Spectral properties of birth-death polynomials

We consider sequences of polynomials that are defined by a three-terms recurrence relation and orthogonal with respect to a positive measure on the nonnegative axis. By a famous result of Karlin and McGregor such sequences are instrumental in the analysis of birth-death processes. Inspired by problems and results in this stochastic setting we present necessary and sufficient conditions in terms of the parameters in the recurrence relation for the smallest or second smallest point in the support of the orthogonalizing measure to be larger than zero, and for the support to be discrete with no finite limit point

An orthogonal-polynomial approach to first-hitting times of birth-death processes

In a recent paper [J. Theor. Probab. 25 (2012) 950-980] Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor's classical results on first-hitting times of a birth-death process on the nonnegative integers by establishing a representation for the Laplace transform $\mathbb{E}[e^{sT_{ij}}]$ of the first-hitting time $T_{ij}$ for {\em any} pair of states $i$ and $j$, as well as asymptotics for $\mathbb{E}[e^{sT_{ij}}]$ when either $i$ or $j$ tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-polynomial toolbox used by Karlin and McGregor, in particular {\em associated polynomials} and {\em Markov's Theorem}

Simulation from quasi-stationary distributions on reducible state spaces

Quasi-stationary distributions (QSDs) arise from stochastic processes that exhibit transient equilibrium behaviour on the way to absorption. QSDs are often mathematically intractable and even drawing samples from them is not straightforward. In this paper the framework of Sequential Monte Carlo samplers is utilized to simulate QSDs and several novel resampling techniques are proposed to accommodate models with reducible state spaces, with particular focus on preserving particle diversity on discrete spaces. Finally an approach is considered to estimate eigenvalues associated with QSDs, such as the decay parameter

Shell polynomials and dual birth-death processes

This paper aims to clarify certain aspects of the relations between birth-death processes, measures solving a Stieltjes moment problem, and sets of parameters defining polynomial sequences that are orthogonal with respect to such a measure. Besides giving an overview of the basic features of these relations, revealed to a large extent by Karlin and McGregor, we investigate a duality concept for birth-death processes introduced by Karlin and McGregor and its interpretation in the context of shell polynomials and the corresponding orthogonal polynomials. This interpretation leads to increased insight in duality, while it suggests a modification of the concept of similarity for birth-death processes

Predicting Health Impacts of the World Trade Center Disaster: 1. Halogenated hydrocarbons, symptom syndromes, secondary victimization, and the burdens of history

The recent attack on the World Trade Center, in addition to direct injury and psychological trauma, has exposed a vast population to dioxins, dibenzofurans, related endocrine disruptors, and a multitude of other physiologically active chemicals arising from the decomposition of the massive quantities of halogenated hydrocarbons and other plastics within the affected buildings. The impacts of these chemical species have been compounded by exposure to asbestos, fiberglass, crushed glass, concrete, plastic, and other irritating dusts. To address the manifold complexities of this incident we combine recent theoretical perspectives on immune, CNS, and sociocultural cognition with empirical studies on survivors of past large toxic fires, other community-scale chemical exposure incidents, and the aftereffects of war. Our analysis suggests the appearance of complex, but distinct and characteristic, spectra of synergistically linked social, psychosocial, psychological and physical symptoms among the 100,000 or so persons most directly affected by the WTC attack. The different 'eigenpatterns' should become increasingly comorbid as a function of exposure. The expected outcome greatly transcends a simple 'Post Traumatic Stress Disorder' model, and may resemble a particularly acute form of Gulf War Syndrome. We explore the role of external social factors in subsequent exacerbation of the syndrome -- secondary victimization -- and study the path-dependent influence of individual and community-level historical patterns of stress. We suggest that workplace and other organizations can act as ameliorating intermediaries. Those without acess to such buffering structures appear to face a particularly bleak future

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory

Coarse Graining of Agent-Based Models and Spatio-Temporal Modeling of Spreading Processes

Agent-based models play a central role in modeling social spreading processes, in part because they allow detailed representation of interactions between individuals while integrating data on real-world processes. However, the resulting models are often too complex for a formal analysis and usually require high simulation e ort. In this thesis, based on general remarks on theoretical concepts such as stochastic dynamics and Markov processes, we have rst presented some new theoretical results on the e cient simulation and model reduction of agent-based models. Among these results are an event-based simulation algorithm for ABMs and a model reduction approach based on a projection of the state space and the utilization of convergence results to approximate agent-based models by less complex metapopulation models that can be simulated with much less e ort. Assuming metastability of the agent system, this approach preserves important model characteristics whith a low approximation error. In relation to this background a number of applications of agent-based models have been discussed. Of these, some are of fundamental structure, including a model to achieve global goals with local information, and others concern concrete spreading processes in prehistoric and contemporary societies. A focus among the applications is the spreading and development of culture and innovations in ancient times, both on a conceptual level and with reference to a concrete application case, the spread of the woolly sheep to Europe. In this context, the presented models have been developed through interdisciplinary cooperation and by taking into account archaeological, anthropological as well as geographical data in order to be able to depict the mobility and interactions of nomads, such as hunter-gatherers or shepherds, as realistically as possible. An important aspect that was discussed is the challenges posed by the prehistoric context, both in model parameterization and in validation and interpretation of the results. The comparison with current modeling scenarios is discussed with reference to the application area of epidemic spreading. Speci cally, the di erences in the assumptions about agent mobility and in the availability and reproducibility of data relevant to the model construction and analysis are highlighted. In the analysis of the models, we focused in particular on the identi cation of metastable processes through the application of clustering methods, including a novel approach that exploits the speci c structure of the agent-based models we have presented. Based on this analysis, possibilities for model reduction were discussed, which allow to generate additional data on macroscopic properties and mesoscopic structures of the models with low e ort. Especially, the generation of relevant statistics about critical transitions and other rare events is enabled by the reduced model complexity.Agentenbasierte Modelle spielen bei der Modellierung sozialer Ausbreitungsprozesse eine zentrale Rolle, da sie unter anderem die Interaktionen zwischen Individuen detailliert abbilden und Daten über reale Prozesse integrieren können. Die resultierenden Modelle sind jedoch häufig zu komplex für eine formale Analyse und in der Regel mit einem hohen Simulationsaufwand verbunden. In dieser Arbeit werden zunächst, aufbauend auf allgemeinen Ausführungen zu theoretischen Konzepten wie stochastischer Dynamik und Markov-Prozessen, einige neue theoretische Ergebnisse zur Simulation und Modellreduktion von agentenbasierten Modellen vorgestellt. Hervorzuheben ist dabei ein auf einer Zustandsraumprojektion basierender Ansatz zur Approximation agentenbasierter Modelle durch weniger komplexe Metapopulationsmodelle, die mit deutlich geringerem Aufwand simuliert werden können. Unter der Voraussetzung der Metastabilität bleiben dabei wichtige Modelleigenschaften bei geringem Approximationsfehler erhalten. In diesem Zusammenhang und im Anschluss daran werden eine Reihe von Anwendungen agentenbasierter Modelle diskutiert. Einige davon sind grundlegender Natur, darunter ein Modell zur Erreichung globaler Ziele mit lokalen Informationen, andere betreffen konkrete Ausbreitungsprozesse in prähistorischen und zeitgenössischen Gesellschaften. Ein Schwerpunkt unter den Anwendungsbereichen ist die Ausbreitung und Entwicklung von Kultur und Innovationen in der Antike, sowohl auf konzeptioneller Ebene als auch in Bezug auf einen konkreten Anwendungsfall, die Ausbreitung des Wollschafs nach Europa. Dabei wurden die vorgestellten Modelle in interdisziplinärer Kooperation und unter Berücksichtigung archäologischer, anthropologischer und geographischer Daten entwickelt, um die Mobilität und Interaktionen von Nomaden wie Jägern und Sammlern oder Hirten möglichst realitätsnah abbilden zu können. Ein wichtiger Aspekt, der diskutiert wird, sind die Herausforderungen, die sich aus dem prähistorischen Kontext sowohl für die Modellparametrisierung als auch für die Validierung und Interpretation der Ergebnisse ergeben. Der Vergleich mit aktuellen Modellierungsszenarien wird in Bezug auf das Anwendungsgebiet der Infektionsausbreitung diskutiert. Dabei werden insbesondere die Unterschiede in den Annahmen zur Mobilität der Agenten und in der Verfügbarkeit und Reproduzierbarkeit der für die Modellkonstruktion und -analyse relevanten Daten hervorgehoben. Bei der Analyse der Modelle liegt ein besonderer Schwerpunkt auf der Identifikation metastabiler Prozesse durch die Anwendung von Clusterverfahren, einschließlich eines neuartigen Ansatzes, der die besondere Struktur agentenbasierter Modelle ausnutzt. Darauf aufbauend werden Möglichkeiten der Modellreduktion diskutiert, die es erlauben, mit geringem Aufwand zusätzliche Daten über makroskopische Eigenschaften und mesoskopische Strukturen der Modelle zu erzeugen. Insbesondere die Generierung relevanter Statistiken über kritische Übergänge und andere seltene Ereignisse wird durch die geringere Modellkomplexität erst ermöglicht

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory

Plasmonic Nanoplatforms for Biochemical Sensing and Medical Applications

Plasmonics, the science of the excitation of surface plasmon polaritons (SPP) at the metal-dielectric interface under intense beam radiation, has been studied for its immense potential for developing numerous nanophotonic devices, optical circuits and lab-on-a-chip devices. The key feature, which makes the plasmonic structures promising is the ability to support strong resonances with different behaviors and tunable localized hotspots, excitable in a wide spectral range. Therefore, the fundamental understanding of light-matter interactions at subwavelength nanostructures and use of this understanding to tailor plasmonic nanostructures with the ability to sustain high-quality tunable resonant modes are essential toward the realization of highly functional devices with a wide range of applications from sensing to switching. We investigated the excitation of various plasmonic resonance modes (i.e. Fano resonances, and toroidal moments) using both optical and terahertz (THz) plasmonic metamolecules. By designing and fabricating various nanostructures, we successfully predicted, demonstrated and analyzed the excitation of plasmonic resonances, numerically and experimentally. A simple comparison between the sensitivity and lineshape quality of various optically driven resonances reveals that nonradiative toroidal moments are exotic plasmonic modes with strong sensitivity to environmental perturbations. Employing toroidal plasmonic metasurfaces, we demonstrated ultrafast plasmonic switches and highly sensitive sensors. Focusing on the biomedical applications of toroidal moments, we developed plasmonic metamaterials for fast and cost-effective infection diagnosis using the THz range of the spectrum. We used the exotic behavior of toroidal moments for the identification of Zika-virus (ZIKV) envelope proteins as the infectious nano-agents through two protocols: 1) direct biding of targeted biomarkers to the plasmonic metasurfaces, and 2) attaching gold nanoparticles to the plasmonic metasurfaces and binding the proteins to the particles to enhance the sensitivity. This led to developing ultrasensitive THz plasmonic metasensors for detection of nanoscale and low-molecular-weight biomarkers at the picomolar range of concentration. In summary, by using high-quality and pronounced toroidal moments as sensitive resonances, we have successfully designed, fabricated and characterized novel plasmonic toroidal metamaterials for the detection of infectious biomarkers using different methods. The proposed approach allowed us to compare and analyze the binding properties, sensitivity, repeatability, and limit of detection of the metasensing device

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory