Approximating multi-dimensional Hamiltonian flows by billiards

Consider a family of smooth potentials $V_{\epsilon}$, which, in the limit $\epsilon\to0$, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as $\epsilon\to0$ to the original billiard, and provide asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the $C^{r}$ norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials

Billiards with polynomial mixing rates

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic -- enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.Comment: 39pages, 11 figue

Convergence of invariant densities in the small-noise limit

This paper presents a systematic numerical study of the effects of noise on the invariant probability densities of dynamical systems with varying degrees of hyperbolicity. It is found that the rate of convergence of invariant densities in the small-noise limit is frequently governed by power laws. In addition, a simple heuristic is proposed and found to correctly predict the power law exponent in exponentially mixing systems. In systems which are not exponentially mixing, the heuristic provides only an upper bound on the power law exponent. As this numerical study requires the computation of invariant densities across more than 2 decades of noise amplitudes, it also provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction

How metal films de-wet substrates - identifying the kinetic pathways and energetic driving forces

We study how single-crystal chromium films of uniform thickness on W(110) substrates are converted to arrays of three-dimensional (3D) Cr islands during annealing. We use low-energy electron microscopy (LEEM) to directly observe a kinetic pathway that produces trenches that expose the wetting layer. Adjacent film steps move simultaneously uphill and downhill relative to the staircase of atomic steps on the substrate. This step motion thickens the film regions where steps advance. Where film steps retract, the film thins, eventually exposing the stable wetting layer. Since our analysis shows that thick Cr films have a lattice constant close to bulk Cr, we propose that surface and interface stress provide a possible driving force for the observed morphological instability. Atomistic simulations and analytic elastic models show that surface and interface stress can cause a dependence of film energy on thickness that leads to an instability to simultaneous thinning and thickening. We observe that de-wetting is also initiated at bunches of substrate steps in two other systems, Ag/W(110) and Ag/Ru(0001). We additionally describe how Cr films are converted into patterns of unidirectional stripes as the trenches that expose the wetting layer lengthen along the W[001] direction. Finally, we observe how 3D Cr islands form directly during film growth at elevated temperature. The Cr mesas (wedges) form as Cr film steps advance down the staircase of substrate steps, another example of the critical role that substrate steps play in 3D island formation

On stochastic sea of the standard map

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $\Omega$ of full Hausdorff dimension. The set $\Omega$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure

Topological entropy and secondary folding

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.Comment: 13 pages, 8 figures. pdfLaTeX with RevTeX4 macro

Highly efficient separation of actinides from lanthanides by a phenanthroline-derived bis-triazine ligand

The synthesis, lanthanide complexation, and solvent ex- traction of actinide(III) and lanthanide(III) radiotracers from nitric acid solutions by a phenanthroline-derived quadridentate bis-triazine ligand are described. The ligand separates Am(III) and Cm(III) from the lanthanides with remarkably high efficiency, high selectivity, and fast extraction kinetics compared to its 2,2'-bipyridine counterpart. Structures of the 1:2 bis-complexes of the ligand with Eu(III) and Yb(III) were elucidated by X-ray crystallography and force field calculations, respec-tively. The Eu(III) bis-complex is the first 1:2 bis-complex of a quadridentate bis-triazine ligand to be characterized by crystallography. The faster rates of extraction were verified by kinetics measurements using the rotating membrane cell technique in several diluents. The improved kinetics of metal ion extraction are related to the higher surface activity of the ligand at the phase interface. The improvement in the ligand's properties on replacing the bipyridine unit with a phenanthroline unit far exceeds what was anticipated based on ligand design alone

Comparative genomics of Dothideomycete fungi

Fungi are a diverse group of eukaryotic micro-organisms particularly suited for comparative genomics analyses. Fungi are important to industry, fundamental science and many of them are notorious pathogens of crops, thereby endangering global food supply. Dozens of fungi have been sequenced in the last decade and with the advances of the next generation sequencing, thousands of new genome sequences will become available in coming years. In this thesis I have used bioinformatics tools to study different biological and evolutionary processes in various genomes with a focus on the genomes of the Dothideomycetefungi Cladosporium fulvum, Dothistroma septosporumand Zymoseptoria tritici. Chapter 1introduces the scientific disciplines of mycology and bioinformatics from a historical perspective. It exemplifies a typical whole-genome sequence analysis of a fungal genome, and focusses in particular on structural gene annotation and detection of transposable elements. In addition it shortly reviews the microRNA pathway as known in animal and plants in the context of the putative existence of similar yet subtle different small RNA pathways in other branches of the eukaryotic tree of life. Chapter 2addresses the novel sequenced genomes of the closely related Dothideomyceteplant pathogenic fungi Cladosporium fulvumand Dothistroma septosporum. Remarkably, it revealed occurrence of a surprisingly high similarity at the protein level combined with striking differences at the DNA level, gene repertoire and gene expression. Most noticeably, the genome of C. fulvumappears to be at least twice as large, which is solely attributable to a much larger content in repetitive sequences. Chapter 3describes a novel alignment-based fungal gene prediction method (ABFGP) that is particularly suitable for plastic genomes like those of fungi. It shows excellent performance benchmarked on a dataset of 7,000 unigene-supported gene models from ten different fungi. Applicability of the method was shown by revisiting the annotations of C. fulvumand D. septosporumand of various other fungal genomes from the first-generation sequencing era. Thousands of gene models were revised in each of the gene catalogues, indeed revealing a correlation to the quality of the genome assembly, and to sequencing strategies used in the sequencing centres, highlighting different types of errors in different annotation pipelines. Chapter 4focusses on the unexpected high number of gene models that were identified by ABFGP that align nicely to informant genes, but only upon toleration of frame shifts and in-frame stop-codons. These discordances could represent sequence errors (SEs) and/or disruptive mutations (DMs) that caused these truncated and erroneous gene models. We revisited the same fungal gene catalogues as in chapter 3, confirmed SEs by resequencing and successively removed those, yielding a high-confidence and large dataset of nearly 1,000 pseudogenes caused by DMs. This dataset of fungal pseudogenes, containing genes listed as bona fide genes in current gene catalogues, does not correspond to various observations previously done on fungal pseudogenes. Moreover, the degree of pseudogenization showing up to a ten-fold variation for the lowest versus the highest affected species, is generally higher in species that reproduce asexually compared to those that in addition reproduce sexually. Chapter 5describes explorative genomics and comparative genomics analyses revealing the presence of introner-like elements (ILEs) in various Dothideomycetefungi including Zymoseptoria triticiin which they had not identified yet, although its genome sequence is already publicly available for several years. ILEs combine hallmark intron properties with the apparent capability of multiplying themselves as repetitive sequence. ILEs strongly associate with events of intron gain, thereby delivering in silico proof of their mobility. Phylogenetic analyses at the intra- and inter-species level showed that most ILEs are related and likely share common ancestry. Chapter 6provides additional evidence that ILE multiplication strongly dominates over other types of intron duplication in fungi. The observed high rate of ILE multiplication followed by rapid sequence degeneration led us to hypothesize that multiplication of ILEs has been the major cause and mechanism of intron gain in fungi, and we speculate that this could be generalized to all eukaryotes. Chapter 7describes a new strategy for miRNA hairpin prediction using statistical distributions of observed biological variation of properties (descriptors) of known miRNA hairpins. We show that the method outperforms miRNA prediction by previous, conventional methods that usually apply threshold filtering. Using this method, several novel candidate miRNAs were assigned in the genomes of Caenorhabditis elegansand two human viruses. Although this chapter is not applied on fungi, the study does provide a flexible method to find evidence for existence of a putative miRNA-like pathway in fungi. Chapter 8provides a general discussion on the advent of bioinformatics in mycological research and its implications. It highlights the necessity of a prioriplanning and integration of functional analysis and bioinformatics in order to achieve scientific excellence, and describes possible scenarios for the near future of fungal (comparative) genomics research. Moreover, it discusses the intrinsic error rate in large-scale, automatically inferred datasets and the implications of using and comparing those.</p