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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.