Class Notes for Math 921/922: Real Analysis, Instructor Mikil Foss

Topics include: Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon- Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration, Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon-Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration.

Isotopic, trace element and nutrient characterization of coastal waters from Ubatuba inner shelf area, south-eastern Brazil

Stable isotopes, tritium, radium isotopes, radon, trace elements and nutrients data were collected during two sampling campaigns in the Ubatuba coastal area (south-eastern Brazil) with the aim of investigating submarine groundwater discharge (SGD) in the region. The isotopic composition (delta D, delta(18)O, (3)H) of submarine waters was characterised by significant variability and heavy isotope enrichment. The stable isotopes and tritium data showed good separation of groundwater and seawater groups. The contribution of groundwater in submarine waters varied from a few % to 17%. Spatial distribution of (222)Rn activity concentration in surface seawater revealed changes between 50 and 200 Bq m(-3) which were in opposite relationship with observed salinities. Time series measurements of (222)Rn activity concentration in Flamengo Bay (from 1 to 5 kBq m(-3)), obtained by in situ underwater gamma-spectrometry showed a negative correlation between the (222)Rn activity concentration and tide/salinity. This may be caused by sea level changes as tide effects induce variations of hydraulic gradients, which increase (222)Rn concentration during lower sea level, and opposite, during high tides where the (222)Rn activity concentration is smaller. The estimated SGD fluxes varied during 22-26 November between 8 and 40 cm d(-1), with an average value of 21 cm d(-1) (the unit is cm(3)/cm(2) per day). The radium isotopes and nutrient data showed scattered distributions with offshore distance and salinity. which implies that in a complex coast with many small bays and islands, the area has been influenced by local currents and groundwater-seawater mixing. SGD in the Ubatuba area is fed by coastal contaminated groundwater and re-circulated seawater (with small admixtures of groundwater). which claims for potential environmental concern with implications on the management of freshwater resources in the region. (C) 2007 Elsevier Ltd. All rights reserved.

Identificação de ondas de Rossby a partir de diagramas de Hovmoller da concentração de clorofila entre as latitudes de 25o S e 45o S do Oceano Atlântico

Ondas de Rossby são ondas planetárias oceânicas de ocorrência quase global com comprimentos de onda da ordem de centenas a milhares de quilômetros, períodos da ordem de meses a anos e amplitudes superficiais da ordem de centímetros, causando anomalia da altura da superfície do mar (AASM). Transportando grandes quantidades de energia e influenciando diretamente a circulação de larga escala e o clima global, sua propagação de fase acontece sempre para oeste e sua velocidade é predominantemente uma função da latitude. É sabido que relações entre AASM e concentração de clorofila (CC) podem ser encontradas em todas as bacias oceânicas, evidenciando uma clara influência de fenômenos físicos em fenômenos biológicos. O presente trabalho visa a identificação de ondas de Rossby através da visualização de dados de CC, que, quando comparados a de AASM, apresentaram comprimento, período e velocidade de fase conforme previsto teoricamente para ondas de Rossby baroclínicas do primeiro modo vertical. A região de estudo, entre as latitudes de 25o S e 45o S, é caracterizada pela presença da Corrente do Atlântico Sul, da convergência subtropical e da faixa de rotacional nulo do vento e é, portanto, sede de importantes fenômenos oceanográficos de larga escala. Ambos os conjuntos de dados foram obtidos por sensoriamento remoto e submetidos a uma cadeia de filtros bidimensionais de resposta impulsiva finita (FIR2D) cujo funcionamento de baseia no princípio de convolução entre a série temporal e uma matriz-filtro. Os sinais filtrados foram decompostos em componentes propagantes e não-propagantes e entre os propagantes foi feita a separação das componentes em diversas bandas espectrais definidas por seus períodos centrais. Diagramas de Hovmoller foram confeccionados e as velocidades de fase foram obtidas através da transformada de Radon. A conclusão do trabalho evidenciou uma clara influência das ondas de Rossby sobre a distribuição de clorofila ao longo da região estudada.

Identificação de ondas de Rossby a partir de diagramas de Hovmoller da concentração de clorofila entre as latitudes de 25o S e 45o S do Oceano Atlântico

Ondas de Rossby são ondas planetárias oceânicas de ocorrência quase global com comprimentos de onda da ordem de centenas a milhares de quilômetros, períodos da ordem de meses a anos e amplitudes superficiais da ordem de centímetros, causando anomalia da altura da superfície do mar (AASM). Transportando grandes quantidades de energia e influenciando diretamente a circulação de larga escala e o clima global, sua propagação de fase acontece sempre para oeste e sua velocidade é predominantemente uma função da latitude. É sabido que relações entre AASM e concentração de clorofila (CC) podem ser encontradas em todas as bacias oceânicas, evidenciando uma clara influência de fenômenos físicos em fenômenos biológicos. O presente trabalho visa a identificação de ondas de Rossby através da visualização de dados de CC, que, quando comparados a de AASM, apresentaram comprimento, período e velocidade de fase conforme previsto teoricamente para ondas de Rossby baroclínicas do primeiro modo vertical. A região de estudo, entre as latitudes de 25o S e 45o S, é caracterizada pela presença da Corrente do Atlântico Sul, da convergência subtropical e da faixa de rotacional nulo do vento e é, portanto, sede de importantes fenômenos oceanográficos de larga escala. Ambos os conjuntos de dados foram obtidos por sensoriamento remoto e submetidos a uma cadeia de filtros bidimensionais de resposta impulsiva finita (FIR2D) cujo funcionamento de baseia no princípio de convolução entre a série temporal e uma matriz-filtro. Os sinais filtrados foram decompostos em componentes propagantes e não-propagantes e entre os propagantes foi feita a separação das componentes em diversas bandas espectrais definidas por seus períodos centrais. Diagramas de Hovmoller foram confeccionados e as velocidades de fase foram obtidas através da transformada de Radon. A conclusão do trabalho evidenciou uma clara influência das ondas de Rossby sobre a distribuição de clorofila ao longo da região estudada

Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

Contenido de radón en aguas subterráneas del noreste de Gran Canaria

[ES] En este trabajo se han determinado los niveles de 222Rn en el agua subterránea en la zona noreste de Gran Canaria a partir de 28 muestras de pozos en bombeo. La concentración de actividad de radón en una muestra de agua se determina mediante un sistema en circuito cerrado que consta de un monitor AlphaGUARD que mide la concentración de radón en aire por medio de una cámara de ionización y un conjunto AquaKIT que se utiliza para transferir el radón disuelto en la muestra de agua al aire del circuito. Los valores de la concentración de radón en agua de las muestras estudiadas varían entre 0.9 y 76.9 Bq/L. Debido a la peligrosidad radiológica del radón, en España se ha establecido un límite de actividad de 100 Bq/L de 222Rn para las aguas de consumo humano. Los valores obtenidos para todas las muestras analizadas se encuentran por debajo de este límite.

Modeling surface-atmosphere exchange of trace gases and energy within and above the Amazon rain forest

Die vorliegende Dissertation untersucht die biogeochemischen Vorgänge in der Vegetationsschicht (Bestand) und die Rückkopplungen zwischen physiologischen und physikalischen Umweltprozessen, die das Klima und die Chemie der unteren Atmosphäre beeinflussen. Ein besondere Schwerpunkt ist die Verwendung theoretischer Ansätze zur Quantifizierung des vertikalen Austauschs von Energie und Spurengasen (Vertikalfluss) unter besonderer Berücksichtigung der Wechselwirkungen der beteiligten Prozesse. Es wird ein differenziertes Mehrschicht-Modell der Vegetation hergeleitet, implementiert, für den amazonischen Regenwald parametrisiert und auf einen Standort in Rondonia (Südwest Amazonien) angewendet, welches die gekoppelten Gleichungen zur Energiebilanz der Oberfläche und CO2-Assimilation auf der Blattskala mit einer Lagrange-Beschreibung des Vertikaltransports auf der Bestandesskala kombiniert. Die hergeleiteten Parametrisierungen beinhalten die vertikale Dichteverteilung der Blattfläche, ein normalisiertes Profil der horizontalen Windgeschwindigkeit, die Lichtakklimatisierung der Photosynthesekapazität und den Austausch von CO2 und Wärme an der Bodenoberfläche. Desweiteren werden die Berechnungen zur Photosynthese, stomatären Leitfähigkeit und der Strahlungsabschwächung im Bestand mithilfe von Feldmessungen evaluiert. Das Teilmodell zum Vertikaltransport wird im Detail unter Verwendung von 222-Radon-Messungen evaluiert. Die ``Vorwärtslösung'' und der ``inverse Ansatz'' des Lagrangeschen Dispersionsmodells werden durch den Vergleich von beobachteten und vorhergesagten Konzentrationsprofilen bzw. Bodenflüssen bewertet. Ein neuer Ansatz wird hergeleitet, um die Unsicherheiten des inversen Ansatzes aus denjenigen des Eingabekonzentrationsprofils zu quantifizieren. Für nächtliche Bedingungen wird eine modifizierte Parametrisierung der Turbulenz vorgeschlagen, welche die freie Konvektion während der Nacht im unteren Bestand berücksichtigt und im Vergleich zu früheren Abschätzungen zu deutlich kürzeren Aufenthaltszeiten im Bestand führt. Die vorhergesagte Stratifizierung des Bestandes am Tage und in der Nacht steht im Einklang mit Beobachtungen in dichter Vegetation. Die Tagesgänge der vorhergesagten Flüsse und skalaren Profile von Temperatur, H2O, CO2, Isopren und O3 während der späten Regen- und Trockenzeit am Rondonia-Standort stimmen gut mit Beobachtungen überein. Die Ergebnisse weisen auf saisonale physiologische Änderungen hin, die sich durch höhere stomatäre Leitfähigkeiten bzw. niedrigere Photosyntheseraten während der Regen- und Trockenzeit manifestieren. Die beobachteten Depositionsgeschwindigkeiten für Ozon während der Regenzeit überschreiten diejenigen der Trockenzeit um 150-250%. Dies kann nicht durch realistische physiologische Änderungen erklärt werden, jedoch durch einen zusätzlichen cuticulären Aufnahmemechanismus, möglicherweise an feuchten Oberflächen. Der Vergleich von beobachteten und vorhergesagten Isoprenkonzentrationen im Bestand weist auf eine reduzierte Isoprenemissionskapazität schattenadaptierter Blätter und zusätzlich auf eine Isoprenaufnahme des Bodens hin, wodurch sich die globale Schätzung für den tropischen Regenwald um 30% reduzieren würde. In einer detaillierten Sensitivitätsstudie wird die VOC Emission von amazonischen Baumarten unter Verwendung eines neuronalen Ansatzes in Beziehung zu physiologischen und abiotischen Faktoren gesetzt. Die Güte einzelner Parameterkombinationen bezüglich der Vorhersage der VOC Emission wird mit den Vorhersagen eines Modells verglichen, das quasi als Standardemissionsalgorithmus für Isopren dient und Licht sowie Temperatur als Eingabeparameter verwendet. Der Standardalgorithmus und das neuronale Netz unter Verwendung von Licht und Temperatur als Eingabeparameter schneiden sehr gut bei einzelnen Datensätzen ab, scheitern jedoch bei der Vorhersage beobachteter VOC Emissionen, wenn Datensätze von verschiedenen Perioden (Regen/Trockenzeit), Blattentwicklungsstadien, oder gar unterschiedlichen Spezies zusammengeführt werden. Wenn dem Netzwerk Informationen über die Temperatur-Historie hinzugefügt werden, reduziert sich die nicht erklärte Varianz teilweise. Eine noch bessere Leistung wird jedoch mit physiologischen Parameterkombinationen erzielt. Dies verdeutlicht die starke Kopplung zwischen VOC Emission und Blattphysiologie.

Indoor and Outdoor Natural Radioactivity in the Vulsini Volcanic District (Central Italy): Estimation of Doses and Radiological Risks

Terrestrial radioactivity for most individual is the major contributor to the total dose and is mostly provided by 238U, 232Th and 40K radionuclides. In particular indoor radioactivity is principally due to 222Rn, a radioactive noble gas descendent of 238U, second cause of lung cancer after cigarettes smoking. Vulsini Volcanic District is a well known quaternary volcanic area located between the northern Latium and southern Tuscany (Central Italy). It is characterized by an high natural radiation background resulting from the high concentrations of 238U, 232Th and 40K in the volcanic products. In this context, subduction-related metasomatic enrichment of incompatible elements in the mantle source coupled with magma differentiation within the upper crust has given rise to U, Th and K enriched melts. Almost every ancient village and town located in this part of Italy has been built with volcanic rocks pertaining to the Vulsini Volcanic District. The radiological risk of living in this area has been estimated considering separately: a. the risk associated with buildings made of volcanic products and built on volcanic rock substrates b. the risk associated to soil characteristics. The former has been evaluated both using direct 222Rn indoor measurements and simulations of “standard rooms” built with the tuffs and lavas from the Vulsini Volcanic District investigated in this work. The latter has been carried out by using in situ measurements of 222Rn activity in the soil gases. A radon risk map for the Bolsena village has been developed using soil radon measurements integrating geological information. Data of airborne radioactivity in ambient aerosol at two elevated stations in Emilia Romagna (North Italy) under the influence of Fukushima plume have been collected, effective doses have been calculated and an extensive comparison between doses associated with artificial and natural sources in different area have been described and discussed.

Maintenance of end-expiratory recruitment with increased respiratory rate after saline-lavage lung injury

Cyclical recruitment of atelectasis with each breath is thought to contribute to ventilator-associated lung injury. Extrinsic positive end-expiratory pressure (PEEPe) can maintain alveolar recruitment at end exhalation, but PEEPe depresses cardiac output and increases overdistension. Short exhalation times can also maintain end-expiratory recruitment, but if the mechanism of this recruitment is generation of intrinsic PEEP (PEEPi), there would be little advantage compared with PEEPe. In seven New Zealand White rabbits, we compared recruitment from increased respiratory rate (RR) to recruitment from increased PEEPe after saline lavage. Rabbits were ventilated in pressure control mode with a fraction of inspired O(2) (Fi(O(2))) of 1.0, inspiratory-to-expiratory ratio of 2:1, and plateau pressure of 28 cmH(2)O, and either 1) high RR (24) and low PEEPe (3.5) or 2) low RR (7) and high PEEPe (14). We assessed cyclical lung recruitment with a fast arterial Po(2) probe, and we assessed average recruitment with blood gas data. We measured PEEPi, cardiac output, and mixed venous saturation at each ventilator setting. Recruitment achieved by increased RR and short exhalation time was nearly equivalent to recruitment achieved by increased PEEPe. The short exhalation time at increased RR, however, did not generate PEEPi. Cardiac output was increased on average 13% in the high RR group compared with the high PEEPe group (P < 0.001), and mixed venous saturation was consistently greater in the high RR group (P < 0.001). Prevention of end-expiratory derecruitment without increased end-expiratory pressure suggests that another mechanism, distinct from intrinsic PEEP, plays a role in the dynamic behavior of atelectasis.

Design of a systematic approach to workplace exposure medical surveillance protocols

Current toxic tort cases have increased national awareness of health concerns and present an important avenue in which public health scientists can perform a vital function: in litigation, and in public health initiatives and promotions which may result. This review presents a systematic approach, using the paradigm of interactive public health disciplines, for the design of a matrix framework for medical surveillance of workers exposed to toxic substances. The matrix framework design addresses the required scientific bases to support the legal remedy of medical monitoring for workers injured as a result of their exposure to toxic agents. A background of recent legal developments which have a direct impact on the use of scientific expertise in litigation is examined in the context of toxic exposure litigation and the attainment of public health goals. The matrix model is applied to five different workplace exposures: dental mercury, firefighting, vinyl chloride manufacture, radon in mining and silica. An exposure matrix designed by the Department of Energy for government nuclear workers is included as a reference comparison to the design matrix. ^

Spatial distribution of orofacial cleft birth defects and uranium-radium concentrations in tap water in Atascosa, Bee, Brooks, Calhoun, Duval, Goliad, Hidalgo, Jim Hogg, Jim Wells, Karnes, Kleberg, Live Oak, McMullen, Nueces, San Patricio, Refugio, Starr, Victoria, Webb, and Zavala Counties, Texas

Background Past and recent evidence shows that radionuclides in drinking water may be a public health concern. Developmental thresholds for birth defects with respect to chronic low level domestic radiation exposures, such as through drinking water, have not been definitely recognized, and there is a strong need to address this deficiency in information. In this study we examined the geographic distribution of orofacial cleft birth defects in and around uranium mining district Counties in South Texas (Atascosa, Bee, Brooks, Calhoun, Duval, Goliad, Hidalgo, Jim Hogg, Jim Wells, Karnes, Kleberg, Live Oak, McMullen, Nueces, San Patricio, Refugio, Starr, Victoria, Webb, and Zavala), from 1999 to 2007. The probable association of cleft birth defect rates by ZIP codes classified according to uranium and radium concentrations in drinking water supplies was evaluated. Similar associations between orofacial cleft birth defects and radium/radon in drinking water were reported earlier by Cech and co-investigators in another of the Gulf Coast region (Harris County, Texas).50, 55 Since substantial uranium mining activity existed and still exists in South Texas, contamination of drinking water sources with radiation and its relation to birth defects is a ground for concern. ^ Methods Residential addresses of orofacial cleft birth defect cases, as well as live births within the twenty Counties during 1999-2007 were geocoded and mapped. Prevalence rates were calculated by ZIP codes and were mapped accordingly. Locations of drinking water supplies were also geocoded and mapped. ZIP codes were stratified as having high combined uranium (≥30μg/L) vs. low combined uranium (<30μg/L). Likewise, ZIP codes having the uranium isotope, Ra-226 in drinking water, were also stratified as having elevated radium (≥3 pCi/L) vs. low radium (<3 pCi/L). A linear regression was performed using STATA® generalized linear model (GLM) program to evaluate the probable association between cleft birth defect rates by ZIP codes and concentration of uranium and radium via domestic water supply. These rates were further adjusted for potentially confounding variables such as maternal age, education, occupation, and ethnicity. ^ Results This study showed higher rates of cleft births in ZIP codes classified as having high combined uranium versus ZIP codes having low combined uranium. The model was further improved by adding radium stratified as explained above. Adjustment for maternal age and ethnicity did not substantially affect the statistical significance of uranium or radium concentrations in household water supplies. ^ Conclusion Although this study lacks individual exposure levels, the findings suggest a significant association between elevated uranium and radium concentrations in tap water and high orofacial birth defect rates by ZIP codes. Future case-control studies that can measure individual exposure levels and adjust for contending risk factors could result in a better understanding of the exposure-disease association.^

Radium concentration in water samples from the Northeast Atlantic

We present field measurements of air-sea gas exchange by the radon deficit method that were carried out during JASIN 1978 (NE Atlantic) and FGGE 1979 (Equatorial Atlantic). Both experiments comprised repeated deficit measurements at fixed position over periods of days or longer, using a previously descibed precise and fast-acquiaition, automatic radon measuring system. The deficit time series exhibit variations that only partly reflect the expected changes in gas transfer. By evaluating averages over each time series we deduce the following gas transfer velocities (average wind velocity and water temperature in parentheses): JASIN phase 1: 1.6 ± 0.8 m/d (at ~6 m/s, 13°C) JASIN phase 2: 4.3 ± 1.2 m/d (at ~8 m/s, 13°C) FGGE: 1.2 ± 0.4 m/d (at ~5 m/s, 28°C) 0.9 ± 0.4 m/d (at ~7 m/s, 28°C) 1.5 ± 0.4 m/d (at ~7 m/s, 28°C) The large difference betwen the JASIN phase 2 and FGGE values despite quite similare average wind velocity becomes even larger when the values are, however, fully compatible with the range of gas transfer velocities observed in laboratory experiments and the conclusion is suggested that their difference is caused by the highly different wind variability in JASIN and FGGE. We conclude that in gas exchange parameterization it is not sufficinent to consider wind velocity only. A comparison of our observations with laboratory results outlines the range of variations of air-sea gas transfer velocities with wind velocity and sea state. We also reformulate the radon deficit method, in the light of our observed deficit variations, to account explicitely for non-stationary and horizontal inhomogeneity in previous radon work introduces considerable uncertainty in deduced gas transfere velocity. We furthermore discuss the observational rewuirements that have to be met for an adequate exploitation of the radon deficit method, of which an observation area of minimum horizontal inhomogeneity and monitoring of the remaining inhomogeneities are thought to be the most stringent ones.