@article{SchirmerPapenbrockKoumarelasetal.2020, author = {Schirmer, Philipp and Papenbrock, Thorsten and Koumarelas, Ioannis and Naumann, Felix}, title = {Efficient discovery of matching dependencies}, series = {ACM transactions on database systems : TODS}, volume = {45}, journal = {ACM transactions on database systems : TODS}, number = {3}, publisher = {Association for Computing Machinery}, address = {New York}, issn = {0362-5915}, doi = {10.1145/3392778}, pages = {33}, year = {2020}, abstract = {Matching dependencies (MDs) are data profiling results that are often used for data integration, data cleaning, and entity matching. They are a generalization of functional dependencies (FDs) matching similar rather than same elements. As their discovery is very difficult, existing profiling algorithms find either only small subsets of all MDs or their scope is limited to only small datasets. We focus on the efficient discovery of all interesting MDs in real-world datasets. For this purpose, we propose HyMD, a novel MD discovery algorithm that finds all minimal, non-trivial MDs within given similarity boundaries. The algorithm extracts the exact similarity thresholds for the individual MDs from the data instead of using predefined similarity thresholds. For this reason, it is the first approach to solve the MD discovery problem in an exact and truly complete way. If needed, the algorithm can, however, enforce certain properties on the reported MDs, such as disjointness and minimum support, to focus the discovery on such results that are actually required by downstream use cases. HyMD is technically a hybrid approach that combines the two most popular dependency discovery strategies in related work: lattice traversal and inference from record pairs. Despite the additional effort of finding exact similarity thresholds for all MD candidates, the algorithm is still able to efficiently process large datasets, e.g., datasets larger than 3 GB.}, language = {en} } @article{MaoutsaReichOpper2020, author = {Maoutsa, Dimitra and Reich, Sebastian and Opper, Manfred}, title = {Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation}, series = {Entropy}, volume = {22}, journal = {Entropy}, number = {8}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e22080802}, pages = {35}, year = {2020}, abstract = {Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.}, language = {en} } @article{BandaraBryan2020, author = {Bandara, Lashi and Bryan, Paul}, title = {Heat kernels and regularity for rough metrics on smooth manifolds}, series = {Mathematische Nachrichten}, volume = {293}, journal = {Mathematische Nachrichten}, number = {12}, publisher = {Wiley-VCH}, address = {Weinheim}, issn = {0025-584X}, doi = {10.1002/mana.201800459}, pages = {2255 -- 2270}, year = {2020}, abstract = {We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Holder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.}, language = {en} } @article{AzzaliPaycha2020, author = {Azzali, Sara and Paycha, Sylvie}, title = {Spectral zeta-invariants lifted to coverings}, series = {Transactions of the American Mathematical Society}, volume = {373}, journal = {Transactions of the American Mathematical Society}, number = {9}, publisher = {American Mathematical Society}, address = {Providence, RI}, issn = {0002-9947}, doi = {10.1090/tran/8067}, pages = {6185 -- 6226}, year = {2020}, abstract = {The canonical trace and the Wodzicki residue on classical pseudo-differential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral zeta-invariants using lifted defect formulae which express discrepancies of zeta-regularised traces in terms of Wodzicki residues. We derive Atiyah's L-2-index theorem as an instance of the Z(2)-graded generalisation of the canonical lift of spectral zeta-invariants and we show that certain lifted spectral zeta-invariants for geometric operators are integrals of Pontryagin and Chern forms.}, language = {en} } @article{BiloLenzner2019, author = {Bil{\`o}, Davide and Lenzner, Pascal}, title = {On the tree conjecture for the network creation game}, series = {Theory of computing systems}, volume = {64}, journal = {Theory of computing systems}, number = {3}, publisher = {Springer}, address = {New York}, issn = {1432-4350}, doi = {10.1007/s00224-019-09945-9}, pages = {422 -- 443}, year = {2019}, abstract = {Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al. (2003) is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bound for the latter conjecture. In particular we show that for alpha > 4n - 13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.}, language = {en} } @article{BlanchardMuecke2020, author = {Blanchard, Gilles and M{\"u}cke, Nicole}, title = {Kernel regression, minimax rates and effective dimensionality}, series = {Analysis and applications}, volume = {18}, journal = {Analysis and applications}, number = {4}, publisher = {World Scientific}, address = {New Jersey}, issn = {0219-5305}, doi = {10.1142/S0219530519500258}, pages = {683 -- 696}, year = {2020}, abstract = {We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.}, language = {en} } @article{PereraBoeckmann2020, author = {Perera, Upeksha and B{\"o}ckmann, Christine}, title = {Solutions of Sturm-Liouville problems}, series = {Mathematics}, volume = {8}, journal = {Mathematics}, number = {11}, publisher = {MDPI}, address = {Basel}, issn = {2227-7390}, doi = {10.3390/math8112074}, pages = {14}, year = {2020}, abstract = {This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm-Liouville problems. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm-Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.}, language = {en} } @article{HydeEvans2022, author = {Hyde, Stephen T. and Evans, Myfanwy E.}, title = {Symmetric tangled Platonic polyhedra}, series = {Proceedings of the National Academy of Sciences of the United States of America}, volume = {119}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, number = {1}, publisher = {National Acad. of Sciences}, address = {Washington}, issn = {0027-8424}, doi = {10.1073/pnas.2110345118}, pages = {10}, year = {2022}, abstract = {Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.}, language = {en} } @techreport{SultanowVolkovCox2017, author = {Sultanow, Eldar and Volkov, Denis and Cox, Sean}, title = {Introducing a Finite State Machine for processing Collatz Sequences}, edition = {1st version}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-399223}, year = {2017}, abstract = {The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine's word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.}, language = {en} } @unpublished{SultanovKalyakinTarkhanov2014, author = {Sultanov, Oskar and Kalyakin, Leonid and Tarkhanov, Nikolai Nikolaevich}, title = {Elliptic perturbations of dynamical systems with a proper node}, volume = {3}, number = {4}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70460}, pages = {12}, year = {2014}, abstract = {The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node.}, language = {en} } @phdthesis{Etzold2021, author = {Etzold, Heiko}, title = {Neue Zug{\"a}nge zum Winkelbegriff}, doi = {10.25932/publishup-50418}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-504187}, school = {Universit{\"a}t Potsdam}, pages = {300}, year = {2021}, abstract = {Die Vielf{\"a}ltigkeit des Winkelbegriffs ist gleichermaßen spannend wie herausfordernd in Hinblick auf seine Zug{\"a}nge im Mathematikunterricht der Schule. Ausgehend von verschiedenen Vorstellungen zum Winkelbegriff wird in dieser Arbeit ein Lehrgang zur Vermittlung des Winkelbegriffs entwickelt und letztlich in konkrete Umsetzungen f{\"u}r den Schulunterricht {\"u}berf{\"u}hrt. Dabei erfolgt zun{\"a}chst eine stoffdidaktische Auseinandersetzung mit dem Winkelbegriff, die von einer informationstheoretischen Winkeldefinition begleitet wird. In dieser wird eine Definition f{\"u}r den Winkelbegriff unter der Fragestellung entwickelt, welche Informationen man {\"u}ber einen Winkel ben{\"o}tigt, um ihn beschreiben zu k{\"o}nnen. So k{\"o}nnen die in der fachdidaktischen Literatur auftretenden Winkelvorstellungen aus fachmathematischer Perspektive erneut abgeleitet und validiert werden. Parallel dazu wird ein Verfahren beschrieben, wie Winkel - auch unter dynamischen Aspekten - informationstechnisch verarbeitet werden k{\"o}nnen, so dass Schlussfolgerungen aus der informationstheoretischen Winkeldefinition beispielsweise in dynamischen Geometriesystemen zur Verf{\"u}gung stehen. Unter dem Gesichtspunkt, wie eine Abstraktion des Winkelbegriffs im Mathematikunterricht vonstatten gehen kann, werden die Grundvorstellungsidee sowie die Lehrstrategie des Aufsteigens vom Abstrakten zum Konkreten miteinander in Beziehung gesetzt. Aus der Verkn{\"u}pfung der beiden Theorien wird ein grunds{\"a}tzlicher Weg abgeleitet, wie im Rahmen der Lehrstrategie eine Ausgangsabstraktion zu einzelnen Winkelaspekten aufgebaut werden kann, was die Generierung von Grundvorstellungen zu den Bestandteilen des jeweiligen Winkelaspekts und zum Operieren mit diesen Begriffsbestandteilen erm{\"o}glichen soll. Hierf{\"u}r wird die Lehrstrategie angepasst, um insbesondere den {\"U}bergang von Winkelsituationen zu Winkelkontexten zu realisieren. Explizit f{\"u}r den Aspekt des Winkelfeldes werden, anhand der Untersuchung der Sichtfelder von Tieren, Lernhandlungen und Forderungen an ein Lernmodell beschrieben, die Sch{\"u}lerinnen und Sch{\"u}ler bei der Begriffsaneignung unterst{\"u}tzen. Die T{\"a}tigkeitstheorie, der die genannte Lehrstrategie zuzuordnen ist, zieht sich als roter Faden durch die weitere Arbeit, wenn nun theoriebasiert Designprinzipien generiert werden, die in die Entwicklung einer interaktiven Lernumgebung m{\"u}nden. Hierzu wird u. a. das Modell der Artifact-Centric Activity Theory genutzt, das das Beziehungsgef{\"u}ge aus Sch{\"u}lerinnen und Sch{\"u}lern, dem mathematischen Gegenstand und einer zu entwickelnden App als vermittelndes Medium beschreibt, wobei der Einsatz der App im Unterrichtskontext sowie deren regelgeleitete Entwicklung Bestandteil des Modells sind. Gem{\"a}ß dem Ansatz der Fachdidaktischen Entwicklungsforschung wird die Lernumgebung anschließend in mehreren Zyklen erprobt, evaluiert und {\"u}berarbeitet. Dabei wird ein qualitatives Setting angewandt, das sich der Semiotischen Vermittlung bedient und untersucht, inwiefern sich die Qualit{\"a}t der von den Sch{\"u}lerinnen und Sch{\"u}lern gezeigten Lernhandlungen durch die Designprinzipien und deren Umsetzung erkl{\"a}ren l{\"a}sst. Am Ende der Arbeit stehen eine finale Version der Designprinzipien und eine sich daraus ergebende Lernumgebung zur Einf{\"u}hrung des Winkelfeldbegriffs in der vierten Klassenstufe.}, language = {de} } @phdthesis{Zass2021, author = {Zass, Alexander}, title = {A multifaceted study of marked Gibbs point processes}, doi = {10.25932/publishup-51277}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-512775}, school = {Universit{\"a}t Potsdam}, pages = {vii, 104}, year = {2021}, abstract = {This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry.}, language = {en} } @phdthesis{Berner2016, author = {Berner, Nadine}, title = {Deciphering multiple changes in complex climate time series using Bayesian inference}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100065}, school = {Universit{\"a}t Potsdam}, pages = {xvi, 135}, year = {2016}, abstract = {Change points in time series are perceived as heterogeneities in the statistical or dynamical characteristics of the observations. Unraveling such transitions yields essential information for the understanding of the observed system's intrinsic evolution and potential external influences. A precise detection of multiple changes is therefore of great importance for various research disciplines, such as environmental sciences, bioinformatics and economics. The primary purpose of the detection approach introduced in this thesis is the investigation of transitions underlying direct or indirect climate observations. In order to develop a diagnostic approach capable to capture such a variety of natural processes, the generic statistical features in terms of central tendency and dispersion are employed in the light of Bayesian inversion. In contrast to established Bayesian approaches to multiple changes, the generic approach proposed in this thesis is not formulated in the framework of specialized partition models of high dimensionality requiring prior specification, but as a robust kernel-based approach of low dimensionality employing least informative prior distributions. First of all, a local Bayesian inversion approach is developed to robustly infer on the location and the generic patterns of a single transition. The analysis of synthetic time series comprising changes of different observational evidence, data loss and outliers validates the performance, consistency and sensitivity of the inference algorithm. To systematically investigate time series for multiple changes, the Bayesian inversion is extended to a kernel-based inference approach. By introducing basic kernel measures, the weighted kernel inference results are composed into a proxy probability to a posterior distribution of multiple transitions. The detection approach is applied to environmental time series from the Nile river in Aswan and the weather station Tuscaloosa, Alabama comprising documented changes. The method's performance confirms the approach as a powerful diagnostic tool to decipher multiple changes underlying direct climate observations. Finally, the kernel-based Bayesian inference approach is used to investigate a set of complex terrigenous dust records interpreted as climate indicators of the African region of the Plio-Pleistocene period. A detailed inference unravels multiple transitions underlying the indirect climate observations, that are interpreted as conjoint changes. The identified conjoint changes coincide with established global climate events. In particular, the two-step transition associated to the establishment of the modern Walker-Circulation contributes to the current discussion about the influence of paleoclimate changes on the environmental conditions in tropical and subtropical Africa at around two million years ago.}, language = {en} } @misc{GoychukKharchenko2013, author = {Goychuk, Igor and Kharchenko, Vasyl O.}, title = {Rocking subdiffusive ratchets}, series = {Mathematical Modelling of Natural Phenomena}, journal = {Mathematical Modelling of Natural Phenomena}, number = {622}, issn = {1866-8372}, doi = {10.1051/mmnp/20138210}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-416138}, pages = {15}, year = {2013}, abstract = {We study origin, parameter optimization, and thermodynamic efficiency of isothermal rocking ratchets based on fractional subdiffusion within a generalized non-Markovian Langevin equation approach. A corresponding multi-dimensional Markovian embedding dynamics is realized using a set of auxiliary Brownian particles elastically coupled to the central Brownian particle (see video on the journal web site). We show that anomalous subdiffusive transport emerges due to an interplay of nonlinear response and viscoelastic effects for fractional Brownian motion in periodic potentials with broken space-inversion symmetry and driven by a time-periodic field. The anomalous transport becomes optimal for a subthreshold driving when the driving period matches a characteristic time scale of interwell transitions. It can also be optimized by varying temperature, amplitude of periodic potential and driving strength. The useful work done against a load shows a parabolic dependence on the load strength. It grows sublinearly with time and the corresponding thermodynamic efficiency decays algebraically in time because the energy supplied by the driving field scales with time linearly. However, it compares well with the efficiency of normal diffusion rocking ratchets on an appreciably long time scale.}, language = {en} } @phdthesis{Oancea2021, author = {Oancea, Marius-Adrian}, title = {Spin Hall effects in general relativity}, doi = {10.25932/publishup-50229}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-502293}, school = {Universit{\"a}t Potsdam}, pages = {vii, 123}, year = {2021}, abstract = {The propagation of test fields, such as electromagnetic, Dirac or linearized gravity, on a fixed spacetime manifold is often studied by using the geometrical optics approximation. In the limit of infinitely high frequencies, the geometrical optics approximation provides a conceptual transition between the test field and an effective point-particle description. The corresponding point-particles, or wave rays, coincide with the geodesics of the underlying spacetime. For most astrophysical applications of interest, such as the observation of celestial bodies, gravitational lensing, or the observation of cosmic rays, the geometrical optics approximation and the effective point-particle description represent a satisfactory theoretical model. However, the geometrical optics approximation gradually breaks down as test fields of finite frequency are considered. In this thesis, we consider the propagation of test fields on spacetime, beyond the leading-order geometrical optics approximation. By performing a covariant Wentzel-Kramers-Brillouin analysis for test fields, we show how higher-order corrections to the geometrical optics approximation can be considered. The higher-order corrections are related to the dynamics of the spin internal degree of freedom of the considered test field. We obtain an effective point-particle description, which contains spin-dependent corrections to the geodesic motion obtained using geometrical optics. This represents a covariant generalization of the well-known spin Hall effect, usually encountered in condensed matter physics and in optics. Our analysis is applied to electromagnetic and massive Dirac test fields, but it can easily be extended to other fields, such as linearized gravity. In the electromagnetic case, we present several examples where the gravitational spin Hall effect of light plays an important role. These include the propagation of polarized light rays on black hole spacetimes and cosmological spacetimes, as well as polarization-dependent effects on the shape of black hole shadows. Furthermore, we show that our effective point-particle equations for polarized light rays reproduce well-known results, such as the spin Hall effect of light in an inhomogeneous medium, and the relativistic Hall effect of polarized electromagnetic wave packets encountered in Minkowski spacetime.}, language = {en} } @unpublished{ConfortiLeonardMurretal.2014, author = {Conforti, Giovanni and L{\´e}onard, Christian and Murr, R{\"u}diger and Roelly, Sylvie}, title = {Bridges of Markov counting processes : reciprocal classes and duality formulas}, volume = {3}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71855}, pages = {12}, year = {2014}, abstract = {Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.}, language = {en} } @unpublished{FradonRoelly2005, author = {Fradon, Myriam and Roelly, Sylvie}, title = {Infinite system of Brownian balls with interaction : the non-reversible case}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51546}, year = {2005}, abstract = {We consider an infinite system of 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 an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures.}, language = {en} } @unpublished{PraLouisMinelli2008, author = {Pra, Paolo Dai and Louis, Pierre-Yves and Minelli, Ida G.}, title = {Complete monotone coupling for Markov processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-18286}, year = {2008}, abstract = {We formalize and analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuoustime but not in discrete-time.}, language = {de} } @unpublished{KleinLeonardRosenberger2012, author = {Klein, Markus and L{\´e}onard, Christian and Rosenberger, Elke}, title = {Agmon-type estimates for a class of jump processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56995}, year = {2012}, abstract = {In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice.}, language = {en} } @unpublished{Kunze2010, author = {Kunze, Simone}, title = {Das Sammelbilderproblem}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51646}, year = {2010}, abstract = {Aus dem Inhalt: 1 Einleitung 2 Entwicklung der L{\"o}sungsans{\"a}tze 3 Martingalansatz 4 Markov-Ketten Ansatz 5 Einbettung in Poisson Prozesse 6 Kombinatorische Ans{\"a}tze 7 Zusammenfassung und Ausblick Literaturverzeichnis}, language = {de} } @phdthesis{Hain2022, author = {Hain, Tobias Martin}, title = {Structure formation and identification in geometrically driven soft matter systems}, doi = {10.25932/publishup-55880}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-558808}, school = {Universit{\"a}t Potsdam}, pages = {xviii, 171}, year = {2022}, abstract = {Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces. Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness. In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate. We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase. The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy.}, language = {en} } @unpublished{LaeuterRamadan2010, author = {L{\"a}uter, Henning and Ramadan, Ayad}, title = {Statistical Scaling of Categorical Data}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49566}, 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 = {en} } @unpublished{Voss2010, author = {Voss, Carola Regine}, title = {Harness-Prozesse}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49651}, year = {2010}, abstract = {Harness-Prozesse finden in der Forschung immer mehr Anwendung. Vor allem gewinnen Harness-Prozesse in stetiger Zeit an Bedeutung. Grundlegende Literatur zu diesem Thema ist allerdings wenig vorhanden. In der vorliegenden Arbeit wird die vorhandene Grundlagenliteratur zu Harness-Prozessen in diskreter und stetiger Zeit aufgearbeitet und Beweise ausgef{\"u}hrt, die bisher nur skizziert waren. Ziel dessen ist die Existenz einer Zerlegung von Harness-Prozessen {\"u}ber Z beziehungsweise R+ nachzuweisen.}, language = {de} } @unpublished{MeleardRoelly2011, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {A host-parasite multilevel interacting process and continuous approximations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51694}, year = {2011}, abstract = {We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We look for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses. The study of the long time behavior of these processes seems very hard and we only develop some simple cases enlightening the difficulties involved.}, language = {en} } @article{BaerMazzeo2021, author = {B{\"a}r, Christian and Mazzeo, Rafe}, title = {Manifolds with many Rarita-Schwinger fields}, series = {Communications in mathematical physics}, volume = {384}, journal = {Communications in mathematical physics}, number = {1}, publisher = {Springer}, address = {Berlin}, issn = {0010-3616}, doi = {10.1007/s00220-021-04030-0}, pages = {533 -- 548}, year = {2021}, abstract = {The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kahler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.}, language = {en} } @misc{GinouxHabib2008, author = {Ginoux, Nicolas and Habib, Georges}, title = {Geometric aspects of transversal Killing spinors on Riemannian flows}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, number = {867}, issn = {1866-8372}, doi = {10.25932/publishup-43478}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-434783}, pages = {69 -- 90}, year = {2008}, abstract = {We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those flows carrying non-trivial solutions.}, language = {en} } @phdthesis{Lewandowski2019, author = {Lewandowski, Max}, title = {Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes}, doi = {10.25932/publishup-43938}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-439381}, school = {Universit{\"a}t Potsdam}, pages = {v, 69}, year = {2019}, abstract = {Quantenfeldtheorie auf gekr{\"u}mmten Raumzeiten ist eine semiklassische N{\"a}herung einer Quantentheorie der Gravitation, im Rahmen derer ein Quantenfeld unter dem Einfluss eines klassisch modellierten Gravitationsfeldes, also einer gekr{\"u}mmten Raumzeit, beschrieben wird. Eine der bemerkenswertesten Vorhersagen dieses Ansatzes ist die Erzeugung von Teilchen durch die gekr{\"u}mmte Raumzeit selbst, wie zum Beispiel durch Hawkings Verdampfen schwarzer L{\"o}cher und den Unruh Effekt. Andererseits deuten diese Aspekte bereits an, dass fundamentale Grundpfeiler der Theorie auf dem Minkowskiraum, insbesondere ein ausgezeichneter Vakuumzustand und damit verbunden der Teilchenbegriff, f{\"u}r allgemeine gekr{\"u}mmte Raumzeiten keine sinnvolle Entsprechung besitzen. Gleichermaßen ben{\"o}tigen wir eine alternative Implementierung von Kovarianz in die Theorie, da gekr{\"u}mmte Raumzeiten im Allgemeinen keine nicht-triviale globale Symmetrie aufweisen. Letztere Problematik konnte im Rahmen lokal-kovarianter Quantenfeldtheorie gel{\"o}st werden, wohingegen die Abwesenheit entsprechender Konzepte f{\"u}r Vakuum und Teilchen in diesem allgemeinen Fall inzwischen sogar in Form von no-go-Aussagen manifestiert wurde. Beim algebraischen Ansatz f{\"u}r eine Quantenfeldtheorie werden zun{\"a}chst Observablen eingef{\"u}hrt und erst anschließend Zust{\"a}nde via Zuordnung von Erwartungswerten. Obwohl die Observablen unter physikalischen Gesichtspunkten konstruiert werden, existiert dennoch eine große Anzahl von m{\"o}glichen Zust{\"a}nden, von denen viele, aus physikalischen Blickwinkeln betrachtet, nicht sinnvoll sind. Dieses Konzept von Zust{\"a}nden ist daher noch zu allgemein und bedarf weiterer physikalisch motivierter Einschr{\"a}nkungen. Beispielsweise ist es nat{\"u}rlich, sich im Falle freier Quantenfeldtheorien mit linearen Feldgleichungen auf quasifreie Zust{\"a}nde zu konzentrieren. Dar{\"u}ber hinaus ist die Renormierung von Erwartungswerten f{\"u}r Produkte von Feldern von zentraler Bedeutung. Dies betrifft insbesondere den Energie-Impuls-Tensor, dessen Erwartungswert durch distributionelle Bil{\"o}sungen der Feldgleichungen gegeben ist. Tats{\"a}chlich liefert J. Hadamard Theorie hyperbolischer Differentialgleichungen Bil{\"o}sungen mit festem singul{\"a}ren Anteil, so dass ein geeignetes Renormierungsverfahren definiert werden kann. Die sogenannte Hadamard-Bedingung an Bidistributionen steht f{\"u}r die Forderung einer solchen Singularit{\"a}tenstruktur und sie hat sich etabliert als nat{\"u}rliche Verallgemeinerung der f{\"u}r flache Raumzeiten formulierten Spektralbedingung. Seit Radzikowskis wegweisenden Resultaten l{\"a}sst sie sich außerdem lokal ausdr{\"u}cken, n{\"a}mlich als eine Bedingung an die Wellenfrontenmenge der Bil{\"o}sung. Diese Formulierung schl{\"a}gt eine Br{\"u}cke zu der von Duistermaat und H{\"o}rmander entwickelten mikrolokalen Analysis, die seitdem bei der {\"U}berpr{\"u}fung der Hadamard-Bedingung sowie der Konstruktion von Hadamard Zust{\"a}nden vielfach Verwendung findet und rasante Fortschritte auf diesem Gebiet ausgel{\"o}st hat. Obwohl unverzichtbar f{\"u}r die Analyse der Charakteristiken von Operatoren und ihrer Parametrizen sind die Methoden und Aussagen der mikrolokalen Analysis ungeeignet f{\"u}r die Analyse von nicht-singul{\"a}ren Strukturen und zentrale Aussagen sind typischerweise bis auf glatte Anteile formuliert. Beispielsweise lassen sich aus Radzikowskis Resultaten nahezu direkt Existenzaussagen und sogar ein konkretes Konstruktionsschema f{\"u}r Hadamard Zust{\"a}nde ableiten, die {\"u}brigen Eigenschaften (Bil{\"o}sung, Kausalit{\"a}t, Positivit{\"a}t) k{\"o}nnen jedoch auf diesem Wege nur modulo glatte Funktionen gezeigt werden. Es ist das Ziel dieser Dissertation, diesen Ansatz f{\"u}r lineare Wellenoperatoren auf Schnitten in Vektorb{\"u}ndeln {\"u}ber global-hyperbolischen Lorentz-Mannigfaltigkeiten zu vollenden und, ausgehend von einer lokalen Hadamard Reihe, Hadamard Zust{\"a}nde zu konstruieren. Beruhend auf Wightmans L{\"o}sung f{\"u}r die d'Alembert-Gleichung auf dem Minkowski-Raum und der Herleitung der avancierten und retardierten Fundamentall{\"o}sung konstruieren wir lokal Parametrizen in Form von Hadamard-Reihen und f{\"u}gen sie zu globalen Bil{\"o}sungen zusammen. Diese besitzen dann die Hadamard-Eigenschaft und wir zeigen anschließend, dass glatte Bischnitte existieren, die addiert werden k{\"o}nnen, so dass die verbleibenden Bedingungen erf{\"u}llt sind.}, language = {en} } @misc{Wallenta2014, author = {Wallenta, Daniel}, title = {A Lefschetz fixed point formula for elliptic quasicomplexes}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, number = {885}, issn = {1866-8372}, doi = {10.25932/publishup-43547}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-435471}, pages = {577 -- 587}, year = {2014}, abstract = {In a recent paper, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.}, language = {en} } @phdthesis{Zadorozhnyi2021, author = {Zadorozhnyi, Oleksandr}, title = {Contributions to the theoretical analysis of the algorithms with adversarial and dependent data}, school = {Universit{\"a}t Potsdam}, pages = {144}, year = {2021}, abstract = {In this work I present the concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under a general functional weak-dependency assumption (the so-called \$\cC-\$mixing). The latter is then used to prove, in the asymptotic framework, excess risk upper bounds of the regularised Hilbert valued statistical learning rules under the τ-mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here, in particular, a question of robustness of the kernel-based forecaster is investigated. Afterwards, in the framework of sequential learning, the multi-armed bandit problem under \$\cC-\$mixing assumption on the arm's outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly, probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma-Hoeffding's and of Burkholder's type) to the partial sums of real-valued weakly dependent random fields (under the type of projective dependence condition).}, language = {en} } @article{MariucciRaySzabo2020, author = {Mariucci, Ester and Ray, Kolyan and Szabo, Botond}, title = {A Bayesian nonparametric approach to log-concave density estimation}, series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, volume = {26}, journal = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, number = {2}, publisher = {International Statistical Institute}, address = {The Hague}, issn = {1350-7265}, doi = {10.3150/19-BEJ1139}, pages = {1070 -- 1097}, year = {2020}, abstract = {The estimation of a log-concave density on R is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.}, language = {en} } @article{Steup2015, author = {Steup, Martin}, title = {Raum und Zahl in der Pflanzenphysiologie}, series = {Raum und Zahl}, journal = {Raum und Zahl}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {77 -- 109}, year = {2015}, language = {de} } @unpublished{FlandoliHoegele2014, author = {Flandoli, Franco and H{\"o}gele, Michael}, title = {A solution selection problem with small stable perturbations}, volume = {3}, number = {8}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71205}, pages = {43}, year = {2014}, abstract = {The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.}, language = {en} } @article{KellerPinchoverPogorzelski2019, author = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix}, title = {Criticality theory for Schr{\"o}dinger operators on graphs}, series = {Journal of spectral theory}, volume = {10}, journal = {Journal of spectral theory}, number = {1}, publisher = {European Mathematical Society}, address = {Z{\"u}rich}, issn = {1664-039X}, doi = {10.4171/JST/286}, pages = {73 -- 114}, year = {2019}, abstract = {We study Schrodinger operators given by positive quadratic forms on infinite graphs. From there, we develop a criticality theory for Schrodinger operators on general weighted graphs.}, language = {en} } @article{KellerMuench2019, author = {Keller, Matthias and M{\"u}nch, Florentin}, title = {A new discrete Hopf-Rinow theorem}, series = {Discrete Mathematics}, volume = {342}, journal = {Discrete Mathematics}, number = {9}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0012-365X}, doi = {10.1016/j.disc.2019.03.014}, pages = {2751 -- 2757}, year = {2019}, abstract = {We prove a version of the Hopf-Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf-Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.}, language = {en} } @article{KellerLenzSchmidtetal.2019, author = {Keller, Matthias and Lenz, Daniel and Schmidt, Marcel and Schwarz, Michael}, title = {Boundary representation of Dirichlet forms on discrete spaces}, series = {Journal de Math{\´e}matiques Pures et Appliqu{\´e}es}, volume = {126}, journal = {Journal de Math{\´e}matiques Pures et Appliqu{\´e}es}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0021-7824}, doi = {10.1016/j.matpur.2018.10.005}, pages = {109 -- 143}, year = {2019}, abstract = {We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods. (C) 2018 Elsevier Masson SAS.}, language = {en} } @article{KellerPinchoverPogorzelski2018, author = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix}, title = {Optimal Hardy inequalities for Schrodinger operators on graphs}, series = {Communications in mathematical physics}, volume = {358}, journal = {Communications in mathematical physics}, number = {2}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-018-3107-y}, pages = {767 -- 790}, year = {2018}, abstract = {For a given subcritical discrete Schrodinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H - lambda w is subcritical in X for all lambda < 1, null-critical in X for lambda = 1, and supercritical near any neighborhood of infinity in X for any lambda > 1. Our results rely on a criticality theory for Schrodinger operators on general weighted graphs.}, language = {en} } @article{KellerPinchoverPogorzelski2018, author = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix}, title = {An improved discrete hardy inequality}, series = {The American mathematical monthly : an official publication of the Mathematical Association of America}, volume = {125}, journal = {The American mathematical monthly : an official publication of the Mathematical Association of America}, number = {4}, publisher = {Taylor \& Francis Group}, address = {Philadelphia}, issn = {0002-9890}, doi = {10.1080/00029890.2018.1420995}, pages = {347 -- 350}, year = {2018}, abstract = {In this note, we prove an improvement of the classical discrete Hardy inequality. Our improved Hardy-type inequality holds with a weight w which is strictly greater than the classical Hardy weight w(H)(n) : 1/(2n)(2), where N.}, language = {en} } @article{SchoppaSiegVogeletal.2020, author = {Schoppa, Lukas and Sieg, Tobias and Vogel, Kristin and Z{\"o}ller, Gert and Kreibich, Heidi}, title = {Probabilistic flood loss models for companies}, series = {Water resources research}, volume = {56}, journal = {Water resources research}, number = {9}, publisher = {American Geophysical Union}, address = {Washington}, issn = {0043-1397}, doi = {10.1029/2020WR027649}, pages = {19}, year = {2020}, abstract = {Flood loss modeling is a central component of flood risk analysis. Conventionally, this involves univariable and deterministic stage-damage functions. Recent advancements in the field promote the use of multivariable and probabilistic loss models, which consider variables beyond inundation depth and account for prediction uncertainty. Although companies contribute significantly to total loss figures, novel modeling approaches for companies are lacking. Scarce data and the heterogeneity among companies impede the development of company flood loss models. We present three multivariable flood loss models for companies from the manufacturing, commercial, financial, and service sector that intrinsically quantify prediction uncertainty. Based on object-level loss data (n = 1,306), we comparatively evaluate the predictive capacity of Bayesian networks, Bayesian regression, and random forest in relation to deterministic and probabilistic stage-damage functions, serving as benchmarks. The company loss data stem from four postevent surveys in Germany between 2002 and 2013 and include information on flood intensity, company characteristics, emergency response, private precaution, and resulting loss to building, equipment, and goods and stock. We find that the multivariable probabilistic models successfully identify and reproduce essential relationships of flood damage processes in the data. The assessment of model skill focuses on the precision of the probabilistic predictions and reveals that the candidate models outperform the stage-damage functions, while differences among the proposed models are negligible. Although the combination of multivariable and probabilistic loss estimation improves predictive accuracy over the entire data set, wide predictive distributions stress the necessity for the quantification of uncertainty.}, language = {en} } @article{MickelssonPaycha2010, author = {Mickelsson, Jouko and Paycha, Sylvie}, title = {The logarithmic residue density of a generalized Laplacian}, series = {Journal of the Australian Mathematical Society}, volume = {90}, journal = {Journal of the Australian Mathematical Society}, number = {1}, publisher = {Cambridge Univ. Press}, address = {Cambridge}, issn = {0263-6115}, doi = {10.1017/S144678871100108X}, pages = {53 -- 80}, year = {2010}, abstract = {We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbell-Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.}, language = {en} } @incollection{Kortenkamp2017, author = {Kortenkamp, Ulrich}, title = {Raum und Form}, series = {Basiswissen Lehrerbildung: Mathematik unterrichten}, booktitle = {Basiswissen Lehrerbildung: Mathematik unterrichten}, editor = {Abshagen, Maike and Barzel, B{\"a}rbel and Kramer, J{\"u}rg and Riecke-Baulecke, Thomas and R{\"o}sken-Winter, Bettina and Selter, Christoph}, publisher = {Kallmeyer}, address = {Seelze}, publisher = {Universit{\"a}t Potsdam}, pages = {99 -- 112}, year = {2017}, language = {de} } @incollection{KiyLuckeSass2015, author = {Kiy, Alexander and Lucke, Ulrike and Sass, K.}, title = {Gewusst was: Mit einer E-Learning-Toolbox die pers{\"o}nliche virtuelle Umgebung gestalten}, series = {DeLFI 2015 - die 13. E-Learning Fachtagung Informatik der Gesellschaft f{\"u}r Informatik e.V. : 1.-4. September 2015 M{\"u}nchen, Deutschland}, booktitle = {DeLFI 2015 - die 13. E-Learning Fachtagung Informatik der Gesellschaft f{\"u}r Informatik e.V. : 1.-4. September 2015 M{\"u}nchen, Deutschland}, number = {247}, editor = {Pongratz, Hans J. and Keil, R.}, publisher = {Gesellschaft f{\"u}r Informatik e.V.}, address = {Bonn}, isbn = {978-3-88579-641-1}, issn = {1617-5468}, publisher = {Universit{\"a}t Potsdam}, pages = {43 -- 55}, year = {2015}, language = {de} } @misc{Dahl2023, type = {Master Thesis}, author = {Dahl, Dorothee Sophie}, title = {Zahlen in den Fingern}, doi = {10.25932/publishup-60762}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-607629}, school = {Universit{\"a}t Potsdam}, pages = {118}, year = {2023}, abstract = {Die Debatte {\"u}ber den Einsatz von digitalen Werkzeugen in der mathematischen Fr{\"u}hf{\"o}rderung ist hoch aktuell. Lernspiele werden konstruiert, mit dem Ziel, mathematisches, informelles Wissen aufzubauen und so einen besseren Schulstart zu erm{\"o}glichen. Doch allein die digitale und spielerische Aufarbeitung f{\"u}hrt nicht zwingend zu einem Lernerfolg. Daher ist es umso wichtiger, die konkrete Implementation der theoretischen Konstrukte und Interaktionsm{\"o}glichkeiten mit den Werkzeugen zu analysieren und passend aufzubereiten. In dieser Masterarbeit wird dazu exemplarisch ein mathematisches Lernspiel namens „Fingu" f{\"u}r den Einsatz im vorschulischen Bereich theoretisch und empirisch im Rahmen der Artifact-Centric Activity Theory (ACAT) untersucht. Dazu werden zun{\"a}chst die theoretischen Hintergr{\"u}nde zum Zahlensinn, Zahlbegriffserwerb, Teil-Ganze-Verst{\"a}ndnis, der Anzahlwahrnehmung und -bestimmung, den Anzahlvergleichen und der Anzahldarstellung mithilfe von Fingern gem{\"a}ß der Embodied Cognition sowie der Verwendung von digitalen Werkzeugen und Multi-Touch-Ger{\"a}ten umfassend beschrieben. Anschließend wird die App Fingu erkl{\"a}rt und dann theoretisch entlang des ACAT-Review-Guides analysiert. Zuletzt wird die selbstst{\"a}ndig durchgef{\"u}hrte Studie mit zehn Vorschulkindern erl{\"a}utert und darauf aufbauend Verbesserungs- und Entwicklungsm{\"o}glichkeiten der App auf wissenschaftlicher Grundlage beigetragen. F{\"u}r Fingu l{\"a}sst sich abschließend festhalten, dass viele Prozesse wie die (Quasi-)Simultanerfassung oder das Z{\"a}hlen gef{\"o}rdert werden k{\"o}nnen, f{\"u}r andere wie das Teil-Ganze-Verst{\"a}ndnis aber noch Anpassungen und/oder die Begleitung durch Erwachsene n{\"o}tig ist.}, language = {de} } @article{Clavier2021, author = {Clavier, Pierre J.}, title = {Borel-{\´E}calle resummation of a two-point function}, series = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, volume = {22}, journal = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, number = {6}, publisher = {Springer}, address = {Cham}, issn = {1424-0637}, doi = {10.1007/s00023-021-01057-w}, pages = {2103 -- 2136}, year = {2021}, abstract = {We provide an overview of the tools and techniques of resurgence theory used in the Borel-ecalle resummation method, which we then apply to the massless Wess-Zumino model. Starting from already known results on the anomalous dimension of the Wess-Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and that its Borel transform is resurgent. The Schwinger-Dyson equation of the model is then used to prove an asymptotic exponential bound for the Borel transformed two-point function on a star-shaped domain of a suitable ramified complex plane. This proves that the two-point function of the Wess-Zumino model is Borel-ecalle summable.}, language = {en} } @phdthesis{Schanner2022, author = {Schanner, Maximilian Arthus}, title = {Correlation based modeling of the archeomagnetic field}, doi = {10.25932/publishup-55587}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-555875}, school = {Universit{\"a}t Potsdam}, pages = {vii, 146}, year = {2022}, abstract = {The geomagnetic main field is vital for live on Earth, as it shields our habitat against the solar wind and cosmic rays. It is generated by the geodynamo in the Earth's outer core and has a rich dynamic on various timescales. Global models of the field are used to study the interaction of the field and incoming charged particles, but also to infer core dynamics and to feed numerical simulations of the geodynamo. Modern satellite missions, such as the SWARM or the CHAMP mission, support high resolution reconstructions of the global field. From the 19 th century on, a global network of magnetic observatories has been established. It is growing ever since and global models can be constructed from the data it provides. Geomagnetic field models that extend further back in time rely on indirect observations of the field, i.e. thermoremanent records such as burnt clay or volcanic rocks and sediment records from lakes and seas. These indirect records come with (partially very large) uncertainties, introduced by the complex measurement methods and the dating procedure. Focusing on thermoremanent records only, the aim of this thesis is the development of a new modeling strategy for the global geomagnetic field during the Holocene, which takes the uncertainties into account and produces realistic estimates of the reliability of the model. This aim is approached by first considering snapshot models, in order to address the irregular spatial distribution of the records and the non-linear relation of the indirect observations to the field itself. In a Bayesian setting, a modeling algorithm based on Gaussian process regression is developed and applied to binned data. The modeling algorithm is then extended to the temporal domain and expanded to incorporate dating uncertainties. Finally, the algorithm is sequentialized to deal with numerical challenges arising from the size of the Holocene dataset. The central result of this thesis, including all of the aspects mentioned, is a new global geomagnetic field model. It covers the whole Holocene, back until 12000 BCE, and we call it ArchKalmag14k. When considering the uncertainties that are produced together with the model, it is evident that before 6000 BCE the thermoremanent database is not sufficient to support global models. For times more recent, ArchKalmag14k can be used to analyze features of the field under consideration of posterior uncertainties. The algorithm for generating ArchKalmag14k can be applied to different datasets and is provided to the community as an open source python package.}, 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} } @misc{Fabian2020, type = {Master Thesis}, author = {Fabian, Melina}, title = {Grundvorstellungen bei Zahlbereichserweiterungen}, doi = {10.25932/publishup-56593}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-565930}, school = {Universit{\"a}t Potsdam}, pages = {95}, year = {2020}, abstract = {Die Erweiterung des nat{\"u}rlichen Zahlbereichs um die positiven Bruchzahlen und die negativen ganzen Zahlen geht f{\"u}r Sch{\"u}lerinnen und Sch{\"u}ler mit großen gedanklichen H{\"u}rden und einem Umbruch bis dahin aufgebauter Grundvorstellungen einher. Diese Masterarbeit tr{\"a}gt wesentliche Ver{\"a}nderungen auf der Vorstellungs- und Darstellungsebene f{\"u}r beide Zahlbereiche zusammen und setzt sich mit den kognitiven Herausforderungen f{\"u}r Lernende auseinander. Auf der Grundlage einer Diskussion traditioneller sowie alternativer Lehrg{\"a}nge der Zahlbereichserweiterung wird eine Unterrichtskonzeption f{\"u}r den Mathematikunterricht entwickelt, die eine parallele Einf{\"u}hrung der Bruchzahlen und der negativen Zahlen vorschl{\"a}gt. Die Empfehlungen der Unterrichtkonzeption erstrecken sich {\"u}ber den Zeitraum von der ersten bis zur siebten Klassenstufe, was der behutsamen Weiterentwicklung und Modifikation des Zahlbegriffs viel Zeit einr{\"a}umt, und enthalten auch didaktische {\"U}berlegungen sowie konkrete Hinweise zu m{\"o}glichen Aufgabenformaten.}, language = {de} } @book{OPUS4-41720, title = {Mathematik mit digitalen Medien - konkret}, series = {Lernen, Lehren und Forschen mit digitalen Medien in der Primarstufe ; 4}, journal = {Lernen, Lehren und Forschen mit digitalen Medien in der Primarstufe ; 4}, editor = {Ladel, Silke and Kortenkamp, Ulrich and Etzold, Heiko}, publisher = {WTM-Verlag}, address = {M{\"u}nster}, isbn = {978-3-95987-078-8}, publisher = {Universit{\"a}t Potsdam}, year = {2018}, abstract = {Neue Medien" war {\"u}ber viele Jahre hinweg das Codewort f{\"u}r Computer, die den Einzug in den Schulunterricht schaffen sollten - wenn es nach den Bef{\"u}rwortern ging. Die Widerst{\"a}nde, gerade in der Grundschule, waren groß und vielf{\"a}ltig. Es ist verst{\"a}ndlich, dass kurz nach der spielerischen Heranf{\"u}hrung an Bildung im Kindergarten, in einer Zeit, in der die Sch{\"u}lerinnen und Sch{\"u}ler auch das soziale Miteinander ein{\"u}ben m{\"u}ssen und auch fein- und grobmotorische F{\"a}higkeiten erwerben sollen, das vereinzelnde Sitzen vor einem Bildschirm nicht zu den obersten Priorit{\"a}ten geh{\"o}rt - und auch unserer Meinung nach nicht geh{\"o}ren sollte. In den letzten Jahren hat sich der Begriff der neuen Medien aber ver{\"a}ndert, und das, was bisher damit verbunden wurde, ist mit der „Digitalisierung" nicht nur des Schulunterrichts, sondern des ganzen Lebens, zu einem Dreh- und Angelpunkt der Bildung geworden. Statt klobigen Computern mit Bildschirmen, die das Miteinander schon {\"u}ber die Ausstattung der Computerr{\"a}ume in die falsche Bahn lenken, haben mobile Ger{\"a}te in der Hand der Sch{\"u}lerinnen und Sch{\"u}ler {\"u}bernommen. Diese k{\"o}nnen nun gemeinsam an einem Ger{\"a}t arbeiten, sie k{\"o}nnen direkt mit den Bildschirminhalten interagieren, sie k{\"o}nnen die Kameras, Mikrophone und Sensoren nutzen, um authentische Daten zu erfassen und zu verarbeiten, sie k{\"o}nnen auch außerhalb des Klassenraums oder der Schule damit arbeiten und haben inzwischen fast jederzeit das ganze Wissen des Internets mit dabei. Schwerpunkt dieses Bandes ist daher der Umgang mit Tablets und den darauf laufenden „Apps" im Mathematikunterricht. In f{\"u}nf Beitr{\"a}gen werden konkrete Unterrichtsvorschl{\"a}ge gemacht, die als Blaupausen f{\"u}r App-gest{\"u}tzten Unterricht dienen k{\"o}nnen. Erg{\"a}nzt wird dieser Band durch einen allgemeinen Leitfaden zur Beurteilung von Apps f{\"u}r den Mathematikunterricht samt Beispielen.}, language = {de} } @phdthesis{Hannes2022, author = {Hannes, Sebastian}, title = {Boundary Value Problems for the Lorentzian Dirac Operator}, doi = {10.25932/publishup-54839}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-548391}, school = {Universit{\"a}t Potsdam}, pages = {67}, year = {2022}, abstract = {The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting. In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. B{\"a}r and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case. Back in the Riemannian setting, B{\"a}r and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement. This thesis aims at applying the boundary conditions found by B{\"a}r and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made. The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters. Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space. Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space. In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting.}, language = {en} } @masterthesis{Engelhardt2021, type = {Bachelor Thesis}, author = {Engelhardt, Max Angel Ronan}, title = {Zwischen Simulation und Beweis - eine mathematische Analyse des Bienaym{\´e}-Galton-Watson-Prozesses und sein Einsatz innerhalb des Mathematikunterrichts}, doi = {10.25932/publishup-52447}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-524474}, school = {Universit{\"a}t Potsdam}, pages = {117}, year = {2021}, abstract = {Die Bienaym{\´e}-Galton-Watson Prozesse k{\"o}nnen f{\"u}r die Untersuchung von speziellen und sich entwickelnden Populationen verwendet werden. Die Populationen umfassen Individuen, welche sich identisch, zuf{\"a}llig, selbstst{\"a}ndig und unabh{\"a}ngig voneinander fortpflanzen und die jeweils nur eine Generation existieren. Die n-te Generation ergibt sich als zuf{\"a}llige Summe der Individuen der (n-1)-ten Generation. Die Relevanz dieser Prozesse begr{\"u}ndet sich innerhalb der Historie und der inner- und außermathematischen Bedeutung. Die Geschichte der Bienaym{\´e}-Galton-Watson-Prozesse wird anhand der Entwicklung des Konzeptes bis heute dargestellt. Dabei werden die Wissenschaftler:innen verschiedener Disziplinen angef{\"u}hrt, die Erkenntnisse zu dem Themengebiet beigetragen und das Konzept in ihren Fachbereichen angef{\"u}hrt haben. Somit ergibt sich die außermathematische Signifikanz. Des Weiteren erh{\"a}lt man die innermathematische Bedeutsamkeit mittels des Konzeptes der Verzweigungsprozesse, welches auf die Bienaym{\´e}-Galton-Watson Prozesse zur{\"u}ckzuf{\"u}hren ist. Die Verzweigungsprozesse stellen eines der aussagekr{\"a}ftigsten Modelle f{\"u}r die Beschreibung des Populationswachstums dar. Dar{\"u}ber hinaus besteht die derzeitige Wichtigkeit durch die Anwendungsm{\"o}glichkeit der Verzweigungsprozesse und der Bienaym{\´e}-Galton-Watson Prozesse innerhalb der Epidemiologie. Es werden die Ebola- und die Corona-Pandemie als Anwendungsfelder angef{\"u}hrt. Die Prozesse dienen als Entscheidungsst{\"u}tze f{\"u}r die Politik und erm{\"o}glichen Aussagen {\"u}ber die Auswirkungen von Maßnahmen bez{\"u}glich der Pandemien. Neben den Prozessen werden ebenfalls der bedingte Erwartungswert bez{\"u}glich diskreter Zufallsvariablen, die wahrscheinlichkeitserzeugende Funktion und die zuf{\"a}llige Summe eingef{\"u}hrt. Die Konzepte vereinfachen die Beschreibung der Prozesse und bilden somit die Grundlage der Betrachtungen. Außerdem werden die ben{\"o}tigten und weiterf{\"u}hrenden Eigenschaften der grundlegenden Themengebiete und der Prozesse aufgef{\"u}hrt und bewiesen. Das Kapitel erreicht seinen H{\"o}hepunkt bei dem Beweis des Kritikalit{\"a}tstheorems, wodurch eine Aussage {\"u}ber das Aussterben des Prozesses in verschiedenen F{\"a}llen und somit {\"u}ber die Aussterbewahrscheinlichkeit get{\"a}tigt werden kann. Die F{\"a}lle werden anhand der zu erwartenden Anzahl an Nachkommen eines Individuums unterschieden. Es zeigt sich, dass ein Prozess bei einer zu erwartenden Anzahl kleiner gleich Eins mit Sicherheit ausstirbt und bei einer Anzahl gr{\"o}ßer als Eins, die Population nicht in jedem Fall aussterben muss. Danach werden einzelne Beispiele, wie der linear fractional case, die Population von Fibroblasten (Bindegewebszellen) von M{\"a}usen und die Entstehungsfragestellung der Prozesse, angef{\"u}hrt. Diese werden mithilfe der erlangten Ergebnisse untersucht und einige ausgew{\"a}hlte zuf{\"a}llige Dynamiken werden im nachfolgenden Kapitel simuliert. Die Simulationen erfolgen durch ein in Python erstelltes Programm und werden mithilfe der Inversionsmethode realisiert. Die Simulationen stellen beispielhaft die Entwicklungen in den verschiedenen Kritikalit{\"a}tsf{\"a}llen der Prozesse dar. Zudem werden die H{\"a}ufigkeiten der einzelnen Populationsgr{\"o}ßen in Form von Histogrammen angebracht. Dabei l{\"a}sst sich der Unterschied zwischen den einzelnen F{\"a}llen best{\"a}tigen und es wird die Anwendungsm{\"o}glichkeit der Bienaym{\´e}-Galton-Watson Prozesse bei komplexeren Problemen deutlich. Histogramme bekr{\"a}ftigen, dass die einzelnen Populationsgr{\"o}ßen nur endlich oft vorkommen. Diese Aussage wurde von Galton aufgeworfen und in der Extinktions-Explosions-Dichotomie verwendet. Die dargestellten Erkenntnisse {\"u}ber das Themengebiet und die Betrachtung des Konzeptes werden mit einer didaktischen Analyse abgeschlossen. Die Untersuchung beinhaltet die Ber{\"u}cksichtigung der Fundamentalen Ideen, der Fundamentalen Ideen der Stochastik und der Leitidee „Daten und Zufall". Dabei ergibt sich, dass in Abh{\"a}ngigkeit der gew{\"a}hlten Perspektive die Anwendung der Bienaym{\´e}-Galton-Watson Prozesse innerhalb der Schule plausibel ist und von Vorteil f{\"u}r die Sch{\"u}ler:innen sein kann. F{\"u}r die Behandlung wird exemplarisch der Rahmenlehrplan f{\"u}r Berlin und Brandenburg analysiert und mit dem Kernlehrplan Nordrhein-Westfalens verglichen. Die Konzeption des Lehrplans aus Berlin und Brandenburg l{\"a}sst nicht den Schluss zu, dass die Bienaym{\´e}-Galton-Watson Prozesse angewendet werden sollten. Es l{\"a}sst sich feststellen, dass die zugrunde liegende Leitidee nicht vollumf{\"a}nglich mit manchen Fundamentalen Ideen der Stochastik vereinbar ist. Somit w{\"u}rde eine Modifikation hinsichtlich einer st{\"a}rkeren Orientierung des Lehrplans an den Fundamentalen Ideen die Anwendung der Prozesse erm{\"o}glichen. Die Aussage wird durch die Betrachtung und {\"U}bertragung eines nordrhein-westf{\"a}lischen Unterrichtsentwurfes f{\"u}r stochastische Prozesse auf die Bienaym{\´e}-Galton-Watson Prozesse unterst{\"u}tzt. Dar{\"u}ber hinaus werden eine Concept Map und ein Vernetzungspentagraph nach von der Bank konzipiert um diesen Aspekt hervorzuheben.}, language = {de} } @misc{Piaskowski2021, type = {Master Thesis}, author = {Piaskowski, Birgit}, title = {Denkh{\"u}rden in den rationalen Zahlen}, doi = {10.25932/publishup-53277}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-532777}, school = {Universit{\"a}t Potsdam}, pages = {V, 141}, year = {2021}, abstract = {Das Professionswissen von Lehrkr{\"a}ften geh{\"o}rt zu den bedeutendsten Stellschrauben der Bildung an den Schulen. Seine Kernbereiche sind fachwissenschaftliches Wissen und fachdidaktisches Wissen, welche haupts{\"a}chlich in der universit{\"a}ren Ausbildung erworben werden. Die vorliegende Arbeit verfolgt das Ziel, einen Beitrag zur stetigen Verbesserung und Sicherung der Qualit{\"a}t der Lehrerausbildung an der Universit{\"a}t Potsdam zu leisten, und stellt die Frage: {\"U}ber welches fachwissenschaftliche und fachdidaktische Wissen verf{\"u}gen die Lehramtsstudierenden im Fach Mathematik nach Besuch der Lehrveranstaltung Arithmetik und ihre Didaktik I und II? Untersucht wurde exemplarisch das Wissen der Lehramtsstudierenden im Bereich der rationalen Zahlen mit dem Fokus auf dem Verst{\"a}ndnis der Dichte von Bruchzahlen. Die Dichte stellt eines der am schwierigsten zu erwerbenden Konzepte im Bruchzahlerwerb dar und fordert ein konzeptionelles Umdenken sowie die Reorganisation bereits erworbener Vorstellungen. Um die Forschungsfrage zu beantworten, wurden in einer qualitativen Studie 112 Lehramtsstudierende hinsichtlich ihres Wissens zu dem Thema Dichte von rationalen Zahlen schriftlich getestet. Um Denkprozesse der Studierenden zu verstehen und Denkh{\"u}rden zu identifizieren, wurden zus{\"a}tzlich qualitative Interviews in Form von Gruppendiskussionen gef{\"u}hrt. Die Daten wurden mithilfe der Qualitativen Inhaltsanalyse computergest{\"u}tzt ausgewertet. Es zeigte sich eine große Bandbreite verschiedener Wissensbest{\"a}nde. Die Ergebnisse im fachdidaktischen Wissen blieben hinter den Ergebnissen im fachwissenschaftlichen Wissen zur{\"u}ck. Am schwierigsten fiel den Studierenden die Gegen{\"u}berstellung von wesentlichen Eigenschaften der rationalen und nat{\"u}rlichen Zahlen auf der metakognitiven Ebene. Neben positiven Ergebnissen, welche f{\"u}r die Effektivit{\"a}t der Konzeption der Lehrveranstaltung sprechen, zeigten sich diverse Denkh{\"u}rden. Defizite im Fachwissen wie ein mangelndes Verst{\"a}ndnis von {\"a}quivalenten Br{\"u}chen oder Fehler im Erweitern von Br{\"u}chen enth{\"u}llen unzul{\"a}nglich ausgebildete Grundvorstellungen im Bereich der rationalen Zahlen seitens der Studierenden. Schwierigkeiten in den fachdidaktischen Aufgaben wie die Formulierung einer kindgerechten Erkl{\"a}rung oder die anschauliche Darstellung des mathematischen Inhalts auf bildlicher Ebene lassen sich urs{\"a}chlich auf die Defizite im Fachwissen zur{\"u}ckf{\"u}hren. Zus{\"a}tzlich stellten sich Einschr{\"a}nkungen seitens der Studierenden in der Motivation und Relevanzzuschreibung heraus. Die Ergebnisse f{\"u}hren zu gezielten {\"A}nderungsvorschl{\"a}gen bez{\"u}glich der Konzeption der Lehrveranstaltung. Es wird empfohlen, verschiedene Lernangebote wie Hausaufgaben und w{\"o}chentliche Selbsttests zur individuellen Lernzielkontrolle f{\"u}r alle Teilnehmenden der Lehrveranstaltung verpflichtend zu gestalten und motivationale Aspekte verst{\"a}rkt aufzugreifen. Zus{\"a}tzlich wird der Ausbau von konkreten {\"U}bungen auf der enaktiven Ebene empfohlen, um den Aufbau von notwendigen Grundvorstellungen im Bereich der rationalen Zahlen zu f{\"o}rdern und somit Denkh{\"u}rden gezielt zu begegnen.}, language = {de} }