@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} }