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We study the possibility of obtaining a computational turbulence model by means of non-dissipative regularisation of the compressible atmospheric equations for climate-type applications. We use an -regularisation (Lagrangian averaging) of the atmospheric equations. For the hydrostatic and compressible atmospheric equations discretised using a finite volume method on unstructured grids, deterministic and non-deterministic numerical experiments are conducted to compare the individual solutions and the statistics of the regularised equations to those of the original model. The impact of the regularisation parameter is investigated. Our results confirm the principal compatibility of -regularisation with atmospheric dynamics and encourage further investigations within atmospheric model including complex physical parametrisations.
We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.
In this paper a technique to obtain a first approximation for singular inverse Sturm-Liouville problems with a symmetrical potential is introduced. The singularity, as a result of unbounded domain (-infinity, infinity), is treated by considering numerically the asymptotic limit of the associated problem on a finite interval (-L, L). In spite of this treatment, the problem has still an ill-conditioned structure unlike the classical regular ones and needs regularization techniques. Direct computation of eigenvalues in iterative solution procedure is made by means of pseudospectral methods. A fairly detailed description of the numerical algorithm and its applications to specific examples are presented to illustrate the accuracy and convergence behaviour of the proposed approach.
Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point
(2015)
The Dirichlet problem for the heat equation in a bounded domain aS, a"e (n+1) is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane t = c, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii's paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.
We study systematically the estimation of Earth's core angular momentum (CAM) variation between 1962.0 and 2008.0 by using core surface flow models derived from the recent geomagnetic field model C(3)FM2. Various flow models are derived by changing four parameters that control the least squares flow inversion. The parameters include the spherical harmonic (SH) truncation degree of the flow models and two Lagrange multipliers that control the weights of two additional constraints. The first constraint forces the energy spectrum of the flow solution to follow a power law l-p, where l is the SH degree and p is the fourth parameter. The second allows to modulate the solution continuously between the dynamical states of tangential geostrophy (TG) and tangential magnetostrophy (TM). The calculated CAM variations are examined in reference to two features of the observed length-of-day (LOD) variation, namely, its secular trend and 6year oscillation. We find flow models in either TG or TM state for which the estimated CAM trends agree with the LOD trend. It is necessary for TM models to have their flows dominate at planetary scales, whereas TG models should not be of this scale; otherwise, their CAM trends are too steep. These two distinct types of flow model appear to correspond to the separate regimes of previous numerical dynamos that are thought to be applicable to the Earth's core. The phase of the subdecadal CAM variation is coherently determined from flow models obtained with extensively varying inversion settings. Multiple sources of model ambiguity need to be allowed for in discussing whether these phase estimates properly represent that of Earth's CAM as an origin of the observed 6year LOD oscillation.
We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L-2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L-2-index formulas.
As applications, we prove a local L-2-index theorem for families of signature operators and an L-2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L-2-eta forms and L-2-torsion forms as transgression forms.
We find necessary conditions for a second order ordinary differential equation to be equivalent to the Painleve III equation under a general point transformation. Their sufficiency is established by reduction to known results for the equations of the form y ' = f (x, y). We consider separately the generic case and the case of reducibility to an autonomous equation. The results are illustrated by the primary resonance equation.
Microsaccades
(2015)
The first thing we do upon waking is open our eyes. Rotating them in our eye sockets, we scan our surroundings and collect the information into a picture in our head. Eye movements can be split into saccades and fixational eye movements, which occur when we attempt to fixate our gaze. The latter consists of microsaccades, drift and tremor. Before we even lift our eye lids, eye movements – such as saccades and microsaccades that let the eyes jump from one to another position – have partially been prepared in the brain stem. Saccades and microsaccades are often assumed to be generated by the same mechanisms. But how saccades and microsaccades can be classified according to shape has not yet been reported in a statistical manner. Research has put more effort into the investigations of microsaccades’ properties and generation only since the last decade. Consequently, we are only beginning to understand the dynamic processes governing microsaccadic eye movements. Within this thesis, the dynamics governing the generation of microsaccades is assessed and the development of a model for the underlying processes. Eye movement trajectories from different experiments are used, recorded with a video-based eye tracking technique, and a novel method is proposed for the scale-invariant detection of saccades (events of large amplitude) and microsaccades (events of small amplitude). Using a time-frequency approach, the method is examined with different experiments and validated against simulated data. A shape model is suggested that allows for a simple estimation of saccade- and microsaccade related properties. For sequences of microsaccades, in this thesis a time-dynamic Markov model is proposed, with a memory horizon that changes over time and which can best describe sequences of microsaccades.