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This work incorporates three treatises which are commonly concerned with a stochastic theory of the Lyapunov exponents. With the help of this theory universal scaling laws are investigated which appear in coupled chaotic and disordered systems. First, two continuous-time stochastic models for weakly coupled chaotic systems are introduced to study the scaling of the Lyapunov exponents with the coupling strength (coupling sensitivity of chaos). By means of the the Fokker-Planck formalism scaling relations are derived, which are confirmed by results of numerical simulations. Next, coupling sensitivity is shown to exist for coupled disordered chains, where it appears as a singular increase of the localization length. Numerical findings for coupled Anderson models are confirmed by analytic results for coupled continuous-space Schrödinger equations. The resulting scaling relation of the localization length resembles the scaling of the Lyapunov exponent of coupled chaotic systems. Finally, the statistics of the exponential growth rate of the linear oscillator with parametric noise are studied. It is shown that the distribution of the finite-time Lyapunov exponent deviates from a Gaussian one. By means of the generalized Lyapunov exponents the parameter range is determined where the non-Gaussian part of the distribution is significant and multiscaling becomes essential.
Concerns have been raised that anthropogenic climate change could lead to large-scale singular climate events, i.e., abrupt nonlinear climate changes with repercussions on regional to global scales. One central goal of this thesis is the development of models of two representative components of the climate system that could exhibit singular behavior: the Atlantic thermohaline circulation (THC) and the Indian monsoon. These models are conceived so as to fulfill the main requirements of integrated assessment modeling, i.e., reliability, computational efficiency, transparency and flexibility. The model of the THC is an interhemispheric four-box model calibrated against data generated with a coupled climate model of intermediate complexity. It is designed to be driven by global mean temperature change which is translated into regional fluxes of heat and freshwater through a linear down-scaling procedure. Results of a large number of transient climate change simulations indicate that the reduced-form THC model is able to emulate key features of the behavior of comprehensive climate models such as the sensitivity of the THC to the amount, regional distribution and rate of change in the heat and freshwater fluxes. The Indian monsoon is described by a novel one-dimensional box model of the tropical atmosphere. It includes representations of the radiative and surface fluxes, the hydrological cycle and surface hydrology. Despite its high degree of idealization, the model satisfactorily captures relevant aspects of the observed monsoon dynamics, such as the annual course of precipitation and the onset and withdrawal of the summer monsoon. Also, the model exhibits the sensitivity to changes in greenhouse gas and sulfate aerosol concentrations that are known from comprehensive models. A simplified version of the monsoon model is employed for the identification of changes in the qualitative system behavior against changes in boundary conditions. The most notable result is that under summer conditions a saddle-node bifurcation occurs at critical values of the planetary albedo or insolation. Furthermore, the system exhibits two stable equilibria: besides the wet summer monsoon, a stable state exists which is characterized by a weak hydrological cycle. These results are remarkable insofar, as they indicate that anthropogenic perturbations of the planetary albedo such as sulfur emissions and/or land-use changes could destabilize the Indian summer monsoon. The reduced-form THC model is employed in an exemplary integrated assessment application. Drawing on the conceptual and methodological framework of the tolerable windows approach, emissions corridors (i.e., admissible ranges of CO2- emissions) are derived that limit the risk of a THC collapse while considering expectations about the socio-economically acceptable pace of emissions reductions. Results indicate, for example, a large dependency of the width of the emissions corridor on climate and hydrological sensitivity: for low values of climate and/or hydrological sensitivity, the corridor boundaries are far from being transgressed by any plausible emissions scenario for the 21st century. In contrast, for high values of both quantities low non-intervention scenarios leave the corridor already in the early decades of the 21st century. This implies that if the risk of a THC collapse is to be kept low, business-as-usual paths would need to be abandoned within the next two decades. All in all, this thesis highlights the value of reduced-form modeling by presenting a number of applications of this class of models, ranging from sensitivity and bifurcation analysis to integrated assessment. The results achieved and conclusions drawn provide a useful contribution to the scientific and policy debate about the consequences of anthropogenic climate change and the long-term goals of climate protection. --- Anmerkung: Die Autorin ist Trägerin des von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Potsdam vergebenen Michelson-Preises für die beste Promotion des Jahres 2003/2004.
Ecological and evolutionary dynamics can occur on similar timescales. However, theoretical predictions of how rapid evolution can affect ecological dynamics are inconclusive and often depend on untested model assumptions. Here we report that rapid prey evolution in response to oscillating predator density affects predator-prey (rotifer-algal) cycles in laboratory microcosms. Our experiments tested explicit predictions from a model for our system that allows prey evolution. We verified the predicted existence of an evolutionary tradeoff between algal competitive ability and defence against consumption, and examined its effects on cycle dynamics by manipulating the evolutionary potential of the prey population. Single-clone algal cultures (lacking genetic variability) produced short cycle periods and typical quarter-period phase lags between prey and predator densities, whereas multi-clonal (genetically variable) algal cultures produced long cycles with prey and predator densities nearly out of phase, exactly as predicted. These results confirm that prey evolution can substantially alter predator-prey dynamics, and therefore that attempts to understand population oscillations in nature cannot neglect potential effects from ongoing rapid evolution.
From the stem bark of Erythrina burttii, a new isoflavone, 5,2',4'-trihydroxy-7-methoxy-6-(3- methylbut-2-enyl)isoflavone (trivial name, 7-O-methylluteone) and a new flavanone, 5,7-dihydroxy-4'-methoxy- 3'-(3-methylbutadienyl)-5'-(3-methylbut-2-enyl)flavanone (trivial name, burttinonedehydrate) along with three known isoflavonoids (8-prenylluteone, 3-O-methylcalopocarpin and genistein) were isolated. The structures were detd. on the basis of spectroscopic evidence.
We present the first separate spectra of both components of the small-separation double QSO HE 0512-3329 obtained with HST/STIS in the optical and near UV. The similarities especially of the emission line profiles and redshifts strongly suggest that this system really consists of two lensed images of one and the same source. The emission line flux ratios are assumed to be unaffected by microlensing and are used to study the differential extinction effects caused by the lensing galaxy. Fits of empirical laws show that the extinction properties seem to be different on both lines of sight. With our new results, HE 0512-3329 becomes one of the few extragalactic systems which show the 2175 Å absorption feature, although the detection is only marginal. We then correct the continuum flux ratio for extinction to obtain the differential microlensing signal. Since this may still be significantly affected by variability and time-delay effects, no detailled analysis of the microlensing is possible at the moment. This is the first time that differential extinction and microlensing could be separated unambiguously. We show that, at least in HE 0512-3329, both effects contribute significantly to the spectral differences and one cannot be analysed without taking into account the other. For lens modelling purposes, the flux ratios can only be used after correcting for both effects.
We present a determination of the optical/UV AGN luminosity function and its evolution, based on a large sample of faint (R < 24) QSOs identified in the COMBO-17 survey. Using multi-band photometry in 17 filters within 350 nm <~ lambdaobs <~ 930 nm, we could simultaneously determine photometric redshifts with an accuracy of sigmaz <0.03 and obtain spectral energy distributions. The redshift range covered by the sample is 1.2 < z < 4.8, which implies that even at z =~ 3, the sample reaches below luminosities corresponding to MB = -23, conventionally employed to distinguish between Seyfert galaxies and quasars. We clearly detect a broad plateau-like maximum of quasar activity around z =~ 2 and map out the smooth turnover between z =~ 1 and z =~ 4. The shape of the LF is characterised by some mild curvature, but no sharp ``break'' is present within the range of luminosities covered. Using only the COMBO-17 data, the evolving LF can be adequately described by either a pure density evolution (PDE) or a pure luminosity evolution (PLE) model. However, the absence of a strong L*-like feature in the shape of the LF inhibits a robust distinction between these modes. We present a robust estimate for the integrated UV luminosity generation by AGN as a function of redshift. We find that the LF continues to rise even at the lowest luminosities probed by our survey, but that the slope is sufficiently shallow that the contribution of low-luminosity AGN to the UV luminosity density is negligible. Although our sample reaches much fainter flux levels than previous data sets, our results on space densities and LF slopes are completely consistent with extrapolations from recent major surveys such as SDSS and 2QZ.
Green formulae for elliptic cone differential operators are established. This is achieved by an accurate description of the maximal domain of an elliptic cone differential operator and its formal adjoint; thereby utilizing the concept of a discrete asymptotic type. From this description, the singular coefficients replacing the boundary traces in classical Green formulas are deduced.
Integral-field spectrophotometry of the quadruple QSO HE 0435-1223 : Evidence for microlensing
(2003)
We present the first spatially resolved spectroscopic observations of the recently discovered quadruple QSO and gravitational lens HE 0435-1223. Using the Potsdam Multi-Aperture Spectrophotometer (PMAS), we show that all four QSO components have very similar but not identical spectra. In particular, the spectral slopes of components A, B, and D are indistinguishable, implying that extinction due to dust plays no major role in the lensing galaxy. While also the emission line profiles are identical within the error bars, as expected from lensing, the equivalent widths show significant differences between components. Most likely, microlensing is responsible for this phenomenon. This is also consistent with the fact that component D, which shows the highest relative continuum level, has brightened by 0.07 mag since Dec. 2001. We find that the emission line flux ratios between the components are in better agreement with simple lens models than broad band or continuum measurements, but that the discrepancies between model and data are still unacceptably large. Finally, we present a detection of the lensing galaxy, although this is close to the limits of the data. Comparing with a model galaxy spectrum, we obtain a redshift estimate of zlens=0.44+/- 0.02.