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Given two weighted graphs (X, b(k), m(k)), k = 1, 2 with b(1) similar to b(2) and m(1) similar to m(2), we prove a weighted L-1-criterion for the existence and completeness of the wave operators W-+/- (H-2, H-1, I-1,I-2), where H-k denotes the natural Laplacian in l(2)(X, m(k)) w.r.t. (X, b(k), m(k)) and I-1,I-2 the trivial identification of l(2)(X, m(1)) with l(2) (X, m(2)). In particular, this entails a general criterion for the absolutely continuous spectra of H-1 and H-2 to be equal.
We prove that if u is a locally Lipschitz continuous function on an open set chi subset of Rn + 1 satisfying the nonlinear heat equation partial derivative(t)u = Delta(vertical bar u vertical bar(p-1) u), p > 1, weakly away from the zero set u(-1) (0) in chi, then u is a weak solution to this equation in all of chi.
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 für den pädagogischen Kontext wenig klassifiziert und für Lehrende kaum zugänglich gemacht worden.
Daher zielt diese Bachelorarbeit darauf ab, eine systematische Strukturierung und Aufarbeitung von Gamification sowie innovative Ansätze für die Verwendung spieltypischer Elemente im Unterricht, konkret dem Mathematikunterricht, zu präsentieren. Dies kann eine Grundlage für andere Fachgebiete, aber auch andere Lehrformen bieten und so die Umsetzbarkeit von Gamification in eigenen Lehrveranstaltungen aufzeigen.
In der Arbeit wird begründet, weshalb und mithilfe welcher Elemente Gamification die Motivation und Leistungsbereitschaft der Lernenden langfristig erhöhen, die Sozial- und Personalkompetenzen fördern sowie die Lernenden zu mehr Aktivität anregen kann. Zudem wird Gamification explizit mit grundlegenden mathematikdidaktischen Prinzipien in Verbindung gesetzt und somit die Relevanz für den Mathematikunterricht hervorgehoben.
Anschließend werden die einzelnen Elemente von Gamification wie Punkte, Level, Abzeichen, Charaktere und Rahmengeschichte entlang einer eigens für den pädagogischen Kontext entwickelten Klassifikation „FUN“ (Feedback – User specific elements – Neutral elements) schematisch beschrieben, ihre Funktionen und Wirkung dargestellt sowie Einsatzmöglichkeiten im Unterricht aufgezeigt. Dies beinhaltet Ideen zu lernförderlichem Feedback, Differenzierungsmöglichkeiten und Unterrichtsrahmengestaltung, die in Lehrveranstaltungen aller Art umsetzbar sein können. Die Bachelorarbeit umfasst zudem ein spezifisches Beispiel, einen Unterrichtsentwurf einer gamifizierten Mathematikstunde inklusive des zugehörigen Arbeitsmaterials, anhand dessen die Verwendung von Gamification deutlich wird.
Gamification offeriert oftmals Vorteile gegenüber dem traditionellen Unterricht, muss jedoch wie jede Methode an den Inhalt und die Zielgruppe angepasst werden. Weiterführende Forschung könnte sich mit konkreten motivationalen Strukturen, personenspezifischen Unterschieden sowie mit mathematischen Inhalten wie dem Problemlösen oder dem Wechsel zwischen verschiedenen Darstellungen hinsichtlich gamifizierter Lehrformen beschäftigen.
In the last decades, there was a notable progress in solving the well-known Boolean satisfiability (Sat) problem, which can be witnessed by powerful Sat solvers. One of the reasons why these solvers are so fast are structural properties of instances that are utilized by the solver’s interna. This thesis deals with the well-studied structural property treewidth, which measures the closeness of an instance to being a tree. In fact, there are many problems parameterized by treewidth that are solvable in polynomial time in the instance size when parameterized by treewidth.
In this work, we study advanced treewidth-based methods and tools for problems in knowledge representation and reasoning (KR). Thereby, we provide means to establish precise runtime results (upper bounds) for canonical problems relevant to KR. Then, we present a new type of problem reduction, which we call decomposition-guided (DG) that
allows us to precisely monitor the treewidth when reducing from one problem to another problem. This new reduction type will be the basis for a long-open lower bound result for quantified Boolean formulas and allows us to design a new methodology for establishing runtime lower bounds for problems parameterized by treewidth.
Finally, despite these lower bounds, we provide an efficient implementation of algorithms that adhere to treewidth. Our approach finds suitable abstractions of instances, which are subsequently refined in a recursive fashion, and it uses Sat solvers for solving subproblems. It turns out that our resulting solver is quite competitive for two canonical counting problems related to Sat.
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.
In the present paper, we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs, we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference.
We analyze a general class of difference operators Hε=Tε+Vε on ℓ2((εZ)d), where Vε is a multi-well potential and ε is a small parameter. We derive full asymptotic expansions of the prefactor of the exponentially small eigenvalue splitting due to interactions between two “wells” (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both the case where there is a single minimal geodesic (with respect to the natural Finsler metric induced by the leading symbol h0(x,ξ) of Hε) connecting the two minima and the case where the minimal geodesics form an ℓ+1 dimensional manifold, ℓ≥1. These results on the tunneling problem are as sharp as the classical results for the Schrödinger operator in Helffer and Sjöstrand (Commun PDE 9:337–408, 1984). Technically, our approach is pseudo-differential and we adapt techniques from Helffer and Sjöstrand [Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique), Mémoires de la S.M.F., 2 series, tome 34, pp 1–113, 1988)] and Helffer and Parisse (Ann Inst Henri Poincaré 60(2):147–187, 1994) to our discrete setting.
Contributions to the theoretical analysis of the algorithms with adversarial and dependent data
(2021)
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).
One of the crucial components in seismic hazard analysis is the estimation of the maximum earthquake magnitude and associated uncertainty. In the present study, the uncertainty related to the maximum expected magnitude mu is determined in terms of confidence intervals for an imposed level of confidence. Previous work by Salamat et al. (Pure Appl Geophys 174:763-777, 2017) shows the divergence of the confidence interval of the maximum possible magnitude m(max) for high levels of confidence in six seismotectonic zones of Iran. In this work, the maximum expected earthquake magnitude mu is calculated in a predefined finite time interval and imposed level of confidence. For this, we use a conceptual model based on a doubly truncated Gutenberg-Richter law for magnitudes with constant b-value and calculate the posterior distribution of mu for the time interval T-f in future. We assume a stationary Poisson process in time and a Gutenberg-Richter relation for magnitudes. The upper bound of the magnitude confidence interval is calculated for different time intervals of 30, 50, and 100 years and imposed levels of confidence alpha = 0.5, 0.1, 0.05, and 0.01. The posterior distribution of waiting times T-f to the next earthquake with a given magnitude equal to 6.5, 7.0, and7.5 are calculated in each zone. In order to find the influence of declustering, we use the original and declustered version of the catalog. The earthquake catalog of the territory of Iran and surroundings are subdivided into six seismotectonic zones Alborz, Azerbaijan, Central Iran, Zagros, Kopet Dagh, and Makran. We assume the maximum possible magnitude m(max) = 8.5 and calculate the upper bound of the confidence interval of mu in each zone. The results indicate that for short time intervals equal to 30 and 50 years and imposed levels of confidence 1 - alpha = 0.95 and 0.90, the probability distribution of mu is around mu = 7.16-8.23 in all seismic zones.
We study the Volterra property of a class of anisotropic pseudo-differential operators on R x B for a manifold B with edge Y and time-variable t. This exposition belongs to a program for studying parabolicity in such a situation. In the present consideration we establish non-smoothing elements in a subalgebra with anisotropic operator-valued symbols of Mellin type with holomorphic symbols in the complex Mellin covariable from the cone theory, where the covariable t of t extends to symbolswith respect to t to the lower complex v half-plane. The resulting space ofVolterra operators enlarges an approach of Buchholz (Parabolische Pseudodifferentialoperatoren mit operatorwertigen Symbolen. Ph. D. thesis, Universitat Potsdam, 1996) by necessary elements to a new operator algebra containing Volterra parametrices under an appropriate condition of anisotropic ellipticity. Our approach avoids some difficulty in choosing Volterra quantizations in the edge case by generalizing specific achievements from the isotropic edge-calculus, obtained by Seiler (Pseudodifferential calculus on manifolds with non-compact edges, Ph. D. thesis, University of Potsdam, 1997), see also Gil et al. (in: Demuth et al (eds) Mathematical research, vol 100. Akademic Verlag, Berlin, pp 113-137, 1997; Osaka J Math 37: 221-260, 2000).
If (T-t) is a semigroup of Markov operators on an L-1-space that admits a nontrivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t -> infinity. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.
Im Zuge der Covid-19 Pandemie werden zwei Werte täglich diskutiert: Die zuletzt gemeldete Zahl der neu Infizierten und die sogenannte Reproduktionsrate. Sie gibt wieder, wie viele weitere Menschen ein an Corona erkranktes Individuum im Durchschnitt ansteckt. Für die Schätzung dieses Wertes gibt es viele Möglichkeiten - auch das Robert Koch-Institut gibt in seinem täglichen Situationsbericht stets zwei R-Werte an: Einen 4-Tage-R-Wert und einen weniger schwankenden 7-Tage-R-Wert. Diese Arbeit soll eine weitere Möglichkeit vorstellen, einige Aspekte der Pandemie zu modellieren und die Reproduktionsrate zu schätzen.
In der ersten Hälfte der Arbeit werden die mathematischen Grundlagen vorgestellt, die man für die Modellierung benötigt. Hierbei wird davon ausgegangen, dass der Leser bereits ein Basisverständnis von stochastischen Prozessen hat. Im Abschnitt Grundlagen werden Verzweigungsprozesse mit einigen Beispielen eingeführt und die Ergebnisse aus diesem Themengebiet, die für diese Arbeit wichtig sind, präsentiert. Dabei gehen wir zuerst auf einfache Verzweigungsprozesse ein und erweitern diese dann auf Verzweigungsprozesse mit mehreren Typen. Um die Notation zu erleichtern, beschränken wir uns auf zwei Typen. Das Prinzip lässt sich aber auf eine beliebige Anzahl von Typen erweitern.
Vor allem soll die Wichtigkeit des Parameters λ herausgestellt werden. Dieser Wert kann als durchschnittliche Zahl von Nachfahren eines Individuums interpretiert werden und bestimmt die Dynamik des Prozesses über einen längeren Zeitraum. In der Anwendung auf die Pandemie hat der Parameter λ die gleiche Rolle wie die Reproduktionsrate R.
In der zweiten Hälfte dieser Arbeit stellen wir eine Anwendung der Theorie über Multitype Verzweigungsprozesse vor. Professor Yanev und seine Mitarbeiter modellieren in ihrer Veröffentlichung Branching stochastic processes as models of Covid-19 epidemic development die Ausbreitung des Corona Virus' über einen Verzweigungsprozess mit zwei Typen. Wir werden dieses Modell diskutieren und Schätzer daraus ableiten: Ziel ist es, die Reproduktionsrate zu ermitteln. Außerdem analysieren wir die Möglichkeiten, die Dunkelziffer (die Zahl nicht gemeldeter Krankheitsfälle) zu schätzen. Wir wenden die Schätzer auf die Zahlen von Deutschland an und werten diese schließlich aus.
The size structure of autotroph communities - the relative abundance of small vs. large individuals - shapes the functioning of ecosystems. Whether common mechanisms underpin the size structure of unicellular and multicellular autotrophs is, however, unknown. Using a global data compilation, we show that individual body masses in tree and phytoplankton communities follow power-law distributions and that the average exponents of these individual size distributions (ISD) differ. Phytoplankton communities are characterized by an average ISD exponent consistent with three-quarter-power scaling of metabolism with body mass and equivalence in energy use among mass classes. Tree communities deviate from this pattern in a manner consistent with equivalence in energy use among diameter size classes. Our findings suggest that whilst universal metabolic constraints ultimately underlie the emergent size structure of autotroph communities, divergent aspects of body size (volumetric vs. linear dimensions) shape the ecological outcome of metabolic scaling in forest vs. pelagic ecosystems.
Let (M-i, g(i))(i is an element of N) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov-Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D-B on B. We give an explicit description of D-B and characterize the special case where D-B equals the Dirac operator on B.
We show that elliptic complexes of (pseudo) differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi, and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.
We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions and the goal is to infer these base distributions. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine three popular machine learning problems that arise in this general setting: multiclass classification with label noise, demixing of mixed membership models, and classification with partial labels. In each case, we give sufficient conditions for identifiability and present algorithms for the infinite and finite sample settings, with associated performance guarantees.
In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincare inequality.
Packungen aus Kreisscheiben
(2019)
Der englische Seefahrer Sir Walter Raleigh fragte sich einst, wie er in seinem Schiffsladeraum moeglichst viele Kanonenkugeln stapeln koennte. Johannes Kepler entwickelte daraufhin 1611 eine Vermutung ueber die optimale Anordnung der Kugeln. Diese Vermutung sollte sich als eine der haertesten mathematischen Nuesse der Geschichte erweisen. Selbst in der Ebene sind dichteste Packungen kongruenter Kreise eine Herausforderung. 1892 und 1910 veroeffentlichte Axel Thue (kritisierte) Beweise, dass die hexagonale Kreispackung optimal sei. Erst 1940 lieferte Laszlo Fejes Toth schliesslich einen wasserdichten Beweis fuer diese Tatsache. Eine Variante des Problems verlangt,
Packungen mit endlich vielen kongruenten Kugeln zu finden, die eine gewisse quadratische Energie minimieren: Diese spannende geometrische Aufgabe wurde 1967 von Toth gestellt. Sie ist auch heute noch nicht vollstaendig gelaest. In diesem Beitrag schlagen die Autorinnen eine originelle wahrscheinlichkeitstheoretische Methode vor, um in der Ebene Näherungen der Lösung zu konstruieren.
We provide explicit examples of positive and power-bounded operators on c(0) and l(infinity) which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if T is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of T .