Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in t-direction. Hence, a numerical method would have to use infinitely many points.\ud \ud To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using Radial Basis Functions

Kernel-based discretisation for solving matrix-valued PDEs

In this paper, we discuss the numerical solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these pproximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical system

Construction of a contraction metric by meshless collocation

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium. The contraction metric is described by a matrix-valued function M(x) such that M(x) is positive definite and F(M)(x) is negative definite, where F denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation F(M)(x) = −C(x). We then use a construction method based on meshless collocation, developed in the companion paper [12], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples

Koppelung agrarökonomischer und hydrologischer Modelle

The objective of the paper is to show the potential of interdisciplinary model networks consisting of agricultural economic and natural science models performing integrated impact assessments of agri-environmental policies. The potentials are exemplified using a coupling of the agricultural sector model RAUMIS with the hydrological models GROWA98 and WEKU in the scope of diffuse agricultural nitrogen immission into waterbodies. A brief classification of applied indicators is given. A key element for policy impact analyses are the relationships between the environmental indicator categories Driving Force, State and Response. The major relationships within this environmental scope as well as the modeling are explained. The presented model network represents a first area wide, regionally differentiated, consistent link-up between the Driving Force indicator “nitrogen balance surplusâ€, the State indicator “nitrogen(nitrate)concentration†and the response indicators. The results show that due to the local characteristics quite different shares of the agricultural nitrogen surplus finally enters the ground and surface waters. Hence, assessing the trade-offs of agri-environmental water protection measures requires an inclusion of natural local characteristics. The integrated consideration of the complex relationships and a comprehensive implementation of local characteristics allows to develop more efficient water pollution control measures.model-based policy consulting, interdisciplinary model networks, agricultural sector model, diffuse agricultural nitrogen immissions, Agricultural and Food Policy, Environmental Economics and Policy,

MÖGLICHKEITEN UND GRENZEN DER POLITIKANALYSE ZUR UMSETZUNG DER WASSERRAHMENRICHTLINIE -- ANWENDUNG EINES HYDRO-ÖKONOMISCHEN MODELLVERBUNDES FÜR DAS WESER EINZUGSGEBIET

EU-Wasserrahmenrichtlinie, landwirtschaftliche Einträge in die Gewässer, hydroökonomische Modellierung, RAUMIS, kosteneffiziente Maßnahmen, Agricultural and Food Policy, Environmental Economics and Policy, Land Economics/Use,

Regulation of polarised growth in fungi

Polarised growth in fungi occurs through the delivery of secretory vesicles along tracks formed by cytoskeletal elements to specific sites on the cell surface where they dock with a multiprotein structure called the exocyst before fusing with the plasmamembrane. The budding yeast, Saccharomyces cerevisiae has provided a useful model to investigate the mechanisms involved and their control. Cortical markers, provided by bud site selection pathways during budding, the septin ring during cytokinesis or the stimulation of the pheromone response receptors during mating, act through upstream signalling pathways to localise Cdc24, the GEF for the rho family GTPase, Cdc42. Cdc42 in its GTP-bound activates a multiprotein protein complex called the polarisome which nucleates actin cables along which the secretory vesicles are transported to the cell surface. Hyphae can elongate at a rate orders of magnitude faster than the extension of a yeast bud, so understanding hyphal growth will require substantial modification of the yeast paradigm. The rapid rate of hyphal growth is driven by a structure called the Spitzenkörper, located just behind the growing tip and which is rich in secretory vesicles. It is thought that secretory vesicles are delivered to the apical region where they accumulate in the Spitzenkörper. The Spitzenkörper then acts as vesicle supply centre in which vesicles exit the Spitzenkörper in all directions, but because of its proximity, the tip receives a greater concentration of vesicles per unit area than subapical regions. There are no obvious equivalents to the bud site selection pathway to provide a spatial landmark for polarised growth in hyphae. However, an emerging model is the way that the site of polarised growth in the fission yeast, Schizosaccharomyces pombe, is marked by delivery of the kelch repeat protein, Tea1, along microtubules. The relationship of the Spitzenkörper to the polarisome and the mechanisms that promote its formation are key questions that form the focus of current research

Grid refinement in the construction of Lyapunov functions using radial basis functions

Lyapunov functions are a main tool to determine the domain of attraction of equilibria in dynamical systems. Recently, several methods have been presented to construct a Lyapunov function for a given system. In this paper, we improve the construction method for Lyapunov functions using Radial Basis Functions. We combine this method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of data points needed to construct Lyapunov functions. Finally, we give numerical examples to illustrate our algorithms

High and Low Molecular Weight Fluorescein Isothiocyanate (FITC)–Dextrans to Assess Blood-Brain Barrier Disruption: Technical Considerations

This note is to report how histological preparation techniques influence the extravasation pattern of the different molecular sizes of fluorescein isothiocyanate (FITC)–dextrans, typically used as markers for blood-brain barrier leakage. By using appropriate preparation methods, false negative results can be minimized. Wistar rats underwent a 2-h middle cerebral artery occlusion and magnetic resonance imaging. After the last imaging scan, Evans blue and FITC–dextrans of 4, 40, and 70 kDa molecular weight were injected. Different histological preparation methods were used. Sites of blood-brain barrier leakage were analyzed by fluorescence microscopy. Extravasation of Evans blue and high molecular FITC–dextrans (40 and 70 kDa) in the infarcted region could be detected with all preparation methods used. If exposed directly to saline, the signal intensity of these FITC–dextrans decreased. Extravasation of the 4-kDa low molecular weight FITC–dextran could only be detected using freshly frozen tissue sections. Preparations involving paraformaldehyde and sucrose resulted in the 4-kDa FITC–dextran dissolving in these reactants and being washed out, giving the false negative result of no extravasation. FITC–dextrans represent a valuable tool to characterize altered blood-brain barrier permeability in animal models. Diffusion and washout of low molecular weight FITC–dextran can be avoided by direct immobilization through immediate freezing of the tissue. This pitfall needs to be known to avoid the false impression that there was no extravasation of low molecular weight FITC–dextrans

UCS protein function is partially restored in the Saccharomyces cerevisiae she4 mutant with expression of the human UNC45-GC, but not UNC45-SM

A dedicated UNC45, Cro1, She4 (UCS) domain-containing protein assists in the Hsp90-mediated folding of the myosin head. Only weak sequence conservation exists between the single UCS protein of simple eukaryotes (She4 in budding yeast) and the two UCS proteins of higher organisms (the general cell and striated muscle UNC45s; UNC45-GC and UNC45-SM, respectively). In vertebrates, UNC45-GC facilitates cytoskeletal functions, whereas the 55% identical UNC45-SM assists assembly of the contractile apparatus of cardiac and skeletal muscles. A Saccharomyces cerevisiae she4Δ mutant, totally lacking any UCS protein, was engineered to express as its sole Hsp90 either the Hsp90α or the Hsp90β isoforms of human cytosolic Hsp90. A transient induction of the human UNC45-GC, but not UNC45-SM, could rescue the defective endocytosis in these she4Δ cells at 39 °C, irrespective of whether they possessed Hsp90α or Hsp90β. UNC45-GC-mediated rescue of the localisation of a Myo5-green fluorescent protein (GFP) fusion to cortical patches at 39 °C was more efficient in the yeast containing Hsp90α, though this may relate to more efficient functioning of Hsp90α as compared to Hsp90β in these strains. Furthermore, inducible expression of UNC45-GC, but not UNC45-SM, could partially rescue survival at a more extreme temperature (45 °C) that normally causes she4Δ mutant yeast cells to lyse. The results indicate that UCS protein function has been most conserved-yeast to man-in the UNC45-GC, not UNC45-SM. This may reflect UNC45-GC being the vertebrate UCS protein that assists formation of the actomyosin complexes needed for cytokinesis, cell morphological change, and organelle trafficking-events also facilitated by the myosins in yeast

Retrotransposons Are the Major Contributors to the Expansion of the \u3ci\u3eDrosophila ananassae\u3c/i\u3e Muller F Element

The discordance between genome size and the complexity of eukaryotes can partly be attributed to differences in repeat density. The Muller F element (∼5.2 Mb) is the smallest chromosome in Drosophila melanogaster, but it is substantially larger (\u3e18.7 Mb) in D. ananassae. To identify the major contributors to the expansion of the F element and to assess their impact, we improved the genome sequence and annotated the genes in a 1.4-Mb region of the D. ananassae F element, and a 1.7-Mb region from the D element for comparison. We find that transposons (particularly LTR and LINE retrotransposons) are major contributors to this expansion (78.6%), while Wolbachia sequences integrated into the D. ananassae genome are minor contributors (0.02%). Both D. melanogaster and D. ananassae F-element genes exhibit distinct characteristics compared to D-element genes (e.g., larger coding spans, larger introns, more coding exons, and lower codon bias), but these differences are exaggerated in D. ananassae. Compared to D. melanogaster, the codon bias observed in D. ananassae F-element genes can primarily be attributed to mutational biases instead of selection. The 5′ ends of F-element genes in both species are enriched in dimethylation of lysine 4 on histone 3 (H3K4me2), while the coding spans are enriched in H3K9me2. Despite differences in repeat density and gene characteristics, D. ananassae F-element genes show a similar range of expression levels compared to genes in euchromatic domains. This study improves our understanding of how transposons can affect genome size and how genes can function within highly repetitive domains