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- aerosol size distribution (2)
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Contents: 1 Introduction 1.1 Tikhanov-Phillips Regularization of Ill-Posed Problems 1.2 A Compact Course to Wavelets 2 A Multilevel Iteration for Tikhonov-Phillips Regularization 2.1 Multilevel Splitting 2.2 The Multilevel Iteration 2.3 Multilevel Approach to Cone Beam Reconstuction 3 The use of approximating operators 3.1 Computing approximating families {Ah}
We have numerically studied the bifurcations and transition to chaos in a two-dimensional fluid for varying values of the Reynolds number. These investigations have been motivated by experiments in fluids, where an array of vortices was driven by an electromotive force. In these experiments, successive changes leading to a complex motion of the vortices, due to increased forcing, have been explored [Tabeling, Perrin, and Fauve, J. Fluid Mech. 213, 511 (1990)]. We model this experiment by means of two-dimensional Navier-Stokes equations with a special external forcing, driving a linear chain of eight counter-rotating vortices, imposing stress-free boundary conditions in the vertical direction and periodic boundary conditions in the horizontal direction. As the strength of the forcing or the Reynolds number is raised, the original stationary vortex array becomes unstable and a complex sequence of bifurcations is observed. Several steady states and periodic branches and a period doubling cascade appear on the route to chaos. For increasing values of the Reynolds number, shear flow develops, for which the spatial scale is large compared to the scale of the forcing. Furthermore, we have investigated the influence of the aspect ratio of the container as well as the effect of no-slip boundary conditions at the top and bottom, on the bifurcation scenario.
We investigate the cognitive control in polyrhythmic hand movements as a model paradigm for bimanual coordination. Using a symbolic coding of the recorded time series, we demonstrate the existence of qualitative transitions induced by experimental manipulation of the tempo. A nonlinear model with delayed feedback control is proposed, which accounts for these dynamical transitions in terms of bifurcations resulting from variation of the external control parameter. Furthermore, it is shown that transitions can also be observed due to fluctuations in the timing control level. We conclude that the complexity of coordinated bimanual movements results from interactions between nonlinear control mechanisms with delayed feedback and stochastic timing components.
We have studied the bifurcations in a three-dimensional incompressible magnetofluid with periodic boundary conditions and an external forcing of the Arnold-Beltrami-Childress (ABC) type. Bifurcation-analysis techniques have been applied to explore the qualitative behavior of solution branches. Due to the symmetry of the forcing, the equations are equivariant with respect to a group of transformations isomorphic to the octahedral group, and we have paid special attention to symmetry-breaking effects. As the Reynolds number is increased, the primary nonmagnetic steady state, the ABC flow, loses its stability to a periodic magnetic state, showing the appearance of a generic dynamo effect; the critical value of the Reynolds number for the instability of the ABC flow is decreased compared to the purely hydrodynamic case. The bifurcating magnetic branch in turn is subject to secondary, symmetry-breaking bifurcations. We have traced periodic and quasi- periodic branches until they end up in chaotic states. In particular detail we have analyzed the subgroup symmetries of the bifurcating periodic branches, which are closely related to the spatial structure of the magnetic field.
We have shown that the two-dimensional complex Ginzburg-Landau equation exhibits supertransient chaos in a certain parameter range. Using numerical methods this behavior is found near the transition line separating frozen spiral solutions from turbulence. Supertransient chaos seems to be a common phenomenon in extended spatiotemporal systems. These supertransients are characterized by an average transient lifetime which depends exponentially on the size of the system and are due to an underlying nonattracting chaotic set.
The stability of the quiescent ground state of an incompressible viscous fluid sheet bounded by two parallel planes, with an electrical conductivity varying across the sheet, and driven by an external electric field tangential to the boundaries is considered. It is demonstrated that irrespective of the conductivity profile, as magnetic and kinetic Reynolds numbers (based on the Alfvén velocity) are raised from small values, two-dimensional perturbations become unstable first.
In single photon emission computed tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. The data gathered suffer from attenuation due to the tissue density µ. Each imaged slice incorporates noisy sample values of the nonlinear attenuated Radon transform (formular at this place in the original abstract) Traditional theory for SPECT reconstruction treats µ as a known parameter. In practical applications, however, µ is not known, but either crudely estimated, determined in costly additional measurements or plainly neglected. We demonstrate that an approximation of both f and µ from SPECT data alone is feasible, leading to quantitatively more accurate SPECT images. The result is based on nonlinear Tikhonov regularization techniques for parameter estimation problems in differential equations combined with Gauss-Newton-CG minimization.
The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed.
The stability of the quiescent ground state of an incompressible, viscous and electrically conducting fluid sheet, bounded by stress-free parallel planes and driven by an external electric field tangential to the boundaries, is studied numerically. The electrical conductivity varies as cosh–2(x1/a), where x1 is the cross-sheet coordinate and a is the half width of a current layer centered about the midplane of the sheet. For a <~ 0.4L, where L is the distance between the boundary planes, the ground state is unstable to disturbances whose wavelengths parallel to the sheet lie between lower and upper bounds depending on the value of a and on the Hartmann number. Asymmetry of the configuration with respect to the midplane of the sheet, modelled by the addition of an externally imposed constant magnetic field to a symmetric equilibrium field, acts as a stabilizing factor.
Using a special technique of data analysis, we have found out 34 grand minima of solar activity obtained from a 7,700 years long Δ14C record. The method used rests on a proper filtering of the Δ14C record and the extrapolation of verifiable results for the later history back in time. Additionally, we use a method of nonlinear dynamics, the recurrence rate, to back up the results. Our findings are not contradictory to the record of solar maxima resp. minima by Eddy [5], but constitute a considerable extension. Hence, it has become possible to look closer at the validity of models. This way, we have tested several models for solar activity, esp. the model of Barnes et al. [1]. There are hints for that the grand minima might solely be driven by the 209 year period found in the Δ14C record.