Stochastic Neural Networks with the Weighted Hebb Rule

Neural networks with synaptic weights constructed according to the weighted Hebb rule, a variant of the familiar Hebb rule, are studied in the presence of noise(finite temperature), when the number of stored patterns is finite and in the limit that the number of neurons $N\rightarrow \infty$. The fact that different patterns enter the synaptic rule with different weights changes the configuration of the free energy surface. For a general choice of weights not all of the patterns are stored as {\sl global} minima of the free energy function. However, as for the case of the usual Hebb rule, there exists a temperature range in which only the stored patterns are minima of the free energy. In particular, in the presence of a single extra pattern stored with an appropriate weight in the synaptic rule, the temperature at which the spurious minima of the free energy are eliminated is significantly lower than for a similar network without this extra pattern. The convergence time of the network, together with the overlaps of the equilibria of the network with the stored patterns, can thereby be improved considerably.Comment: 14 pages, OKHEP 93-00

Estimation of near-surface Air temperature during day and night-time from MODIS over Different LC/LU Using machine learning methods in Berlin

Urbanization is manifest by changes in the physical structure of the land surface, owing to extensive construction features such as buildings, street canyons, changes in the thermal structure because of materials of different thermal properties and also intensive human activities. Urban areas are generally also characterized by higher surface air temperatures as compared to the rural surroundings. This temperature excess can be up to 10-12°C and more and is referred to as the urban heat island(UHI)phenomenon. Since residents living in cities are especially affected by extreme temperature events, urban climate studies are gaining in importance. Currently, more than half of the world s population already lives in urban areas, which accentuates the major role agglomerations must play in mitigation and adaptation to climate change. Recommendations regarding behavioural patterns during heat stress situations and urban planning measures require a comprehensive understanding of the inner urban temperature distribution including the identification of thermal hot spots. Both very cold and very hot temperatures could affect the human health. Excessive exposure to heat is referred to as heat stress and excessive exposure to cold is referred to as cold stress. Urban temperature data (2 m temperature data) is very important for all investigations on the urban heat island (UHI) effect, human health. They are usually either based on remote sensing techniques or air temperature measurements or from models. Remote sensing data like infra-red surface temperature from airborne measuring instruments may have a very high spatial resolution and are presently available for many urban areas, but only in clear sky cases. This spatial resolution is appropriate to exhibit typical urban structures that are expected to cause the UHI effect. Nevertheless, information on surface temperature cannot replace air temperature data, since beside the problem that the former is typically only available for single days, there is no fixed relation between surface and air temperatures. Especially for systematic analyses of the relationship between urban structures and 2m temperatures for different weather situations a large data basis is desirable. Air temperature data can be obtained from mobile measurements and measurement at permanent or temporary weather stations. On the one hand, the use of weather stations provides high data accuracy using a well-known standard technology. On the other hand, the spatial representation of weather station data within the urban environment, which is characterized by the surface composition including buildings, infrastructure and different types of land use, is very limited. Consequently, since the beginning of the 20th century, many efforts have been made to identify temperature patterns in urban areas with high spatial resolution instead of only using single point information. In this regard, in this study Air temperature (T2m) or Tair measurements from 20 ground weather stations in Berlin were used to estimate the relationship between air temperature and the remotely sensed land surface temperature (LST) measured by Moderate Resolution Imaging Spectroradiometer over different land-cover types (LCT). Knowing this relationship enables a better understanding of the magnitude and pattern of Urban Heat Island (UHI), by considering the contribution of land cover in the formation of UHI. In order to understand the seasonal behaviour of this relationship, the influence of the normalized difference vegetation index (NDVI) as an indicator of degree of vegetation on LST over different LCT was investigated. Next to it, to evaluate the influence of LCT, a regression analysis between LST and NDVI was made. The results demonstrate that the slope of regression depends on the LCT. It depicts a negative correlation between LST and NDVI over all LCTs. Our analysis indicates that the strength of correlations between LST and NDVI depends on the season, time of day, and land cover. This statistical analysis can also be used to assess the variation of the LST– T2m relationship during day- and night-time over different land covers. The results show that LSTDay and LSTNight are correlated significantly (p = 0.0001) with T2mDay(daytime air temperature) and T2mNight(night-time air temperature). The correlation (r) between LSTDay and TDay is higher in cold seasons than in warm seasons. Moreover, during cold seasons over every LCT, a higher correlation was observed during daytime than during night-time. In contrast, a reverse relationship was observed during warm seasons. It was found that in most cases, during daytime and in cold seasons, LST is lower than T2m. In warm seasons, however, a reverse relationship was observed over all land-cover types. In every season, LSTNight was lower than or close to T2mNight . Air temperature (Tair or T2m) is an important climatological variable for forest biosphere processes and climate change research. Due to the low density and the uneven distribution of weather stations, traditional ground-based observations cannot accurately capture the spatial distribution of Tair. Therefore, it is necessary to develop a method for the estimation of air temperature with reasonable accuracy and spatial and temporal resolution in the urban areas with low temperature gauge density. But since the estimation of meteorological variables using various statistical techniques (such as linear regression models or combined regression and kriging techniques for T interpolation) have been examined by many researchers and they came to conclusion that an appropriate machine learning technique could be a robust computational technique which has been used for the estimation of meteorological data as a function of the corresponding data of one or more reference stations. In this research, Tair in Berlin is estimated during the day and night-time over six land cover/land use (LC/LU) types by satellite remote sensing data over a large domain and a relatively long period (7 years). Aqua and Terra MODIS (Moderate Resolution Imaging Spectroradiometer) data and meteorological data for the period from 2007 to 2013 were collected to estimate Tair. Twelve environmental variables (land surface temperature (LST), normalized difference vegetation index (NDVI), Julian day, latitude, longitude, Emissivity31, Emissivity32, altitude, albedo, wind speed, wind direction and air pressure) were selected as predictors. Moreover, a comparison between LST from MODIS Terra and Aqua with daytime and night-time air temperatures (TDay , TNight) was done respectively and in addition, the spatial variability of LST and Tair relationship by applying a varying window size on the MODIS LST grid was examined. An analysis of the relationship between the observed Tair and the spatially averaged remotely sensed LST, indicated that 3 × 3 and 1 × 1 pixel size was the optimal window size for the statistical model estimating Tair from MODIS data during the day and night time, respectively. Three supervised learning methods (Adaptive Neuro Fuzzy Inference system (ANFIS), Artificial Neural Network (ANN) and Support vector machine (SVR)) were used to estimate Tair during the day and night-time, and their performances were validated by cross-validation for each LC/LU. by applying each technique, a estimator model of air temperature had been generated. The comparison between these methods has been done and finally we evaluated the accuracy of each model and choose the best one for the high-resolution temperature estimation. Moreover, tuning the hyper parameters of some models like SVR and ANN were investigated. For tuning the hyper parameters of SVR, Simulated Annealing (SA) was applied (SA-SVR model) and a multiple-layer feed-forward (MLF) neural networks with three layers and variable nodes in hidden layers had been applied with Levenberg-Marquardt back-propagation (LM-BP), in order to achieve higher accuracy in the estimation of Tair . Results indicated that the ANN model achieved better accuracy (RMSE=2.16°C, MAE =1.69°C, R2 =0.95) than SA-SVR model (RMSE= 2.50°C, MAE =1.92°C, R2=0.91) and ANFIS model (RMSE=2.88°C, MAE=2.2°C, R2=0.89) over six LC/LU during the day and night time. The Q-Q diagram of SA-SVR, ANFIS and ANN show that all three models slightly tended to underestimate and overestimate the extreme and low temperatures for all LC/LU classes during the day and night-time. The weak performance in the extreme and low temperatures are a consequence of the small numbers of data in these temperatures. These satisfactory results indicate that this approach is proper for estimating air temperature and spatial window size is an important factor that should be considered in the estimation of air temperature. Moreover, for better understanding the relationship between LST and Tair in Berlin during day and night-time, over six land LC/LU types namely airport, agriculture, urban area, forest, industrial and needle leaf trees, two input variable selection methods were applied. Input variable selection is an essential step in environmental, biological, industrial and climatological applications. One approach which help us in better understanding data, decreasing computation effort, the impact of curse of dimensionality and improving the estimator performance. Through input variable selection the irrelevant or redundant variables will be to eliminated therefore a suitable subset of variables is identified as the input of a model. Meanwhile, the complexity of the model structure is simplified, and the computational efficiency is improved. In this work, the two input variable selection methods, including brute force search and greedy best search algorithm using artificial neural network (ANN) were considered for estimating of near surface air temperature from MODIS over six LC/LU types. The motivation behind this research was to formulate a more efficient way of choosing input variables using ANN models of environmental processes. Moreover, AIC, BIC and RMSE are considered for ranking the features and finding a subset of potential variables which improves the overall estimation performance. In this study, Aqua and Terra MODIS data and meteorological data for the period from 2007 to 2013 were collected to estimate Tair .Moreover, twelve environmental variables LST, normalized difference vegetation index (NDVI), Julian day, latitude, longitude, Emis31, Emis32, altitude, albedo, wind speed, wind direction and air pressure were selected as predictors. The results show that the LC/LU has a key factor in the relationship between Tair and LST. The results show that the effectiveness of optimal models in estimation Tair is varied in different LC/LU because of the specific heat capacities of different LC/LU. Air temperature mainly rely on the heat transfer process which was significantly affected by the local radiation budget. Generally, air is heated much quicker over barren land than forest because, barren land has lower heat capacity than forest. Vegetation can cause to latent heat flux, such as enhancing or reducing transpiration and cool the Tair in forests. In this study, the cooling effect was not taken into account because of roughly distribution of meteorological stations across different vegetation types. Therefore, it was difficult to consider the vegetation type in our models. However, land cover also affected land surface albedo, thus, the influence of LU/LC on estimating Tair was conditional and time dependent because different variables are selected for the same LU/LC during day and night-time. Moreover, another issue that we tried to find an answer was, what is the pitfall of using the global model and what is the advantage of features selection? It has been debated that inferencing from a model with all the features which thought to be important is simple and avoid the complications of model selection.Urbanisierung stellt eine Veränderungen in der physikalischen Struktur der Landoberfläche durch umfangreiche Konstruktionsmerkmale, wie Gebäude und Straßenschluchten, dar. Die damit verbundenen Änderungen der thermischen Struktur durch Verwendung von Materialien mit unterschiedlichen thermischen Eigenschaften sowie intensive menschliche Aktivität spielen hierbei eine wichtige Rolle. Urbane Gebiete sind im Allgemeinen durch eine höhere Oberflächentemperatur im Vergleich zur ländlichen Umgebung gekennzeichnet. Der Temperaturüberschuss kann bis zu 10-12°C und mehr betragen und wird als Phänomen der städtischen Wärmeinsel (UHI) bezeichnet. Da in Städten lebende Menschen besonders stark von extremen Temperaturereignissen betroffen sind, gewinnen Studien zum urbanen Klima vermehrt an Bedeutung. Derzeit lebt mehr als die Hälfte der Weltbevölkerung in urbanen Gebieten. Dies unterstreicht die wichtige Rolle die Ballungsräume in Bezug auf Minderung und Anpassung an den Klimawandel darstellt. Empfehlungen bezüglich des Verhaltens während Hitzestresssituationen sowie städtebauliche Maßnahmen erfordern ein umfangreiches Verständnis der innerstädtischen Temperaturverteilung einschließlich der Identifizierung von thermischen Hotspots. Sehr kalte wie auch sehr heiße Temperaturen können gleichermaßen die menschliche Gesundheit beeinträchtigen. Übermäßige Hitzebelastung wird als Hitzestress bezeichnet, übermäßige Kältebelastung als Kältestress. Urbane Temperaturdaten (2m Temperaturdaten) sind wichtig für alle Untersuchungen bezüglich des urbanen Wärmeinseleffekts (UHI), der menschlichen Gesundheit. Normalerweise basieren die Daten entweder auf Fernerkundungstechniken oder auf Messungen oder Simulationen der Lufttemperatur. Fernerkundungsdaten, wie die der Infrarot-Oberflächentemperatur von satellitengestützen Messinstrumenten können eine sehr hohe räumliche Auflösung haben und sind gegenwärtig für viele urbane Gebiete zugänglich, doch nur im Fall von wolkenfreiem Himmel. Die räumliche Auflösung ist dafür geeignet typische urbane Strukturen zu erkennen, die den UHI-Effekt auslösen. Dennoch können Informationen der Oberflächentemperatur, die Lufttemperaturdaten nicht ersetzen, da neben dem Problem, dass die Oberflächentemperatur in der Regel nur für einzelne Tage zur Verfügung steht, es keinen festen Zusammenhang zwischen Oberflächentemperatur und Lufttemperatur besteht. Besonders für systematische Analysen der Zusammenhänge zwischen urbanen Strukturen und der 2m-Temperatur unterschiedlicher Wettersituationen ist eine hohe Datenbasis wünschenswert. Daten der Lufttemperatur können von mobilen Messungen und permanenten Messstationen oder von temporären Wetterstationen erhalten werden. Einerseits bieten die Wetterstationen eine hohe Datenqualität durch Verwendung von bekannten Standard-Technologien; andererseits ist die räumliche Verteilung der Wetterstationsdaten in der urbanen Umgebung, die durch die oberflächliche Komposition von Gebäuden, Infrastruktur und verschiedenen Landnutzungsklassen charakterisiert ist, sehr eingeschränkt. Seit Beginn des 20. Jahrhunderts konnten somit viele Vorzüge bei der Identifizierung von Temperaturmustern in urbanen Gebieten mit hoher räumlicher Auflösung erzielt werden anstatt nur einzelne Punktinformationen zu nutzen. Folglich werden für diese Studie Lufttemperatur (T2m) oder Tair Messungen von 20 Bodenwetterstationen in Berlin verwendet, um den Zusammenhang zwischen Lufttemperatur und Fernerkundungsdaten der Oberflächentemperatur (LST) gemessen vom Moderate Resolution Imaging Spectroradiometer (MODIS) über verschiedene Landnutzungstypen (LCT). Die Kenntnis über diesen Zusammenhang ermöglicht ein besseres Verständnis der Stärke und Muster von urbanen Wärmeinseln (UHI) durch Beachtung der Verteilung der Oberflächenbeschaffenheit bei Ausbildung von UHI. Um das saisonale Verhalten dieses Zusammenhangs zu verstehen, wurde der Einfluss des normalisierten Differenzvegetationsindex (NDVI) als ein Indikator für den Vegetationsgrad auf LST über verschiedene LCT untersucht. Darüber hinaus wurde eine Regressionsanalyse zwischen der LST und dem NDVI durchgeführt, um den Einfluss der LCT zu bewerten. Die Ergebnisse zeigen, dass die Steigung der Regressionsgeraden von der LST abhängt. Es besteht eine negative Korrelation zwischen LST und NDVI über alle LCTs. Unsere Analyse signalisiert, dass die Stärke der Korrelation zwischen LST und NDVI von der Jahreszeit, der Tageszeit sowie der Landnutzung abhängig ist. Die statistische Analyse kann auch verwendet werden, um die Variation der LST-T2m-Beziegung während der Tages- und Nachtzeit über verschiedene Bodenbedeckungen zu bewerten. Die Ergebnisse zeigen eine signifikante Korrelation (p=0.0001) von LSTday und LSTnight mit der T2mDay (Lufttemperatur tagsüber) und der T2mNight (Lufttemperatur nachts). Zwischen LSTday und Tday ist die Korrelation (r) in der kalten Jahreszeit höher als in der warmen. Darüber hinaus wurde eine höhere Korrelation während der kalten Jahreszeit über alle LCTs am Tag beobachtet als in der Nacht. In der warmen Jahreszeit wurde im Gegensatz dazu ein umgekehrter Zusammenhang festgestellt. Es wurde beobachtet, dass in den meisten Fällen, tagsüber und in kalten Jahreszeiten, die LST niedriger ist als die T2m. In warmen Jahreszeiten wurde jedoch ein umgekehrter Zusammenhang über alle Landbedeckungsarten beobachtet. In jeder Saison war die LSTNight niedriger oder fast gleich wie die T2mNight. Die Lufttemperatur (Tair oder T2m) ist eine wichtige klimatologischen Variable für Prozesse der Waldbiosphäre und die Erforschung des Klimawandels. Aufgrund der geringen Dichte und der ungleichmäßigen Verteilung von Wetterstationen können herkömmliche bodengebundene Beobachtungen die räumliche Verteilung von Tair nicht genau erfassen. Daher ist es notwendig, eine Methode zur Abschätzung der Lufttemperatur mit angemessener Genauigkeit sowie räumlicher und zeitlicher Auflösung in urbanen Gebieten mit niedriger Temperaturmessdichte zu entwickeln. Da aber die Abschätzung meteorologischer Variablen mit verschiedenen statistischen Techniken (wie linearen Regressionsmodellen und kombinierten Regressions- und Krigingtechniken für die T-Interpolation) von vielen Forschern untersucht wurde, kamen sie zu dem Schluss, dass eine geeignete ‚machine learning‘ Technik eine robuste Rechentechnik sein könnte, die für die Abschätzung meteorologischer Daten in Abhängigkeit von entsprechenden Daten einer oder mehrerer Referenzstationen verwendete. In dieser Studie wird Tair in Berlin tagsüber sowie nachts über sechs Landbedeckungs/Landnutzungsarten (LC/LU) mittels Satelliten-Fernerkundungsdaten über einen großen Bereich und einem relativ langen Zeitraum (7Jahre) geschätzt. Daten des ‚Terra und Aqua MODIS‘ (Moderate Resolution Imaging Spectroradiometer) und meteorologische Daten für den Zeitraum von 2007 bis 2013 wurden gesammelt, um Tair zu bestimmen. Als Prädikatoren wurden zwölf Umweltvariablen (Landoberflächentemperatur (LST), normalisierter Differenzvegetationsindex (NDVI), Julianischer Tag, Breitengrad, Längengrad, Emissionsgrad 31, Emissionsgrad 32, Höhe, Albedo, Windgeschwindigkeit, Windrichtung und Luftdruck) ausgewählt. Drüber hinaus wurde ein Vergleich zwischen LST von MODIS Terra und Aqua mit Tages- und Nachtlufttemperaturen (TDay , TNight) durchgeführt bzw. zusätzlich die räumliche Variabilität des Zusammenhangs von LST und Tair durch Anwendung einer variierenden Fenstergröße auf das MODIS LST-Gitter untersucht. Eine Analyse der Beziehung zwischen der beobachteten Tair und dem räumlich gemittelten Fernerkundungs-LST ergab, dass die Größe 3 x 3 und 1 x 1 Pixel die optimale Fenstergröße für das statistische Modell war, das Tair aus den MODIS-Daten während Tages- bzw. Nachtzeit schätzte. Drei überwachte Lernmethoden (Adaptive Neuro Fuzzy Inference system (ANFIS), künstliches neuronales Netzwerk (ANN) und Support vector machine (SVR)) wurden verwendet, um Tair währen des Tages und der Nacht zu schätzen. Die Leistungen wurden durch Kreuzvalidierung für jede LC/LU validiert. Durch die Anwendung jeder Technik wurde ein Schätzmodell ausgewertet und das Beste für die hochauflösende Temperaturschätzung ausgewählt. Darüber hinaus wurde die Einstellung der Hyperparameter einiger Modelle wie SVR und ANN untersucht. Für die Einstellung der Hyperparameter von SVR wurde ‚Simulated Annealing‘ (SA) angewendet (SA-SVR Modell). Mit der Levenberg-Marquardt Backpropagation (LM-BP) wurde ein mehrschichtiges Feed-forward (MLF) neuronales Netzwerk mit drei Schichten und variablen Knoten in versteckten Schichten angewendet, um eine höhere Genauigkeit bei der Schätzung von Tair zu erreichen. Die Ergebnisse zeigten, dass das ANN-Modell über sechs LC/LU, tags sowie nachts, eine höhere Genauigkeit erreichte (RMSE=2.16°C, MAE =1.69°C, R2 =0.95) als das SA-SVR-Modell (RMSE= 2.50°C, MAE =1.92°C, R2=0.91) und das ANFIS-Modell (RMSE=2.88°C, MAE=2.2°C, R2=0.89). Das Q-Q-Diagramm von SA-SVR, ANFIS und ANN zeigt, dass alle drei Modelle die extrem hohen und niedrigen Temperaturen für alle LC/LU-Klassen tagsüber sowie nachts leicht unterschätzen und überschätzen. Die schwache Leistung bei extrem hohen und niedrigen Temperaturen ist eine Folge der geringen Datenmenge bei diesen Temperaturen. Um den Zusammenhang zwischen LST und Tair in Berlin bei Tag und Nacht besser verstehen zu können, wurden über sechs Land-LC/LU-Typen (Flughafen, Landwirtschaft, urbanes Gebiet, Wald-, Industrie- und Nadelblattbäume) zwei Auswahlmethoden für die Eingangsvariablen angewendet. Die Auswahl dieser Variablen ist ein wesentlicher Schritt in ökologischen, biologischen, industriellen und klimatologischen Anwendungen. Ein Ansatz, der uns hilft, Daten besser zu verstehen, den Rechenaufwand zu verringern, die Auswirkungen des Fluches der Dimensionalität und die Lei

Rare-earth ion doped planar waveguides for integrated quantum photonics

This thesis presents a spectroscopic study of Pr3+ ions in a novel passive waveguide architecture. To make this structure, the high refractive glass TeO2 was selected as the thin film and it was deposited on a Pr3+:Y2SiO5 crystal. In this waveguide, the 3H4 to 1D2 transition of Pr3+ ions were probed by the optical evanescent field extending into the substrate. The main concern in assessing the suitability of this material for quantum information applications was ensuring that the coherence properties of rare-earth ions, making them suitable for quantum information purposes, are preserved in this architecture. After which, to make low loss devices with these waveguides, efficient coupling techniques had to be developed. To prove that the coherence properties of the rare-earth ion doped crystal were preserved, the critical parameters of inhomogeneous linewidth, the absorption of the ions, the coherence time and spin lifetimes of the Pr3+ ions were studied. The inhomogeneous linewidth of evanescent coupled ions was about 10.0 GHz, which was consistent with the linewidth of bulk samples with the same Pr3+ doping concentration (Hedges, 2011). The absorption due to the evanescent coupling was 9.38 dB, approximately 90% of what was expected with respect to the bulk crystal with the same doping concentration. Therefore, despite using the evanescent field, the absorption is high enough for quantum memory applications. An optical coherence time of about 121 microseconds was measured, which corresponded to a homogeneous linewidth of about 2.6 kHz. This is very close to bulk sample measurements of 111 microseconds, with the same temperature and doping concentration (R. W. Equall, 1995). The spin state lifetime observed was about 9.8 s, which is also very close to the bulk sample measurement of 8.67 s (Mieth, 2016). Initial Stark shifting experiments were performed to determine whether the active ions in the substrate of the passive waveguides could be electrically controlled by applying a small voltage to electrodes on the thin film. In these experiments with a voltage change of 100 mV, the measured holewidth broadening was increasing about 0.55 MHz, which was similar to the calculated values of 0.45 MHz. The Stark coefficient for site 1 was 51.6 kHzcm/V along the D2 axis of the crystal (site 1 will be explained in Section 4.3). (F.R. Graf, 1997). Prism coupling and grating coupling were used to couple light to the ions in the substrate. Prism coupling is an easy and quick method to couple light into a waveguide and observe the properties of the system. However, grating coupling is much more practical when moving towards building a device using this method. The measurements described above indicated that the properties of ions interacting with the evanescent tail of the waveguide mode were consistent with those of bulk ions. This investigation also showed that depositing a glass thin film on a rare-earth ion doped crystal was not affecting the good coherence properties of the substrate. These results establish the foundation for large, integrated, controllable and high performance rare earth ion quantum waveguide systems

Hybrid PDE-ABM models: from oncology to virology

This PhD thesis presents three new mathematical models aimed at advancing the current understanding of viral dynamics and cancer. The models are implemented in the HAL system and are applied to address current significant questions in the field. The first model, described in Chapter 3, is focused on the evolution of sensitive and resistant tumor cells and considers Darwinian selection, Lamarckian induction, and microvesicle transfer as mechanisms for resistance. The model is compared to an agent-based model and provides insight into the impact of microvesicles on the spread of cancer cells. Chapter 4 presents a hybrid partial differential equation - agent-based (PDE-ABM) model to describe the spatiotemporal viral dynamics in a cell population. The virus concentration is modeled as a continuous variable, while changes in the states of cells are represented by a stochastic agent-based model. The two subsystems are intertwined and the model is applied to predict SARS-CoV-2 and influenza infections. The results are compared to a classical ODE system of viral dynamics and demonstrate the importance of considering spatial patterns in predicting the spread of infections. In Chapter 6, the pharmacometric features of the novel antiviral drug Paxlovid for SARS-CoV-2 are explored using a hybrid multiscale mathematical approach. The results of the evaluation match clinical expectations and provide insight into the importance of early interventions and the sensitivity of the results to the diffusion coefficient of the virus. Overall, this PhD thesis makes a significant contribution to the current understanding of viral dynamics and cancer and demonstrates the potential of mathematical models to provide insight into these complex systems

Isotope Planetology

The quest for in-depth knowledge of the formation and early evolution of the Earth-Moon system is a cornerstone of the planetary sciences. Virtually all chemical studies that address these key questions rely on the availability of extremely ancient rock samples (>4 billion years ago). On Earth's surface, active plate tectonics, weathering, and volcanism have destroyed nearly all ancient samples. Samples from the Moon are sufficiently old but strongly limited in quantity and cover only a small portion of the lunar surface. The Moon is thought to have formed from the residue of an impact (or impacts) into the proto-Earth. There are two schools of thought as to when this occurred, one advocating an "old Moon" forming prior to 100 million years after solar system formation (SSF) and one supporting a "young Moon" forming later than 100 million years after SSF. This debate and the clear definition of the processes involved have continued unabated for the 50 years since lunar samples were first returned by the Apollo missions. A key to deepening our knowledge of these issues resides in understanding the extinct 182Hf-182W decay system in lunar and terrestrial rocks. To address this question, we analyzed a suite of 29 lunar samples from the Apollo missions to better understand the elemental Hf and W budgets of the moon. We used new high-precision, high field strength element (HFSE) analyses, combined with isotopic and experimental partitioning data in line with the lunar magma ocean (LMO) model. Through these methods it is possible to observe lunar mantle-wide heterogeneities in ratios of highly incompatible elements such as U/W, which are traditionally assumed to be invariant. This observation, in conjunction with 182W isotope data for lunar rocks, supports the hypothesis of a Moon covered by a magma ocean after its formation. Crystallization and mixing of this LMO produced different hybrid cumulate sources; thus forming the sources of the distinct rock types found in the lunar sample suite. Under the low oxygen fugacity conditions during lunar mantle partial melting, the low-Ti mare basalt source preferentially retains tungsten (W) over hafnium (Hf). The measured Hf/W values of low-Ti mare basalts thus provide a minimum for the Hf/W of the low-Ti source and by extension of the silicate Moon. We find that the Hf/W of the silicate Moon should lie between 30 to 50, significantly higher than the silicate Earth's modeled Hf/W of 25.8. Combined with a recently reported “global, uniform” 182W excess in lunar samples, we find that in-situ decay of 182Hf, in the time range between 40 to 60 million years after SSF is a superior explanation of the lunar 182W excess instead of a previously suggested disproportionate “late accretion” of extraterrestrial material to the Moon and the Earth. Our finding lends clear support for an "old Moon." We expanded our work on lunar samples to include the KREEP-rich gabbroic meteorite Northwest Africa (NWA) 6950. This meteorite yields new insight into the history of the KREEP reservoir which formed as the final residual melt of the LMO. A previous study had dated the meteorite to 3100 million years ago through Pb-Pb dating of baddeleyite grains. This marks the NWA 6950 meteorite to be the youngest KREEP-like sample available and thus decisive for constraining lunar evolution. We obtained Lu-Hf, Sm-Nd, and Rb-Sr mineral isochrons for this meteorite. Through Lu-Hf dating we found an age of 3103 ± 39 million years ago, perfectly overlapping the Pb-Pb age and underpinning the significance of this meteorite’s isotope systematics to anchor the evolution of KREEP. A Sm-Nd isochron of clean, hand-picked minerals yielded a compatible age of 3052 ± 57 million years ago. Inclusion of all mineral fractions that might have suffered later disturbance yields a young Sm-Nd isochron age of 2900 ± 200 million years ago that is closely akin to previous ages found via Ar-Ar (2800), Rb-Sr (2900), and Sm-Nd (2900) which dates younger resetting. In addition, the Rb-Sr isochron provides an even younger age of ca. 1450 million years ago, although this may bear no geological relevance. The significance of finding these young ages becomes clear considering that several Sm-Nd and Rb-Sr studies aimed to date related meteorites whose history might thus have been characterized incorrectly. The initial εHf of NWA 6950 is the youngest anchor of the KREEP evolution line, from which we determined a time of KREEP formation at 4514 million years ago, or ca. 55 million years after SSF. We therefore found, through an entirely different line of research, independent support for an "old Moon" formation. To calibrate this methodology, we investigated multiple peridotites from the West Eifel volcanic field of Germany that exhibit similarly low abundances of Lu, Hf, Sm, Nd, Rb, and Sr. For this project, three different ion exchange separation techniques were investigated as part of the calibration. Mineral isochrons of Lu-Hf, Sm-Nd, and Rb-Sr all provided a functionally modern age, indicative of a resetting event during the Quaternary. We also found that whole rock, host rock, and mineral compositions argue against equilibration of the host magma and the peridotite xenoliths. The observation that whole rock samples plot off the horizontal isochrons, in contrast, is explained by melt infiltration and grain boundary entrainment which likely postdated the resetting of the isochrons. One peridotite examined in a companion study supervised by myself (M.M. Thiemens) yielded four distinct ages. The Lu-Hf system was reset by a Quaternary age event, while the Hf isotope signature was highly radiogenic, indicative of differentiation from a modern mantle source between 1.22 and 1.76 Ga. Rb-Sr isochron data yielded an age of ca. 635 Ma, and a Sm-Nd age of 235 Ma corresponds with regional uplift. Our findings reveal that fine scaled isotope investigations are potent tools to unravel evolutionary complexities. The wealth of fine scaled information gained from the Eifel peridotite xenoliths once again underlines the stark contrast between the extremely dynamic evolution of the Earth’s lithosphere and mantle when compared to the largely static lunar evolution following LMO crystallization