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In the model of randomly perturbed graphs we consider the union of a deterministic graph G with minimum degree αn and the binomial random graph G(n, p). This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Pósa and Korshunov on the threshold in G(n, p). In this note we extend this result in G ∪G(n, p) to sparser graphs with α = o(1). More precisely, for any ε > 0 and α: N ↦→ (0, 1) we show that a.a.s. G ∪ G(n, β/n) is Hamiltonian, where β = −(6 + ε) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part G(n, p) is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into G(n, p).
For a class of Cannings models we prove Haldane’s formula, π(sN)∼2sNρ2, for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for sN∼N−b and 0<b<1/2. Here, sN is the selective advantage of an individual carrying the beneficial type, and ρ2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.
[Nachruf] Wolfgang Schwarz
(2013)
The problem of unconstrained or constrained optimization occurs in many branches of mathematics and various fields of application. It is, however, an NP-hard problem in general. In this thesis, we examine an approximation approach based on the class of SAGE exponentials, which are nonnegative exponential sums. We examine this SAGE-cone, its geometry, and generalizations. The thesis consists of three main parts:
1. In the first part, we focus purely on the cone of sums of globally nonnegative exponential sums with at most one negative term, the SAGE-cone. We ex- amine the duality theory, extreme rays of the cone, and provide two efficient optimization approaches over the SAGE-cone and its dual.
2. In the second part, we introduce and study the so-called S-cone, which pro- vides a uniform framework for SAGE exponentials and SONC polynomials. In particular, we focus on second-order representations of the S-cone and its dual using extremality results from the first part.
3. In the third and last part of this thesis, we turn towards examining the con- ditional SAGE-cone. We develop a notion of sublinear circuits leading to new duality results and a partial characterization of extremality. In the case of poly- hedral constraint sets, this examination is simplified and allows us to classify sublinear circuits and extremality for some cases completely. For constraint sets with certain conditions such as sets with symmetries, conic, or polyhedral sets, various optimization and representation results from the unconstrained setting can be applied to the constrained case.
The aim of this bachelor thesis is to compare and empirically test the use of classification to improve the topic models Latent Dirichlet Allocation (LDA) and Author Topic Modeling
(ATM) in the context of the social media platform Twitter. For this purpose, a corpus was classified with the Dewey Decimal Classification (DDC) and then used to train the topic models. A second dataset, the unclassified corpus, was used for comparison. The assumption that the use of classification could improve the topic models did not prove true for the LDA topic model. Here, a sufficiently good improvement of the models could not be achieved. The ATM model, on the other hand, could be improved by using the classification. In general, the ATM model performed significantly better than the LDA model. In the context of the social media platform Twitter, it can thus be seen that the ATM model is superior to the LDA model and can additionally be improved by classifying the data.
We provide extensions of the dual variational method for the nonlinear Helmholtz equation from Evéquoz and Weth. In particular we prove the existence of dual ground state solutions in the Sobolev critical case, extend the dual method beyond the standard Stein Tomas and Kenig Ruiz Sogge range and generalize the method for sign changing nonlinearities.
We study continuous dually epi-translation invariant valuations on certain cones of convex functions containing the space of finite-valued convex functions. Using the homogeneous decomposition of this space, we associate a certain distribution to any homogeneous valuation similar to the Goodey-Weil embedding for translation invariant valuations on convex bodies. The support of these distributions induces a corresponding notion of support for the underlying valuations, which imposes certain restrictions on these functionals, and we study the relation between the support of a valuation and its domain. This gives a partial answer to the question which dually epi-translation invariant valuations on finite-valued convex functions can be extended to larger cones of convex functions.
We also study topological properties of spaces of valuations with support contained in a fixed compact set. As an application of these results, we introduce the class of smooth valuations on convex functions and show that the subspace of smooth dually epi-translation invariant valuations is dense in the space of continuous dually epi-translation invariant valuation on finite-valued convex functions. These smooth valuations are given by integrating certain smooth differential forms over the graph of the differential of a convex function. We use this construction to give a characterization of a dense subspace of all continuous valuations on finite-valued convex functions that are rotation invariant as well as dually epi-translation invariant.
Using results from Alesker's theory of smooth valuations on convex bodies, we also show that any smooth valuation can be written as a convergent sum of mixed Hessian valuations. In particular, mixed Hessian valuations span a dense subspace, which is a version of McMullen’s conjecture for valuations on convex functions.
In this paper we deal with an implementation as well as numerical experiments for the coupling of interior and exterior problems of the elastodynamic wave equation with transparent boundary conditions in 3D as described in a previous paper by this author. In more detail, the FEM‐BEM‐coupling as well as the time discretization by using leapfrog and convolution quadrature is considered. Our aim is to provide an insight into the necessary steps of the implementation. Based on this, we present numerical experiments for a non‐convex domain and analyze the errors.
We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.
Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.
The specific temporal evolution of bacterial and phage population sizes, in particular bacterial depletion and the emergence of a resistant bacterial population, can be seen as a kinetic fingerprint that depends on the manifold interactions of the specific phage–host pair during the course of infection. We have elaborated such a kinetic fingerprint for a human urinary tract Klebsiella pneumoniae isolate and its phage vB_KpnP_Lessing by a modeling approach based on data from in vitro co-culture. We found a faster depletion of the initially sensitive bacterial population than expected from simple mass action kinetics. A possible explanation for the rapid decline of the bacterial population is a synergistic interaction of phages which can be a favorable feature for phage therapies. In addition to this interaction characteristic, analysis of the kinetic fingerprint of this bacteria and phage combination revealed several relevant aspects of their population dynamics: A reduction of the bacterial concentration can be achieved only at high multiplicity of infection whereas bacterial extinction is hardly accomplished. Furthermore the binding affinity of the phage to bacteria is identified as one of the most crucial parameters for the reduction of the bacterial population size. Thus, kinetic fingerprinting can be used to infer phage–host interactions and to explore emergent dynamics which facilitates a rational design of phage therapies.
Wir konnten unseren eigenen Weg gehen, jeder von uns hatte am Ende ein anderes Ergebnis und es war keines falsch. Das macht für mich die Qualität beim Lernen aus, dass mir genug Platz für meine Gedanken gegeben wird und ich ernst genommen werde. […] Dieses Gefühl ist bis heute nicht verloren gegangen und der Gedanke, wie es sein könnte, hilft mir, aus mir raus zukommen und andere zu motivieren, das ebenfalls zu tun, um auch um mich herum anregende Gespräche zu führen, die an die während der Akademie geführten heranreichen. (Feedback einer Teilnehmerin der HSAKA-M 2018)
Bildung durch Wissenschaft im Sinne des Forschenden Lernens ist ein zentrales Thema schulischer Bildung und findet beispielsweise im Konzept Kultur.Forscher! eine didaktische, schulische Umsetzung und wird vom Wissenschaftsrat als Leitgedanke ebenfalls für Universitäten mit dem Ziel empfohlen, Studium und Lehre deutlicher an der Forschung auszurichten.
Die Erfahrung, "…dass alles auch ganz anders sein könnte" ist die wohl wichtigste Erfahrung in Bildungsprozessen. Die Entdeckung von Möglichkeiten, Perspektivwechseln und transformatorischen Selbst-Bildungsprozessen ist zentral für eine gelungene kulturelle Bildungssituation. (Birgit Mandel, 2005).
Die Hessischen Schülerakademien zur Förderung besonders engagierter und begabter junger Menschen wurden bewusst als ein Unterfangen des Forschenden Lernens gegründet und fühlen sich diesem Leitgedanken im Kontext kultureller Bildung verpflichtet. Dieser Satz klingt zunächst einmal gut und zeitgemäß. Doch was steckt genau dahinter?
Das Zusammentreffen zu Beginn der Sommerferien von 60 wissbegierigen und experimentierfreudigen Schülerinnen und Schülern mit einem ebensolchen Team aus Hochschullehrenden und Kulturschaffenden, versprach wie immer eine intensive und aufregende Zeit zu werden. Diese positive Erwartung wurde auch voll erfüllt und gipfelte am Gästenachmittag mit Eltern, Verwandten, Freunden und interessierten Besuchern in einen feierlich-fröhlichen Abschluss mit spannenden und auch überraschenden Werkschauen der Kurse. Ein besonderes Highlight war die großformatige Gestaltung eines Modells der BURG FÜRSTENECK als interdisziplinäres Ergebnis des Hauptkurses Mathematik und des Wahlkurses Modellbau.
Als wir im Herbst 2015 auf den Homepages von BURG FÜRSTENECK und der Schülerakademie unsere Ausschreibung für die Akademie 2016 veröffentlichten, ahnten wir noch nicht, dass wir uns weitere Werbung mit dem jährlichen Flyer, den wir zum Jahreswechsel an die hessischen Gymnasien und Gesamtschulen mit gymnasialen Zweig versenden, hätten (fast) sparen können. Zu unserer Überraschung und großer Freude zählten wir bereits im Februar 2016 "58" Anmeldungen von Schülerinnen und Schülern. Die Werbung hat uns im Anschluss über 20 weitere Bewerbungen beschert und in die unangenehme Situation gebracht, (zu) vielen Schülerinnen und Schülern absagen bzw. sie auf das nächste Jahr vertrösten zu müssen.