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We consider Walsh’s conformal map from the exterior of a compact set E ⊆ C onto a lemniscatic domain. If E is simply connected, the lemniscatic domain is the exterior of a circle, while if E has several components, the lemniscatic domain is the exterior of a generalized lemniscate and is determined by the logarithmic capacity of E and by the exponents and centers of the generalized lemniscate. For general E, we characterize the exponents in terms of the Green’s function of Ec. Under additional symmetry conditions on E, we also locate the centers of the lemniscatic domain. For polynomial pre-images E = P−1(Ω) of a simply-connected infinite compact set Ω, we explicitly determine the exponents in the lemniscatic domain and derive a set of equations to determine the centers of the lemniscatic domain. Finally, we present several examples where we explicitly obtain the exponents and centers of the lemniscatic domain, as well as the conformal map.
Given a manifold with a string structure, we construct a spinor bundle on its loop space. Our construction is in analogy with the usual construction of a spinor bundle on a spin manifold, but necessarily makes use of tools from infinite dimensional geometry. We equip this spinor bundle on loop space with an action of a bundle of Clifford algebras. Given two smooth loops in our string manifold that share a segment, we can construct a third loop by deleting this segment. If this third loop is smooth, then we say that the original pair of loops is a pair of compatible loops. It is well-known that this operation of fusing compatible loops is important if one wants to understand the geometry of a manifold through its loop space. In this work, we explain in detail how the spinor bundle on loop space behaves with respect to fusion of compatible loops. To wit, we construct a family of fusion isomorphisms indexed by pairs of compatible loops in our string manifold. Each of these fusion isomorphisms is an isomorphism from the relative tensor product of the fibres of the spinor bundle over its index pair of compatible loops to the fibre over the loop that is the result of fusing the index pair. The construction of a spinor bundle on loop space equipped with a fusion product as above was proposed by Stolz and Teichner with the goal of studying the Dirac operator on loop space". Our construction combines facets of the theory of bimodules for von Neumann algebras, infinite dimensional manifolds, and Lie groups and their representations. We moreover place our spinor bundle on loop space in the context of bundle gerbes and bundle gerbe modules.
As the tree of life is populated with sequenced genomes ever more densely, the new challenge is the accurate and consistent annotation of entire clades of genomes. In my dissertation, I address this problem with a new approach to comparative gene finding that takes a multiple genome alignment of closely related species and simultaneously predicts the location and structure of protein-coding genes in all input genomes, thereby exploiting negative selection and sequence conservation. The model prefers potential gene structures in the different genomes that are in agreement with each other, or—if not—where the exon gains and losses are plausible given the species tree. The multi-species gene finding problem is formulated as a binary labeling problem on a graph. The resulting optimization problem is NP hard, but can be efficiently approximated using a subgradient-based dual decomposition approach.
I tested the novel approach on whole-genome alignments of 12 vertebrate and 12 Drosophila species. The accuracy was evaluated for human, mouse and Drosophila melanogaster and compared to competing methods. Results suggest that the new method is well-suited for annotation of a large number of genomes of closely related species within a clade, in particular, when RNA-Seq data are available for many of the genomes. The transfer of existing annotations from one genome to another via the genome alignment is more accurate than previous approaches that are based on protein-spliced alignments, when the genomes are at close to medium distances. The method is implemented in C++ as part of the gene finder AUGUSTUS.
Abstract
The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified for many well‐known processes up to a deterministic time, it was not known whether the required bound holds for random time. In fact, even the simplest case of Brownian motion up to the exit time of a planar disc was open. For this particular case we answer this question using a suitable hyperbolic projection of the expected signature. The projection satisfies a three‐dimensional system of linear PDEs, which (surprisingly) can be solved explicitly, and which allows us to show that the upper bound on the expected signature is not satisfied.
Universal products provide an axiomatic framework to study noncommutative independences general enough to include, besides the well known "single-faced" case (i.e., tensor, free, Boolean, monotone and antimonotone independence), also more recent "multi-faced" examples like bifree independence. Questions concerning classification have been fully answered in the single-faced case, but are in general still open in the multi-faced case. In this thesis we discuss how one can use insights in the relation between universal products and their associated moment-cumulant formula as a starting point towards a combinatorial approach to (multi-faced) universal products. We define certain classes of partitions and discuss why the defining axioms are sufficient to associate to each of them a multi-faced universal product. For the two-faced case we present our result that every positive and symmetric universal product can be produced in this fashion and we outline how these results might contribute to a classification of positive and symmetric universal products.
The study of sow reproduction traits is important in livestock science and production to increase animal survival and economic efficiency. This work deals with the detection of different effects on within-litter variance of birth weight by applying different statistical models with different distributional assumptions. The piglets within one litter were separated by sex. The trait of sow was formed from the sample variances of birth weights within litter separated by sex to consider the sex effect on mean birth weight. A linear mixed model (LMM) approach was fitted to the logarithmized sample variance and the sample standard deviation. A generalized linear mixed model with gamma distributed residuals and log-link function was applied to the untransformed sample variance. Appropriate weights were constructed to account for individual litter sizes. Models were compared by analysing data from Landrace and Large White. The estimates of heritability for the different traits ranged from 6-14%. The LMM for the weighted standard deviation of birth weights was identified as most suitable in terms of residual normality. Furthermore, the impact of piglets´ sex on birth weight variability was tested, but it was only proved for one practical dataset. Additionally, we analysed the influence of including or not including birth weights of stillborn piglets on the estimates of variance components of birth weight variability. With omitted stillborns the estimates of heritability resulted in about 2% higher values than in investigations of total born piglets. We were interested in the presence of the random boar effect on birth weight variability. The corresponding variance component was tested via restricted likelihood ratio test. Among others, the null distribution of the test statistic was approximated by parametric bootstrap simulations which were computational intensive. We picked up a two-parametric approach from literature and proposed a three-parametric approach to approximate the null distribution of the test statistic. We have analysed correlated data in balanced (simulated data) and unbalanced (empirical data) designs. The two-parametric approach using a scaled mixture of chisquare-distributions as well as a three-parametric approach, that uses a mixture of the point mass at zero and a gamma distribution, behaved most solid in all investigations and were most powerful in the simulation study.
Statistical Methods and Applications for Biomarker Discovery Using Large Scale Omics Data Set
(2023)
This thesis focuses on identifying genetic factors associated with human kidney disease progression, with three articles presented. Article I describes the identification of loci associated with UACR through trans-ethnic, European-ancestry-specific, and diabetes-specific meta-analyses. An approximate conditional analysis was performed to identify additional independent UACR-associated variants within identified loci. The genome-wide significance level of 𝛼=5×10−8 is used for both primary GWAS association and conditional analyses. However, unlike primary association tests, conditional tests are limited to specific genomic regions surrounding primary GWAS index signals rather than being applied on a genome-wide scale.
In article II, we hypothesized that the application of 𝛼=5×10−8 is overly strict and results in a loss of power. To address this issue, we developed a quasi-adaptive method within a weighted hypothesis testing framework. This method exploits the type I error (𝛼=0.05) by providing less conservative SNP specific 𝛼-thresholds to select secondary signals in conditional analysis. Through simulation studies and power analyses, we demonstrate that the quasi-adaptive method outperforms the established criterion 𝛼=5×10−8 as well as the equal weighting scheme (the Sidak-correction). Furthermore, our method performs well when applied to real datasets and can potentially reveal previously undetected secondary signals in existing data.
In article III, we extended our quasi-adaptive method to identify plausible multiple independent signals at each locus (a secondary signal, a tertiary signal, a signal of 4th, and beyond) and applied it to the publically available GWAS meta-analysis to detect additional multiple independent eGFR-associated signals. The improved quasi-adaptive method successfully identified additional novel replicated independent SNPs that would have gone undetected by applying too conservative genome-wide significance level of 𝛼=5× 10−8. Colocalization analysis based on the novel independent signals identified potentially functional genes across the kidney and other tissues.
Overall, these articles contribute to the understanding of genetic factors associated with human kidney disease progression and provide novel methods for identifying secondary and multiple independent signals in conditional GWAS analyses.
Self-similar sets are a class of fractals which can be rigorously defined and treated by mathematical methods. Their theory has been developed in n-dimensional space, but we have just a few good examples of self-similar sets in three-dimensional space. This thesis has two different aims. First, to extend fractal constructions from two-dimensional space to three-dimensional space. Second, to study some of the properties of these fractals such as finite type, disk-likeness, ball-likeness, and the Hausdorff dimension of boundaries. We will use the neighbor graph tool for creating new fractals, and studying their properties.