@unpublished{PornsawadBoeckmann2014,
  author    = {Pornsawad, Pornsarp and B{\"o}ckmann, Christine},
  title     = {Modified iterative Runge-Kutta-type methods for nonlinear ill-posed problems},
  volume    = {3},
  number    = {7},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70834},
  pages     = {30},
  year      = {2014},
  abstract  = {This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under H{\"o}lder-type source-wise condition if the Fr{\´e}chet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.},
  language  = {en}
}
@phdthesis{Wichitsanguan2016,
  author    = {Wichitsa-nguan, Korakot},
  title     = {Modifications and extensions of the logistic regression and Cox model},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-90033},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 131},
  year      = {2016},
  abstract  = {In many statistical applications, the aim is to model the relationship between covariates and some outcomes. A choice of the appropriate model depends on the outcome and the research objectives, such as linear models for continuous outcomes, logistic models for binary outcomes and the Cox model for time-to-event data. In epidemiological, medical, biological, societal and economic studies, the logistic regression is widely used to describe the relationship between a response variable as binary outcome and explanatory variables as a set of covariates. However, epidemiologic cohort studies are quite expensive regarding data management since following up a large number of individuals takes long time. Therefore, the case-cohort design is applied to reduce cost and time for data collection. The case-cohort sampling collects a small random sample from the entire cohort, which is called subcohort. The advantage of this design is that the covariate and follow-up data are recorded only on the subcohort and all cases (all members of the cohort who develop the event of interest during the follow-up process). In this thesis, we investigate the estimation in the logistic model for case-cohort design. First, a model with a binary response and a binary covariate is considered. The maximum likelihood estimator (MLE) is described and its asymptotic properties are established. An estimator for the asymptotic variance of the estimator based on the maximum likelihood approach is proposed; this estimator differs slightly from the estimator introduced by Prentice (1986). Simulation results for several proportions of the subcohort show that the proposed estimator gives lower empirical bias and empirical variance than Prentice's estimator. Then the MLE in the logistic regression with discrete covariate under case-cohort design is studied. Here the approach of the binary covariate model is extended. Proving asymptotic normality of estimators, standard errors for the estimators can be derived. The simulation study demonstrates the estimation procedure of the logistic regression model with a one-dimensional discrete covariate. Simulation results for several proportions of the subcohort and different choices of the underlying parameters indicate that the estimator developed here performs reasonably well. Moreover, the comparison between theoretical values and simulation results of the asymptotic variance of estimator is presented. Clearly, the logistic regression is sufficient for the binary outcome refers to be available for all subjects and for a fixed time interval. Nevertheless, in practice, the observations in clinical trials are frequently collected for different time periods and subjects may drop out or relapse from other causes during follow-up. Hence, the logistic regression is not appropriate for incomplete follow-up data; for example, an individual drops out of the study before the end of data collection or an individual has not occurred the event of interest for the duration of the study. These observations are called censored observations. The survival analysis is necessary to solve these problems. Moreover, the time to the occurence of the event of interest is taken into account. The Cox model has been widely used in survival analysis, which can effectively handle the censored data. Cox (1972) proposed the model which is focused on the hazard function. The Cox model is assumed to be λ(t|x) = λ0(t) exp(β^Tx) where λ0(t) is an unspecified baseline hazard at time t and X is the vector of covariates, β is a p-dimensional vector of coefficient. In this thesis, the Cox model is considered under the view point of experimental design. The estimability of the parameter β0 in the Cox model, where β0 denotes the true value of β, and the choice of optimal covariates are investigated. We give new representations of the observed information matrix In(β) and extend results for the Cox model of Andersen and Gill (1982). In this way conditions for the estimability of β0 are formulated. Under some regularity conditions, ∑ is the inverse of the asymptotic variance matrix of the MPLE of β0 in the Cox model and then some properties of the asymptotic variance matrix of the MPLE are highlighted. Based on the results of asymptotic estimability, the calculation of local optimal covariates is considered and shown in examples. In a sensitivity analysis, the efficiency of given covariates is calculated. For neighborhoods of the exponential models, the efficiencies have then been found. It is appeared that for fixed parameters β0, the efficiencies do not change very much for different baseline hazard functions. Some proposals for applicable optimal covariates and a calculation procedure for finding optimal covariates are discussed. Furthermore, the extension of the Cox model where time-dependent coefficient are allowed, is investigated. In this situation, the maximum local partial likelihood estimator for estimating the coefficient function β(·) is described. Based on this estimator, we formulate a new test procedure for testing, whether a one-dimensional coefficient function β(·) has a prespecified parametric form, say β(·; ϑ). The score function derived from the local constant partial likelihood function at d distinct grid points is considered. It is shown that the distribution of the properly standardized quadratic form of this d-dimensional vector under the null hypothesis tends to a Chi-squared distribution. Moreover, the limit statement remains true when replacing the unknown ϑ0 by the MPLE in the hypothetical model and an asymptotic α-test is given by the quantiles or p-values of the limiting Chi-squared distribution. Finally, we propose a bootstrap version of this test. The bootstrap test is only defined for the special case of testing whether the coefficient function is constant. A simulation study illustrates the behavior of the bootstrap test under the null hypothesis and a special alternative. It gives quite good results for the chosen underlying model. References P. K. Andersen and R. D. Gill. Cox's regression model for counting processes: a large samplestudy. Ann. Statist., 10(4):1100{1120, 1982. D. R. Cox. Regression models and life-tables. J. Roy. Statist. Soc. Ser. B, 34:187{220, 1972. R. L. Prentice. A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika, 73(1):1{11, 1986.},
  language  = {en}
}
@book{Jankowski2004,
  author    = {Jankowski, \poundsukasz},
  title     = {Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres. - [{\"u}berarb. Diss.]},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001659},
  publisher      = {Universit{\"a}t Potsdam},
  year      = {2004},
  abstract  = {Kunststofflichtwellenleiter (POFs) stellen ein verh{\"a}ltnism{\"a}ßig neues Medium zur optische Datenkommunikation {\"u}ber kurzen Strecken dar. W{\"a}hrend ihrer Einsatzdauer unterliegen POFs unterschiedlichen Arten von Umweltbeanspruchungen, haupts{\"a}chlich durch hohe Temperatur, hohe Feuchtigkeit und mechanischen Belastungen. Zahlreiche experimentelle Forschungen besch{\"a}ftigten sich mit der standardisierten Pr{\"u}fung der Zuverl{\"a}ssigkeit von im Handel erh{\"a}ltlichen Fasern. Jedoch gab es bisher wenig Erfolg bei der Bem{\"u}hung, zwei grundlegende optische Erscheinungen, Absorption und Streuung, die die Lichtausbreitung in Fasern stark beeinflussen, zu verstehen und praktisch zu modellieren: Diese beiden Effekte beschreiben nicht nur die Qualit{\"a}t neuer Fasern, sondern sie werden auch stark durch die Alterungsprozess beeinflusst. Der Hauptzweck dieser Doktorarbeit war es, ein praktisch verwendbares und theoretisch gut fundiertes Modell der Lichtausbreitung in nicht gealterten und gealterten POFs zu entwickeln und es durch optische Experimente zu verifizieren. Dabei wurden anwendungsorientierte Aspekte mit theoretischer POF-Modellierung kombiniert. Die Arbeit enth{\"a}lt die erste bekannte Anwendung der Wellenanalyse zur Untersuchung der winkelabh{\"a}ngigen Eigenschaften der Streuung in Lichtwellenleitern. F{\"u}r die praktischen Experimente wurden mehrere POF-Proben unterschiedlicher Hersteller k{\"u}nstlich gealtert, indem sie bis 4500 Stunden bei 100 °C gelagert wurden (ohne Feuchtekontrolle). Die Parameter der jeweiligen Simulationen wurden mittels einer systematischen Optimierung an die gemessen optischen Eigenschaften der gealterten Proben angeglichen. Die Resultate deuten an, dass der {\"U}bertragungsverlust der gealterten Fasern in den ersten Tagen und Wochen der Alterung am st{\"a}rksten durch eine wesentliche physikalische Verschlechterung der Kern-Mantel-Grenzfl{\"a}che verursacht wird. Chemische Effekte des Alterungsprozesses scheinen im Faserkernmaterial zuerst nach einigen Monaten aufzutreten.},
  language  = {en}
}
@phdthesis{Jankowski2004,
  author    = {Jankowski, \poundsukasz},
  title     = {Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001649},
  school      = {Universit{\"a}t Potsdam},
  year      = {2004},
  abstract  = {Kunststofflichtwellenleiter (POFs) stellen ein verh{\"a}ltnism{\"a}ßig neues Medium zur optische Datenkommunikation {\"u}ber kurzen Strecken dar. W{\"a}hrend ihrer Einsatzdauer unterliegen POFs unterschiedlichen Arten von Umweltbeanspruchungen, haupts{\"a}chlich durch hohe Temperatur, hohe Feuchtigkeit und mechanischen Belastungen. Zahlreiche experimentelle Forschungen besch{\"a}ftigten sich mit der standardisierten Pr{\"u}fung der Zuverl{\"a}ssigkeit von im Handel erh{\"a}ltlichen Fasern. Jedoch gab es bisher wenig Erfolg bei der Bem{\"u}hung, zwei grundlegende optische Erscheinungen, Absorption und Streuung, die die Lichtausbreitung in Fasern stark beeinflussen, zu verstehen und praktisch zu modellieren: Diese beiden Effekte beschreiben nicht nur die Qualit{\"a}t neuer Fasern, sondern sie werden auch stark durch die Alterungsprozess beeinflusst. Der Hauptzweck dieser Doktorarbeit war es, ein praktisch verwendbares und theoretisch gut fundiertes Modell der Lichtausbreitung in nicht gealterten und gealterten POFs zu entwickeln und es durch optische Experimente zu verifizieren. Dabei wurden anwendungsorientierte Aspekte mit theoretischer POF-Modellierung kombiniert. Die Arbeit enth{\"a}lt die erste bekannte Anwendung der Wellenanalyse zur Untersuchung der winkelabh{\"a}ngigen Eigenschaften der Streuung in Lichtwellenleitern. F{\"u}r die praktischen Experimente wurden mehrere POF-Proben unterschiedlicher Hersteller k{\"u}nstlich gealtert, indem sie bis 4500 Stunden bei 100 °C gelagert wurden (ohne Feuchtekontrolle). Die Parameter der jeweiligen Simulationen wurden mittels einer systematischen Optimierung an die gemessen optischen Eigenschaften der gealterten Proben angeglichen. Die Resultate deuten an, dass der {\"U}bertragungsverlust der gealterten Fasern in den ersten Tagen und Wochen der Alterung am st{\"a}rksten durch eine wesentliche physikalische Verschlechterung der Kern-Mantel-Grenzfl{\"a}che verursacht wird. Chemische Effekte des Alterungsprozesses scheinen im Faserkernmaterial zuerst nach einigen Monaten aufzutreten.},
  language  = {en}
}
@unpublished{Runge2010,
  author    = {Runge, Antonia},
  title     = {Modellierung der Lebensdauer von Systemen},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51674},
  year      = {2010},
  abstract  = {Aus dem Inhalt: Einleitung und Zusammenfassung 1 Grundlagen der Lebensdaueranalyse 2 Systemzuverl{\"a}ssigkeit 3 Zensierung 4 Sch{\"a}tzen in nichtparametrischen Modellen 5 Sch{\"a}tzen in parametrischen Modellen 6 Konfidenzintervalle f{\"u}r Parametersch{\"a}tzungen 7 Verteilung einer gemischten Population 8 Kurze Einf{\"u}hrung: Lebensdauer und Belastung 9 Ausblick A R-Quellcode B Symbole und Abk{\"u}rzungen},
  language  = {de}
}
@unpublished{LaeuterRamadan2010,
  author    = {L{\"a}uter, Henning and Ramadan, Ayad},
  title     = {Modeling and Scaling of Categorical Data},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49572},
  year      = {2010},
  abstract  = {Estimation and testing of distributions in metric spaces are well known. R.A. Fisher, J. Neyman, W. Cochran and M. Bartlett achieved essential results on the statistical analysis of categorical data. In the last 40 years many other statisticians found important results in this field. Often data sets contain categorical data, e.g. levels of factors or names. There does not exist any ordering or any distance between these categories. At each level there are measured some metric or categorical values. We introduce a new method of scaling based on statistical decisions. For this we define empirical probabilities for the original observations and find a class of distributions in a metric space where these empirical probabilities can be found as approximations for equivalently defined probabilities. With this method we identify probabilities connected with the categorical data and probabilities in metric spaces. Here we get a mapping from the levels of factors or names into points of a metric space. This mapping yields the scale for the categorical data. From the statistical point of view we use multivariate statistical methods, we calculate maximum likelihood estimations and compare different approaches for scaling.},
  language  = {de}
}
@phdthesis{Knoechel2019,
  author    = {Kn{\"o}chel, Jane},
  title     = {Model reduction of mechanism-based pharmacodynamic models and its link to classical drug effect models},
  doi       = {10.25932/publishup-44059},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-440598},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 147},
  year      = {2019},
  abstract  = {Continuous insight into biological processes has led to the development of large-scale, mechanistic systems biology models of pharmacologically relevant networks. While these models are typically designed to study the impact of diverse stimuli or perturbations on multiple system variables, the focus in pharmacological research is often on a specific input, e.g., the dose of a drug, and a specific output related to the drug effect or response in terms of some surrogate marker. To study a chosen input-output pair, the complexity of the interactions as well as the size of the models hinders easy access and understanding of the details of the input-output relationship. The objective of this thesis is the development of a mathematical approach, in specific a model reduction technique, that allows (i) to quantify the importance of the different state variables for a given input-output relationship, and (ii) to reduce the dynamics to its essential features -- allowing for a physiological interpretation of state variables as well as parameter estimation in the statistical analysis of clinical data. We develop a model reduction technique using a control theoretic setting by first defining a novel type of time-limited controllability and observability gramians for nonlinear systems. We then show the superiority of the time-limited generalised gramians for nonlinear systems in the context of balanced truncation for a benchmark system from control theory. The concept of time-limited controllability and observability gramians is subsequently used to introduce a state and time-dependent quantity called the input-response (ir) index that quantifies the importance of state variables for a given input-response relationship at a particular time. We subsequently link our approach to sensitivity analysis, thus, enabling for the first time the use of sensitivity coefficients for state space reduction. The sensitivity based ir-indices are given as a product of two sensitivity coefficients. This allows not only for a computational more efficient calculation but also for a clear distinction of the extent to which the input impacts a state variable and the extent to which a state variable impacts the output. The ir-indices give insight into the coordinated action of specific state variables for a chosen input-response relationship. Our developed model reduction technique results in reduced models that still allow for a mechanistic interpretation in terms of the quantities/state variables of the original system, which is a key requirement in the field of systems pharmacology and systems biology and distinguished the reduced models from so-called empirical drug effect models. The ir-indices are explicitly defined with respect to a reference trajectory and thereby dependent on the initial state (this is an important feature of the measure). This is demonstrated for an example from the field of systems pharmacology, showing that the reduced models are very informative in their ability to detect (genetic) deficiencies in certain physiological entities. Comparing our novel model reduction technique to the already existing techniques shows its superiority. The novel input-response index as a measure of the importance of state variables provides a powerful tool for understanding the complex dynamics of large-scale systems in the context of a specific drug-response relationship. Furthermore, the indices provide a means for a very efficient model order reduction and, thus, an important step towards translating insight from biological processes incorporated in detailed systems pharmacology models into the population analysis of clinical data.},
  language  = {en}
}
@unpublished{ShlapunovTarkhanov2004,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Mixed problems with a parameter},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26677},
  year      = {2004},
  abstract  = {Let X be a smooth n -dimensional manifold and D be an open connected set in X with smooth boundary ∂D. Perturbing the Cauchy problem for an elliptic system Au = f in D with data on a closed set Γ ⊂ ∂D we obtain a family of mixed problems depending on a small parameter ε > 0. Although the mixed problems are subject to a non-coercive boundary condition on ∂D\Γ in general, each of them is uniquely solvable in an appropriate Hilbert space DT and the corresponding family {uε} of solutions approximates the solution of the Cauchy problem in DT whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in DT is equivalent to the boundedness of the family {uε}. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing its solution. Examples for Dirac operators in the Euclidean space Rn are considered. In the latter case we obtain a family of mixed boundary problems for the Helmholtz equation.},
  language  = {en}
}
@phdthesis{Friedrich2020,
  author    = {Friedrich, Alexander},
  title     = {Minimizers of generalized Willmore energies and applications in general relativity},
  doi       = {10.25932/publishup-48142},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-481423},
  school      = {Universit{\"a}t Potsdam},
  pages     = {100},
  year      = {2020},
  abstract  = {Das Willmore Funktional ist eine Funktion die jeder Fl{\"a}che in einer Riemannschen Mannigfaltigkeit, ihre totale mittlere Kr{\"u}mmung zuweist. Ein klassisches Problem der Differentialgeometrie ist es geschlossene (kompakt und ohne Rand) Fl{\"a}chen zu finden die das Willmore funktional minimieren, beziehungsweise die kritische Punkte des Willmore Funktionals sind. In dieser Doktorarbeit entwickeln wir ein Konzept von verallgemeinerten Willmore Funktionalen f{\"u}r Fl{\"a}chen in Riemannschen Mannigfaltigkeiten, wobei wir uns von physikalischen Modellen leiten lassen. Insbesondere ist hier die Hawking Energie der allgemeinen Relativit{\"a}tstheorie und die Biegungsenergie von d{\"u}nnen Membranen zu nennen. F{\"u}r dieses verallgemeinerten Willmore Funktionale beweisen wir die Existenz von Minimieren mit vorgeschriebenen Fl{\"a}cheninhalt, in einer geeigneten Klasse von verallgemeinerten Fl{\"a}chen. Insbesondere konstruieren wir Minimierer der oben erw{\"a}hnten Biegungsenergie mit vorgeschrieben Fl{\"a}cheninhalt und vorgeschriebenen, eingeschlossenem Volumen. Außerdem beweisen wir, dass kritische Punkte von verallgemeinerten Willmore Funktionalen mit vorgeschriebenen Fl{\"a}cheninhalt abseits endlich vieler Punkte glatt sind. Dabei st{\"u}tzen wir uns, wie auch im folgenden, auf die bestehende Theorie f{\"u}r das Willmore Funktional. An diese allgemeinen Resultate schließen wir eine detailliertere Analyse der Hawking Energie an. Im Kontext der allgemeinen Relativit{\"a}tstheorie beschreibt die Umgebungsmannigfaltigkeit den Raum zu einem Zeitpunkt. Daher sind wir an dem Wechselspiel zwischen der Hawking Energie und der umgebenden Mannigfaltigkeit interessiert. Wir charakterisieren Punkte in der umgebenden Mannigfaltigkeit f{\"u}r die es in jeder Umgebung eine kritische Fl{\"a}che mit vorgeschriebenem, kleinem Fl{\"a}cheninhalt gibt. Diese Punnkte werden als Konzentrationspunkte der Hawking Energie interpretiert. Außerdem berechnen wir eine Entwicklung der Hawking Energie auf kleinen, runden Sph{\"a}ren. Dadurch k{\"o}nnen wir eine Art Energiedichte der Hawking Energie identifizieren. Hierbei ist anzumerken, dass unsere Resultate im Kontrast zu Ergebnissen in der Literatur stehen. Dort wurde berechnet, dass die Entwicklung der Hawking Energie auf Sph{\"a}ren im Lichtkegel eines Punktes der umgebenden Mannigfaltigkeit in f{\"u}hrender Ordnung proportional zur der klassischen Energiedichte der allgemeinen Relativit{\"a}tstheorie ist. Zu diesem Zeitpunkt ist nicht klar wie diese Diskrepanz zu begr{\"u}nden ist. Ferner betrachten wir asymptotisch Schwarzschild Mannigfaltigkeiten. Sie sind ein Spezialfall von asymptotisch flachen Mannigfaltigkeiten, welche in der allgemeinen Relativit{\"a}tstheorie als Modelle f{\"u}r isolierte Systeme dienen. Die Schwarzschild Raumzeit selbst ist eine rotationssymmetrische Raumzeit die schwarzen Loch beschreibt. In diesen asymptotisch Schwarzschild Mannigfaltigkeiten konstruieren wir eine Bl{\"a}tterung des {\"a}ußeren Bereiches durch kritische Fl{\"a}chen der Hawking Energie mit vorgeschriebenen Fl{\"a}cheninhalt. Diese Bl{\"a}tterung kann in einem verallgemeinertem Sinne als Schwerpunkt des isolierten Systems betrachtet werden. Außerdem zeigen wir, dass die Hawking Energie entlang der Bl{\"a}tterung w{\"a}chst je gr{\"o}ßer die Fl{\"a}chen werden.},
  language  = {en}
}
@phdthesis{Bettenbuehl2015,
  author    = {Bettenb{\"u}hl, Mario},
  title     = {Microsaccades},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-122-6},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72622},
  school      = {Universit{\"a}t Potsdam},
  pages     = {iv, 126},
  year      = {2015},
  abstract  = {The first thing we do upon waking is open our eyes. Rotating them in our eye sockets, we scan our surroundings and collect the information into a picture in our head. Eye movements can be split into saccades and fixational eye movements, which occur when we attempt to fixate our gaze. The latter consists of microsaccades, drift and tremor. Before we even lift our eye lids, eye movements - such as saccades and microsaccades that let the eyes jump from one to another position - have partially been prepared in the brain stem. Saccades and microsaccades are often assumed to be generated by the same mechanisms. But how saccades and microsaccades can be classified according to shape has not yet been reported in a statistical manner. Research has put more effort into the investigations of microsaccades' properties and generation only since the last decade. Consequently, we are only beginning to understand the dynamic processes governing microsaccadic eye movements. Within this thesis, the dynamics governing the generation of microsaccades is assessed and the development of a model for the underlying processes. Eye movement trajectories from different experiments are used, recorded with a video-based eye tracking technique, and a novel method is proposed for the scale-invariant detection of saccades (events of large amplitude) and microsaccades (events of small amplitude). Using a time-frequency approach, the method is examined with different experiments and validated against simulated data. A shape model is suggested that allows for a simple estimation of saccade- and microsaccade related properties. For sequences of microsaccades, in this thesis a time-dynamic Markov model is proposed, with a memory horizon that changes over time and which can best describe sequences of microsaccades.},
  language  = {en}
}
@phdthesis{Samaras2016,
  author    = {Samaras, Stefanos},
  title     = {Microphysical retrieval of non-spherical aerosol particles using regularized inversion of multi-wavelength lidar data},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-396528},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xiv, 190},
  year      = {2016},
  abstract  = {Numerous reports of relatively rapid climate changes over the past century make a clear case of the impact of aerosols and clouds, identified as sources of largest uncertainty in climate projections. Earth's radiation balance is altered by aerosols depending on their size, morphology and chemical composition. Competing effects in the atmosphere can be further studied by investigating the evolution of aerosol microphysical properties, which are the focus of the present work. The aerosol size distribution, the refractive index, and the single scattering albedo are commonly used such properties linked to aerosol type, and radiative forcing. Highly advanced lidars (light detection and ranging) have reduced aerosol monitoring and optical profiling into a routine process. Lidar data have been widely used to retrieve the size distribution through the inversion of the so-called Lorenz-Mie model (LMM). This model offers a reasonable treatment for spherically approximated particles, it no longer provides, though, a viable description for other naturally occurring arbitrarily shaped particles, such as dust particles. On the other hand, non-spherical geometries as simple as spheroids reproduce certain optical properties with enhanced accuracy. Motivated by this, we adapt the LMM to accommodate the spheroid-particle approximation introducing the notion of a two-dimensional (2D) shape-size distribution. Inverting only a few optical data points to retrieve the shape-size distribution is classified as a non-linear ill-posed problem. A brief mathematical analysis is presented which reveals the inherent tendency towards highly oscillatory solutions, explores the available options for a generalized solution through regularization methods and quantifies the ill-posedness. The latter will improve our understanding on the main cause fomenting instability in the produced solution spaces. The new approach facilitates the exploitation of additional lidar data points from depolarization measurements, associated with particle non-sphericity. However, the generalization of LMM vastly increases the complexity of the problem. The underlying theory for the calculation of the involved optical cross sections (T-matrix theory) is computationally so costly, that would limit a retrieval analysis to an unpractical point. Moreover the discretization of the model equation by a 2D collocation method, proposed in this work, involves double integrations which are further time consuming. We overcome these difficulties by using precalculated databases and a sophisticated retrieval software (SphInX: Spheroidal Inversion eXperiments) especially developed for our purposes, capable of performing multiple-dataset inversions and producing a wide range of microphysical retrieval outputs. Hybrid regularization in conjunction with minimization processes is used as a basis for our algorithms. Synthetic data retrievals are performed simulating various atmospheric scenarios in order to test the efficiency of different regularization methods. The gap in contemporary literature in providing full sets of uncertainties in a wide variety of numerical instances is of major concern here. For this, the most appropriate methods are identified through a thorough analysis on an overall-behavior basis regarding accuracy and stability. The general trend of the initial size distributions is captured in our numerical experiments and the reconstruction quality depends on data error level. Moreover, the need for more or less depolarization points is explored for the first time from the point of view of the microphysical retrieval. Finally, our approach is tested in various measurement cases giving further insight for future algorithm improvements.},
  language  = {en}
}
@unpublished{TarkhanovVasilevski2005,
  author    = {Tarkhanov, Nikolai Nikolaevich and Vasilevski, Nikolai},
  title     = {Microlocal analysis of the Bochner-Martinelli integral},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30012},
  year      = {2005},
  abstract  = {In order to characterise the C*-algebra generated by the singular Bochner-Martinelli integral over a smooth closed hypersurfaces in Cn, we compute its principal symbol. We show then that the Szeg{\"o} projection belongs to the strong closure of the algebra generated by the singular Bochner-Martinelli integral.},
  language  = {en}
}
@unpublished{HoegelePavlyukevich2014,
  author    = {H{\"o}gele, Michael and Pavlyukevich, Ilya},
  title     = {Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed L{\´e}vy type noise},
  volume    = {3},
  number    = {5},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70639},
  pages     = {27},
  year      = {2014},
  abstract  = {We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse-Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed L{\´e}vy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a It{\^o}, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.},
  language  = {en}
}
@unpublished{DeDonnoSchulze2003,
  author    = {De Donno, G. and Schulze, Bert-Wolfgang},
  title     = {Meromorphic symbolic structures for boundary value problems on manifolds with edges},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26570},
  year      = {2003},
  abstract  = {We investigate the ideal of Green and Mellin operators with asymtotics for a manifold with edge-corner singularities and boundary which belongs to the structure of parametrices of elliptic boundary value problems on a configuration with corners whose base manifolds have edges.},
  language  = {en}
}
@unpublished{DinesSchulze2003,
  author    = {Dines, Nicoleta and Schulze, Bert-Wolfgang},
  title     = {Mellin-edge representations of elliptic operators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26627},
  year      = {2003},
  abstract  = {We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator-valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator As in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of As as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices Ps of As, interpreted as Mellin-edge representations of P.},
  language  = {en}
}
@article{JansenTsagkarogiannis2020,
  author    = {Jansen, Sabine and Tsagkarogiannis, Dimitrios},
  title     = {Mayer expansion for the Asakura-Oosawa model of colloid theory},
  series = {Lectures in pure and applied mathematics},
  journal   = {Lectures in pure and applied mathematics},
  number    = {6},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-485-2},
  issn      = {2199-4951},
  doi       = {10.25932/publishup-47210},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-472109},
  pages     = {127 -- 134},
  year      = {2020},
  language  = {en}
}
@unpublished{KoppitzMusunthia2012,
  author    = {Koppitz, J{\"o}rg and Musunthia, Tiwadee},
  title     = {Maximal subsemigroups containing a particular semigroup},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57465},
  year      = {2012},
  abstract  = {We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X = N of natural numbers containing a given subsemigroup W of T(X), where each element of a given set U is a generator of T(X) modulo W. This note continues the study of maximal subsemigroups of the monoid of all full transformations on an infinite set.},
  language  = {en}
}
@unpublished{RattanaBoeckmann2012,
  author    = {Rattana, Amornrat and B{\"o}ckmann, Christine},
  title     = {Matrix methods for computing Eigenvalues of Sturm-Liouville problems of order four},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59279},
  year      = {2012},
  abstract  = {This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.},
  language  = {en}
}
@phdthesis{Gopalakrishnan2016,
  author    = {Gopalakrishnan, Sathej},
  title     = {Mathematical modelling of host-disease-drug interactions in HIV disease},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100100},
  school      = {Universit{\"a}t Potsdam},
  pages     = {121},
  year      = {2016},
  abstract  = {The human immunodeficiency virus (HIV) has resisted nearly three decades of efforts targeting a cure. Sustained suppression of the virus has remained a challenge, mainly due to the remarkable evolutionary adaptation that the virus exhibits by the accumulation of drug-resistant mutations in its genome. Current therapeutic strategies aim at achieving and maintaining a low viral burden and typically involve multiple drugs. The choice of optimal combinations of these drugs is crucial, particularly in the background of treatment failure having occurred previously with certain other drugs. An understanding of the dynamics of viral mutant genotypes aids in the assessment of treatment failure with a certain drug combination, and exploring potential salvage treatment regimens. Mathematical models of viral dynamics have proved invaluable in understanding the viral life cycle and the impact of antiretroviral drugs. However, such models typically use simplified and coarse-grained mutation schemes, that curbs the extent of their application to drug-specific clinical mutation data, in order to assess potential next-line therapies. Statistical models of mutation accumulation have served well in dissecting mechanisms of resistance evolution by reconstructing mutation pathways under different drug-environments. While these models perform well in predicting treatment outcomes by statistical learning, they do not incorporate drug effect mechanistically. Additionally, due to an inherent lack of temporal features in such models, they are less informative on aspects such as predicting mutational abundance at treatment failure. This limits their application in analyzing the pharmacology of antiretroviral drugs, in particular, time-dependent characteristics of HIV therapy such as pharmacokinetics and pharmacodynamics, and also in understanding the impact of drug efficacy on mutation dynamics. In this thesis, we develop an integrated model of in vivo viral dynamics incorporating drug-specific mutation schemes learned from clinical data. Our combined modelling approach enables us to study the dynamics of different mutant genotypes and assess mutational abundance at virological failure. As an application of our model, we estimate in vivo fitness characteristics of viral mutants under different drug environments. Our approach also extends naturally to multiple-drug therapies. Further, we demonstrate the versatility of our model by showing how it can be modified to incorporate recently elucidated mechanisms of drug action including molecules that target host factors. Additionally, we address another important aspect in the clinical management of HIV disease, namely drug pharmacokinetics. It is clear that time-dependent changes in in vivo drug concentration could have an impact on the antiviral effect, and also influence decisions on dosing intervals. We present a framework that provides an integrated understanding of key characteristics of multiple-dosing regimens including drug accumulation ratios and half-lifes, and then explore the impact of drug pharmacokinetics on viral suppression. Finally, parameter identifiability in such nonlinear models of viral dynamics is always a concern, and we investigate techniques that alleviate this issue in our setting.},
  language  = {en}
}
@unpublished{Keller2013,
  author    = {Keller, Peter},
  title     = {Mathematical modeling of molecular motors},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63045},
  year      = {2013},
  abstract  = {Amongst the many complex processes taking place in living cells, transport of cargoes across the cytosceleton is fundamental to cell viability and activity. To move cargoes between the different cell parts, cells employ Molecular Motors. The motors operate by transporting cargoes along the so-called cellular micro-tubules, namely rope-like structures that connect, for instance, the cell-nucleus and outer membrane. We introduce a new Markov Chain, the killed Quasi-Random-Walk, for such transport molecules and derive properties like the maximal run length and time. Furthermore we introduce permuted balance, which is a more flexible extension of the ordinary reversibility and introduce the notion of Time Duality, which compares certain passage times pathwise. We give a number of sufficient conditions for Time Duality based on the geometry of the transition graph. Both notions are closely related to properties of the killed Quasi-Random-Walk.},
  language  = {en}
}
@phdthesis{Schindler2023,
  author    = {Schindler, Daniel},
  title     = {Mathematical modeling and simulation of protrusion-driven cell dynamics},
  doi       = {10.25932/publishup-61327},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-613275},
  school      = {Universit{\"a}t Potsdam},
  pages     = {VI, 161},
  year      = {2023},
  abstract  = {Amoeboid cell motility takes place in a variety of biomedical processes such as cancer metastasis, embryonic morphogenesis, and wound healing. In contrast to other forms of cell motility, it is mainly driven by substantial cell shape changes. Based on the interplay of explorative membrane protrusions at the front and a slower-acting membrane retraction at the rear, the cell moves in a crawling kind of way. Underlying these protrusions and retractions are multiple physiological processes resulting in changes of the cytoskeleton, a meshwork of different multi-functional proteins. The complexity and versatility of amoeboid cell motility raise the need for novel computational models based on a profound theoretical framework to analyze and simulate the dynamics of the cell shape. The objective of this thesis is the development of (i) a mathematical framework to describe contour dynamics in time and space, (ii) a computational model to infer expansion and retraction characteristics of individual cell tracks and to produce realistic contour dynamics, (iii) and a complementing Open Science approach to make the above methods fully accessible and easy to use. In this work, we mainly used single-cell recordings of the model organism Dictyostelium discoideum. Based on stacks of segmented microscopy images, we apply a Bayesian approach to obtain smooth representations of the cell membrane, so-called cell contours. We introduce a one-parameter family of regularized contour flows to track reference points on the contour (virtual markers) in time and space. This way, we define a coordinate system to visualize local geometric and dynamic quantities of individual contour dynamics in so-called kymograph plots. In particular, we introduce the local marker dispersion as a measure to identify membrane protrusions and retractions in a fully automated way. This mathematical framework is the basis of a novel contour dynamics model, which consists of three biophysiologically motivated components: one stochastic term, accounting for membrane protrusions, and two deterministic terms to control the shape and area of the contour, which account for membrane retractions. Our model provides a fully automated approach to infer protrusion and retraction characteristics from experimental cell tracks while being also capable of simulating realistic and qualitatively different contour dynamics. Furthermore, the model is used to classify two different locomotion types: the amoeboid and a so-called fan-shaped type. With the complementing Open Science approach, we ensure a high standard regarding the usability of our methods and the reproducibility of our research. In this context, we introduce our software publication named AmoePy, an open-source Python package to segment, analyze, and simulate amoeboid cell motility. Furthermore, we describe measures to improve its usability and extensibility, e.g., by detailed run instructions and an automatically generated source code documentation, and to ensure its functionality and stability, e.g., by automatic software tests, data validation, and a hierarchical package structure. The mathematical approaches of this work provide substantial improvements regarding the modeling and analysis of amoeboid cell motility. We deem the above methods, due to their generalized nature, to be of greater value for other scientific applications, e.g., varying organisms and experimental setups or the transition from unicellular to multicellular movement. Furthermore, we enable other researchers from different fields, i.e., mathematics, biophysics, and medicine, to apply our mathematical methods. By following Open Science standards, this work is of greater value for the cell migration community and a potential role model for other Open Science contributions.},
  language  = {en}
}
@unpublished{Anders2009,
  author    = {Anders, Martin},
  title     = {Martingale, Amarts und das starke Gesetz der Grossen Zahlen},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49494},
  year      = {2009},
  abstract  = {Aus dem Inhalt: Einleitung Kapitel 1. Starke Gesetze der Grossen Zahlen 1. SGGZ unter Wachstumsbedingungen an die p-ten Momente 2. SGGZ f{\"u}r identisch verteilte Zufallsvariablen 3. SGGZ f{\"u}r Prozesse mit *-mixing-Eigenschaft Kapitel 2. Einf{\"u}hrung zu diskreten (Sub-,Super-)Martingalen 1. Vorhersagbarkeit 2. gestoppte (Sub-,Super-)Martingale 3. Upcrossings 4. Konvergenzs{\"a}tze 5. Doob-Zerlegung 6. Eine {\"a}quivalente Definition eines (Sub-)Martingals Kapitel 3. Martingale und gleichgradige Integrierbarkeit 1. Gleichm{\"a}ßige(-f¨ormige,-gradige) Integrierbarkeit 2. gleichgradig integrierbare Martingale Kapitel 4. Martingale und das SGGZ Kapitel 5."reversed" (Sub-,Super-)Martingale 1. Konvergenzs{\"a}tze Kapitel 6. (Sub-,Super-)Martingale mit gerichteter Indexmenge 1. {\"A}quivalente Formulierung eines (Sub-)Martingals 2. Konvergenzs{\"a}tze Kapitel 7. Quasimartingale,Amarts und Semiamarts 1. Konvergenzs{\"a}tze 2. Riesz-Zerlegung 3. Doob-Zerlegung Kapitel 8. Amarts und das SGGZ Kapitel 9."reversed" Amarts und Semiamarts 1. Konvergenzs{\"a}tze 2."Aufw{\"a}rts"- gegen "Abw{\"a}rts"-Adaptiertheit 3. Riesz-Zerlegung 4. Stabilit{\"a}tsanalyse Kapitel 10. Amarts mit gerichteter Indexmenge 1. Konvergenzs{\"a}tze 2. Riesz-Zerlegung Anhang A. zur Existenz einer Folge unabh{\"a}ngiger Zufallsvariablen B. Konvergenz},
  language  = {de}
}
@unpublished{Rafler2008,
  author    = {Rafler, Mathias},
  title     = {Martin-Dynkin Boundaries of the Bose Gas},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51667},
  year      = {2008},
  abstract  = {The Ginibre gas is a Poisson point process defined on a space of loops related to the Feynman-Kac representation of the ideal Bose gas. Here we study thermodynamic limits of different ensembles via Martin-Dynkin boundary technique and show, in which way infinitely long loops occur. This effect is the so-called Bose-Einstein condensation.},
  language  = {en}
}
@unpublished{HieberSchrohe1997,
  author    = {Hieber, Matthias and Schrohe, Elmar},
  title     = {Lρ spectral independence of elliptic operators via commutator estimates},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25047},
  year      = {1997},
  abstract  = {Let {Tsub(p) : q1 ≤ p ≤ q2} be a family of consistent Csub(0) semigroups on Lφ(Ω) with q1, q2 ∈ [1, ∞)and Ω ⊆ IRn open. We show that certain commutator conditions on Tφ and on the resolvent of its generator Aφ ensure the φ independence of the spectrum of Aφ for φ ∈ [q1, q2]. Applications include the case of Petrovskij correct systems with H{\"o}lder continuous coeffcients, Schr{\"o}dinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coeffcients.},
  language  = {en}
}
@unpublished{Galstian2001,
  author    = {Galstian, Anahit},
  title     = {Lp - Lq decay estimates for the equation with exponentially growing coefficient},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26079},
  year      = {2001},
  abstract  = {Contents: 1 Introduction 1 Representation formulas 2 Consideration in the pseudodifferential zone: e up(t) |ξ| ≤ 1 3 Consideration in he hyperbolic zone: e up(t) |ξ| ≥ 1},
  language  = {en}
}
@unpublished{Popivanov2004,
  author    = {Popivanov, Petăr R.},
  title     = {Lorenz transformations and creation of logarithmic singularities to the solutions of some nonstrictly hyperbolic semilinear systems with two space variables},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26710},
  year      = {2004},
  language  = {en}
}
@unpublished{KrainerSchulze2000,
  author    = {Krainer, Thomas and Schulze, Bert-Wolfgang},
  title     = {Long-time asymptotics with geometric singularities in the spatial variables},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25824},
  year      = {2000},
  abstract  = {Content: Introduction 1 Anisotropic operators in a cylinder with a conical base 1.1 Manifolds with conical singularities and opertors of Fuchs type 1.2 Typical operators and symbol structures 2 Weighted wedge Sobolev spaces and edge asymptotics 2.1 Discrete edge asymptotics 2.2 Continuos edge asymptotics with discrete limit at infinity 2.3 Calculus with operator valued symbols 3 Corner asymptotics at infinity 3.1 The structure of singular functions 3.2 Operators with trace and potential conditions 3.3 Asymptotics and (anisotropic) elliptic regularity},
  language  = {en}
}
@unpublished{CattiauxFradonKuliketal.2013,
  author    = {Cattiaux, Patrick and Fradon, Myriam and Kulik, Alexei Michajlovič and Roelly, Sylvie},
  title     = {Long time behavior of stochastic hard ball systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68388},
  year      = {2013},
  abstract  = {We study the long time behavior of a system of two or three Brownian hard balls living in the Euclidean space of dimension at least two, submitted to a mutual attraction and to elastic collisions.},
  language  = {en}
}
@unpublished{NazaikinskiiSchulzeSternin2001,
  author    = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris},
  title     = {Localization problem in index theory of elliptic operators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26175},
  year      = {2001},
  abstract  = {This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.},
  language  = {en}
}
@unpublished{Messina2001,
  author    = {Messina, Francesca},
  title     = {Local solvability for semilinear Fuchsian equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26124},
  year      = {2001},
  abstract  = {Contents: 1 Introduction 2 Differential operators on manifolds with conical singularities 3 When H up(s,y) (B) is an algebra 4 Statement of the main result 5 Proof of the theorem},
  language  = {en}
}
@unpublished{BrauerKarp2006,
  author    = {Brauer, Uwe and Karp, Lavi},
  title     = {Local existence of classical solutions for the Einstein-Euler system using weighted Sobolev spaces of fractional order},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30175},
  year      = {2006},
  abstract  = {We prove the existence of a class of local in time solutions, including static solutions, of the Einstein-Euler system. This result is the relativistic generalisation of a similar result for the Euler-Poisson system obtained by Gamblin [8]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article.},
  language  = {en}
}
@unpublished{Witt2002,
  author    = {Witt, Ingo},
  title     = {Local asymptotic types},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26346},
  year      = {2002},
  abstract  = {The local theory of asymptotic types is elaborated. It appears as coordinate-free version of part of GOHBERG-SIGAL's theory of the inversion of finitely meromorphic, operator-valued functions at a point.},
  language  = {en}
}
@misc{ChampagnatRoelly2008,
  author    = {Champagnat, Nicolas and Roelly, Sylvie},
  title     = {Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-18610},
  year      = {2008},
  abstract  = {A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too .},
  language  = {en}
}
@unpublished{ChampagnatRoelly2007,
  author    = {Champagnat, Nicolas and Roelly, Sylvie},
  title     = {Limit theorems for conditioned multitype Dawson-Watanabe processes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49426},
  year      = {2007},
  abstract  = {A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every nite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. The explicit form of the Laplace functional of the conditioned process is used to obtain several results on the long time behaviour of the mass of the conditioned and unconditioned processes. The general case is considered first, where the mutation matrix which modelizes the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are also analysed.},
  language  = {en}
}
@masterthesis{Dahl2021,
  type      = {Bachelor Thesis},
  author    = {Dahl, Dorothee Sophie},
  title     = {Let's have FUN! Gamification im Mathematikunterricht},
  doi       = {10.25932/publishup-51593},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-515937},
  school      = {Universit{\"a}t Potsdam},
  pages     = {78},
  year      = {2021},
  abstract  = {Spiele und spieltypische Elemente wie das Sammeln von Treuepunkten sind aus dem Alltag kaum wegzudenken. Zudem werden sie zunehmend in Unternehmen oder in Lernumgebungen eingesetzt. Allerdings ist die Methode Gamification bisher f{\"u}r den p{\"a}dagogischen Kontext wenig klassifiziert und f{\"u}r Lehrende kaum zug{\"a}nglich gemacht worden. Daher zielt diese Bachelorarbeit darauf ab, eine systematische Strukturierung und Aufarbeitung von Gamification sowie innovative Ans{\"a}tze f{\"u}r die Verwendung spieltypischer Elemente im Unterricht, konkret dem Mathematikunterricht, zu pr{\"a}sentieren. Dies kann eine Grundlage f{\"u}r andere Fachgebiete, aber auch andere Lehrformen bieten und so die Umsetzbarkeit von Gamification in eigenen Lehrveranstaltungen aufzeigen. In der Arbeit wird begr{\"u}ndet, weshalb und mithilfe welcher Elemente Gamification die Motivation und Leistungsbereitschaft der Lernenden langfristig erh{\"o}hen, die Sozial- und Personalkompetenzen f{\"o}rdern sowie die Lernenden zu mehr Aktivit{\"a}t anregen kann. Zudem wird Gamification explizit mit grundlegenden mathematikdidaktischen Prinzipien in Verbindung gesetzt und somit die Relevanz f{\"u}r den Mathematikunterricht hervorgehoben. Anschließend werden die einzelnen Elemente von Gamification wie Punkte, Level, Abzeichen, Charaktere und Rahmengeschichte entlang einer eigens f{\"u}r den p{\"a}dagogischen Kontext entwickelten Klassifikation „FUN" (Feedback - User specific elements - Neutral elements) schematisch beschrieben, ihre Funktionen und Wirkung dargestellt sowie Einsatzm{\"o}glichkeiten im Unterricht aufgezeigt. Dies beinhaltet Ideen zu lernf{\"o}rderlichem Feedback, Differenzierungsm{\"o}glichkeiten und Unterrichtsrahmengestaltung, die in Lehrveranstaltungen aller Art umsetzbar sein k{\"o}nnen. Die Bachelorarbeit umfasst zudem ein spezifisches Beispiel, einen Unterrichtsentwurf einer gamifizierten Mathematikstunde inklusive des zugeh{\"o}rigen Arbeitsmaterials, anhand dessen die Verwendung von Gamification deutlich wird. Gamification offeriert oftmals Vorteile gegen{\"u}ber dem traditionellen Unterricht, muss jedoch wie jede Methode an den Inhalt und die Zielgruppe angepasst werden. Weiterf{\"u}hrende Forschung k{\"o}nnte sich mit konkreten motivationalen Strukturen, personenspezifischen Unterschieden sowie mit mathematischen Inhalten wie dem Probleml{\"o}sen oder dem Wechsel zwischen verschiedenen Darstellungen hinsichtlich gamifizierter Lehrformen besch{\"a}ftigen.},
  language  = {de}
}
@unpublished{SchulzeTarkhanov1997,
  author    = {Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich},
  title     = {Lefschetz theory on manifolds with edges : introduction},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-24948},
  year      = {1997},
  abstract  = {The aim of this book is to develop the Lefschetz fixed point theory for elliptic complexes of pseudodifferential operators on manifolds with edges. The general Lefschetz theory contains the index theory as a special case, while the case to be studied is much more easier than the index problem. The main topics are: - The calculus of pseudodifferential operators on manifolds with edges, especially symbol structures (inner as well as edge symbols). - The concept of ellipticity, parametrix constructions, elliptic regularity in Sobolev spaces. - Hodge theory for elliptic complexes of pseudodifferential operators on manifolds with edges. - Development of the algebraic constructions for these complexes, such as homotopy, tensor products, duality. - A generalization of the fixed point formula of Atiyah and Bott for the case of simple fixed points. - Development of the fixed point formula also in the case of non-simple fixed points, provided that the complex consists of diferential operarators only. - Investigation of geometric complexes (such as, for instance, the de Rham complex and the Dolbeault complex). Results in this direction are desirable because of both purely mathe matical reasons and applications in natural sciences.},
  language  = {en}
}
@article{PecherskyPirogovYambartsev2020,
  author    = {Pechersky, Eugeny and Pirogov, Sergei and Yambartsev, Anatoly},
  title     = {Large emissions},
  series = {Lectures in pure and applied mathematics},
  journal   = {Lectures in pure and applied mathematics},
  number    = {6},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-485-2},
  issn      = {2199-4951},
  doi       = {10.25932/publishup-47204},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-472049},
  pages     = {77 -- 86},
  year      = {2020},
  language  = {en}
}
@article{KeverpuetzKueppers2018,
  author    = {Keverp{\"u}tz, Claudia and K{\"u}ppers, Bastian},
  title     = {Konsistente Lehr-Lern-Prozesse in der dualen IT-Ausbildung},
  series = {Commentarii informaticae didacticae},
  journal   = {Commentarii informaticae didacticae},
  number    = {12},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-416341},
  pages     = {91 -- 100},
  year      = {2018},
  abstract  = {In der dualen IT-Ausbildung als Verbindung von beruflicher und akademischer Qualifikation werden die berufstypischen Werkzeuge, wie z. B. Laptops, ebenso in den Lehr-Lern-Prozessen der akademischen Unterrichtseinheiten eingesetzt. Im Pr{\"u}fungswesen wird oft auf klassische Papierklausuren zur{\"u}ckgegriffen. Unterrichtseinheiten mit hohem Blended-Learning-Anteil ohne E-Pr{\"u}fung werden dabei als „nicht konsistent" wahrgenommen. In diesem Artikel wird eine empirische Studie dargelegt, die untersucht, welche Einfl{\"u}sse aus der pers{\"o}nlichen Lernbiografie bei den Lehrenden in einer dualen IT-Ausbildung dazu f{\"u}hren k{\"o}nnen, die M{\"o}glichkeiten eines E-Assessments als summative Modulpr{\"u}fung anzunehmen oder abzulehnen. Beispielhaft wurden in der dargelegten Studie Interviews mit Dozenten gef{\"u}hrt und diese hinsichtlich der Verbindung zwischen Lernbiografie, Gestaltung der Didaktik der Lehr-Lern-Prozesse, Zufriedenheit und Ver{\"a}nderungsbereitschaft untersucht.},
  language  = {de}
}
@unpublished{PrenovTarkhanov2001,
  author    = {Prenov, B. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Kernel spikes of singular problems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26195},
  year      = {2001},
  abstract  = {Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This paper is aimed at describing refined asymptotics in the Dirichlet problem in a ball. The beauty of explicit formulas highlights the structure of asymptotic expansions in the calculi on singular varieties.},
  language  = {en}
}
@unpublished{Shlapunov1999,
  author    = {Shlapunov, Alexander},
  title     = {Iterations of self-adjoint operators and their applications to elliptic systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25401},
  year      = {1999},
  abstract  = {Let Hsub(0), Hsub(1) be Hilbert spaces and L : Hsub(0) -> Hsub(1) be a linear bounded operator with ||L|| ≤ 1. Then L*L is a bounded linear self-adjoint non-negative operator in the Hilbert space Hsub(0) and one can use the Neumann series ∑∞sub(v=0)(I - L*L)v L*f in order to study solvability of the operator equation Lu = f. In particular, applying this method to the ill-posed Cauchy problem for solutions to an elliptic system Pu = 0 of linear PDE's of order p with smooth coefficients we obtain solvability conditions and representation formulae for solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauchy-Riemann system in C the summands of the Neumann series are iterations of the Cauchy type integral. We also obtain similar results 1) for the equation Pu = f in Sobolev spaces, 2) for the Dirichlet problem and 3) for the Neumann problem related to operator P*P if P is a homogeneous first order operator and its coefficients are constant. In these cases the representations involve sums of series whose terms are iterations of integro-differential operators, while the solvability conditions consist of convergence of the series together with trivial necessary conditions.},
  language  = {en}
}
@misc{KolbeEvans2020,
  author    = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.},
  title     = {Isotopic tiling theory for hyperbolic surfaces},
  series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-54428},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-544285},
  pages     = {30},
  year      = {2020},
  abstract  = {In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.},
  language  = {en}
}
@unpublished{AirapetyanWitt1997,
  author    = {Airapetyan, Ruben and Witt, Ingo},
  title     = {Isometric properties of the Hankel Transformation in weighted sobolev spaces},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25001},
  year      = {1997},
  abstract  = {It is shown that the Hankel transformation Hsub(v) acts in a class of weighted Sobolev spaces. Especially, the isometric mapping property of Hsub(v) which holds on L²(IRsub(+),rdr) is extended to spaces of arbitrary Sobolev order. The novelty in the approach consists in using techniques developed by B.-W. Schulze and others to treat the half-line Rsub(+) as a manifold with a conical singularity at r = 0. This is achieved by pointing out a connection between the Hankel transformation and the Mellin transformation.The procedure proposed leads at the same time to a short proof of the Hankel inversion formula. An application to the existence and higher regularity of solutions, including their asymptotics, to the 1-1-dimensional edge-degenerated wave equation is given.},
  language  = {en}
}
@unpublished{LevendorskiiBoyarchenko1998,
  author    = {Levendorskii, Sergei Z. and Boyarchenko, Svetlana I.},
  title     = {Investment under uncertainty when shocks are non-gaussian},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25201},
  year      = {1998},
  language  = {en}
}
@misc{Tschisgale2020,
  type      = {Master Thesis},
  author    = {Tschisgale, Paul},
  title     = {Introduction to the Glauber dynamics for the Curie-Weiss Potts model},
  doi       = {10.25932/publishup-48676},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-486769},
  school      = {Universit{\"a}t Potsdam},
  pages     = {104},
  year      = {2020},
  abstract  = {This thesis aims at presenting in an organized fashion the required basics to understand the Glauber dynamics as a way of simulating configurations according to the Gibbs distribution of the Curie-Weiss Potts model. Therefore, essential aspects of discrete-time Markov chains on a finite state space are examined, especially their convergence behavior and related mixing times. Furthermore, special emphasis is placed on a consistent and comprehensive presentation of the Curie-Weiss Potts model and its analysis. Finally, the Glauber dynamics is studied in general and applied afterwards in an exemplary way to the Curie-Weiss model as well as the Curie-Weiss Potts model. The associated considerations are supplemented with two computer simulations aiming to show the cutoff phenomenon and the temperature dependence of the convergence behavior.},
  language  = {en}
}
@book{Kulik2015,
  author    = {Kulik, Alexei Michajlovič},
  title     = {Introduction to Ergodic rates for Markov chains and processes},
  editor    = {Roelly, Sylvie},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-338-1},
  issn      = {2199-4951},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-79360},
  publisher      = {Universit{\"a}t Potsdam},
  pages     = {ix, 122 S.},
  year      = {2015},
  abstract  = {The present lecture notes aim for an introduction to the ergodic behaviour of Markov Processes and addresses graduate students, post-graduate students and interested readers. Different tools and methods for the study of upper bounds on uniform and weak ergodic rates of Markov Processes are introduced. These techniques are then applied to study limit theorems for functionals of Markov processes. This lecture course originates in two mini courses held at University of Potsdam, Technical University of Berlin and Humboldt University in spring 2013 and Ritsumameikan University in summer 2013. Alexei Kulik, Doctor of Sciences, is a Leading researcher at the Institute of Mathematics of Ukrainian National Academy of Sciences.},
  language  = {en}
}
@phdthesis{Solms2017,
  author    = {Solms, Alexander Maximilian},
  title     = {Integrating nonlinear mixed effects and physiologically-based modeling approaches for the analysis of repeated measurement studies},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-397070},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 141},
  year      = {2017},
  abstract  = {During the drug discovery \& development process, several phases encompassing a number of preclinical and clinical studies have to be successfully passed to demonstrate safety and efficacy of a new drug candidate. As part of these studies, the characterization of the drug's pharmacokinetics (PK) is an important aspect, since the PK is assumed to strongly impact safety and efficacy. To this end, drug concentrations are measured repeatedly over time in a study population. The objectives of such studies are to describe the typical PK time-course and the associated variability between subjects. Furthermore, underlying sources significantly contributing to this variability, e.g. the use of comedication, should be identified. The most commonly used statistical framework to analyse repeated measurement data is the nonlinear mixed effect (NLME) approach. At the same time, ample knowledge about the drug's properties already exists and has been accumulating during the discovery \& development process: Before any drug is tested in humans, detailed knowledge about the PK in different animal species has to be collected. This drug-specific knowledge and general knowledge about the species' physiology is exploited in mechanistic physiological based PK (PBPK) modeling approaches -it is, however, ignored in the classical NLME modeling approach. Mechanistic physiological based models aim to incorporate relevant and known physiological processes which contribute to the overlying process of interest. In comparison to data--driven models they are usually more complex from a mathematical perspective. For example, in many situations, the number of model parameters outrange the number of measurements and thus reliable parameter estimation becomes more complex and partly impossible. As a consequence, the integration of powerful mathematical estimation approaches like the NLME modeling approach -which is widely used in data-driven modeling -and the mechanistic modeling approach is not well established; the observed data is rather used as a confirming instead of a model informing and building input. Another aggravating circumstance of an integrated approach is the inaccessibility to the details of the NLME methodology so that these approaches can be adapted to the specifics and needs of mechanistic modeling. Despite the fact that the NLME modeling approach exists for several decades, details of the mathematical methodology is scattered around a wide range of literature and a comprehensive, rigorous derivation is lacking. Available literature usually only covers selected parts of the mathematical methodology. Sometimes, important steps are not described or are only heuristically motivated, e.g. the iterative algorithm to finally determine the parameter estimates. Thus, in the present thesis the mathematical methodology of NLME modeling is systemically described and complemented to a comprehensive description, comprising the common theme from ideas and motivation to the final parameter estimation. Therein, new insights for the interpretation of different approximation methods used in the context of the NLME modeling approach are given and illustrated; furthermore, similarities and differences between them are outlined. Based on these findings, an expectation-maximization (EM) algorithm to determine estimates of a NLME model is described. Using the EM algorithm and the lumping methodology by Pilari2010, a new approach on how PBPK and NLME modeling can be combined is presented and exemplified for the antibiotic levofloxacin. Therein, the lumping identifies which processes are informed by the available data and the respective model reduction improves the robustness in parameter estimation. Furthermore, it is shown how apriori known factors influencing the variability and apriori known unexplained variability is incorporated to further mechanistically drive the model development. Concludingly, correlation between parameters and between covariates is automatically accounted for due to the mechanistic derivation of the lumping and the covariate relationships. A useful feature of PBPK models compared to classical data-driven PK models is in the possibility to predict drug concentration within all organs and tissue in the body. Thus, the resulting PBPK model for levofloxacin is used to predict drug concentrations and their variability within soft tissues which are the site of action for levofloxacin. These predictions are compared with data of muscle and adipose tissue obtained by microdialysis, which is an invasive technique to measure a proportion of drug in the tissue, allowing to approximate the concentrations in the interstitial fluid of tissues. Because, so far, comparing human in vivo tissue PK and PBPK predictions are not established, a new conceptual framework is derived. The comparison of PBPK model predictions and microdialysis measurements shows an adequate agreement and reveals further strengths of the presented new approach. We demonstrated how mechanistic PBPK models, which are usually developed in the early stage of drug development, can be used as basis for model building in the analysis of later stages, i.e. in clinical studies. As a consequence, the extensively collected and accumulated knowledge about species and drug are utilized and updated with specific volunteer or patient data. The NLME approach combined with mechanistic modeling reveals new insights for the mechanistic model, for example identification and quantification of variability in mechanistic processes. This represents a further contribution to the learn \& confirm paradigm across different stages of drug development. Finally, the applicability of mechanism--driven model development is demonstrated on an example from the field of Quantitative Psycholinguistics to analyse repeated eye movement data. Our approach gives new insight into the interpretation of these experiments and the processes behind.},
  language  = {en}
}
@inproceedings{AudinDucourtiouxOuedraogoetal.2017,
  author    = {Audin, Mich{\`e}le and Ducourtioux, Catherine and Ou{\´e}draogo, Fran{\c{c}}oise and Schulz, Ren{\´e} and Delgado, Julio and Ruzhansky, Michael and Lebeau, Gilles},
  title     = {Integral Fourier operators},
  editor    = {Paycha, Sylvie},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-413-5},
  issn      = {2199-4951},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-402657},
  pages     = {xxi, 229},
  year      = {2017},
  abstract  = {This volume of contributions based on lectures delivered at a school on Fourier Integral Operators held in Ouagadougou, Burkina Faso, 14-26 September 2015, provides an introduction to Fourier Integral Operators (FIO) for a readership of Master and PhD students as well as any interested layperson. Considering the wide spectrum of their applications and the richness of the mathematical tools they involve, FIOs lie the cross-road of many a field. This volume offers the necessary background, whether analytic or geometric, to get acquainted with FIOs, complemented by more advanced material presenting various aspects of active research in that area.},
  language  = {en}
}
@unpublished{FradonRoelly2009,
  author    = {Fradon, Myriam and Roelly, Sylvie},
  title     = {Infinitely many Brownian globules with Brownian radii},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49552},
  year      = {2009},
  abstract  = {We consider an infinite system of non overlaping globules undergoing Brownian motions in R3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinitedimensional Stochastic Differential Equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.},
  language  = {en}
}
@unpublished{FradonRoelly2005,
  author    = {Fradon, Myriam and Roelly, Sylvie},
  title     = {Infinite system of Brownian Balls: Equilibrium measures are canonical Gibbs},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51594},
  year      = {2005},
  abstract  = {We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.},
  language  = {en}
}
@unpublished{RoellyFradon2006,
  author    = {Roelly, Sylvie and Fradon, Myriam},
  title     = {Infinite system of Brownian balls : equilibrium measures are canonical Gibbs},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6720},
  year      = {2006},
  abstract  = {We consider a system of infinitely many hard balls in R<sup>d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.},
  language  = {en}
}