegan diets to climate change mitigation and resource conservation. Primary data were collected by way of a cross-sectional, Internet-based survey. All Accreditation Council for Education in Nutrition and Dietetics (ACEND) program directors in the U.S. (N = 574) were sent a 37-question survey and invited to participate in the study. Outcome measures included the prevalence of vegetarian and vegan nutrition instruction, quantifying if relationships exist among variables, and the frequency of connecting vegetarian and vegan diets to environmental impact. Descriptive and inferential statistics were utilized. Respondents (n = 205) indicated that over 51% of programs teach vegetarian nutrition while 49% teach vegan nutrition. There were significant differences between program type and the prevalence of vegetarian (p = 0.00005) and vegan (p = 0.00005) nutrition instruction. Over 90% of program directors believe that vegetarian and vegan nutrition should be taught. Over 50% of programs identify the connection between vegetarian and vegan diets in climate change mitigation and resource conservation. Most ACEND program directors believe vegetarian and vegan nutrition should be taught and half connect diet to environmental concern. Nevertheless, there is a discrepancy between beliefs and practice behaviors. These results suggest the need for increased collaboration and the use of novel techniques that better incorporate vegan and vegetarian nutrition throughout dietetics education

Evaluating Airsoft Electric Guns for Control of Invasive Brown Treesnakes

Firearms are often used in lethal control of invasive vertebrates, but safety and regulatory aspects limit the circumstances under which they can be used. During August 2016 at the Brown Treesnake Project laboratory on Guam, we evaluated hobby‐grade Airsoft Electric Guns (AEGs; a lower powered, less‐hazardous, and less‐regulated alternative to firearms) for capture and control of small animals, with specific emphasis on invasive brown treesnakes (Boiga irregularis). Tests of AEGs differing in power with ammunition (plastic pellets) masses ranging from 0.20 to 0.39 g, fired at gelatin blocks from distances of 4, 8, and 12 m, showed that heavy ammunition is of overriding importance for maximizing lethality: 0.39‐g pellets penetrated more deeply at 12 m than did 0.20‐g pellets at 4 m. Inspection of tissue damage in brown treesnake carcasses subjected to fire with the 0.39‐g ammunition from the same distances suggested that injuries sustained by a direct hit from 12 m would often be lethal, and snakes would be unlikely to survive multiple hits from automatic fire discharged at approximately 17/s. Limited trials with live snakes helped us to understand behavioral responses in a snake hit by ≥1 pellets, including distance traveled over time. Based on these factors, we assessed the risk that a snake injured by pellet fire might evade subsequent capture by rapid responders in the proximity. We also discuss ethical considerations and regulatory advantages of using AEGs