The Maxwell-Landau-Lifshitz-Gilbert System: Mathematical Theory and Numerical Approximation

This thesis deals with the mathematical theory and numerical approximation of the Landau--Lifshitz--Gilbert equation coupled to the Maxwell equations without artificial boundary conditions. As a starting point, the physical equations are stated on the unbounded three dimensional space and reformulated in a mathematically precise way to a coupled partial differential -- boundary integral system. We derive a weak form of the whole coupled system, state the relation to the strong form and show uniqueness of the Maxwell part of the solution. A numerical algorithm is proposed based on the tangent plane scheme for the LLG part and using a finite element and boundary element coupling as spatial discretization and the backward Euler method and Convolution Quadrature as time discretization for the interior Maxwell part and the boundary, respectively. Under minimal assumptions on the regularity of solutions, we present well-posedness and convergence of the numerical algorithm. For the pure Maxwell equations without the coupling to the LLG equation, we are able to show stronger results than in the coupled case. We derive a weak form for the Maxwell transmission problem and demonstrate existence and uniqueness of the weak solutions as well as equivalence with a strong solution. The proposed algorithm of finite-element/ boundary-element coupling via Convolution Quadrature converges with only minimal assumptions on the regularity of the input data. Again for the full Maxwell--LLG system, we show a-priori error bounds in the situation of a sufficiently regular solution. This is done by a combination of the known linearly implicit backward difference formula time discretizations with higher order non-conforming finite element space discretizations for the LLG equation and the leapfrog and Convolution Quadrature time discretization with higher order discontinuous Galerkin elements and continuous boundary elements for the boundary integral formulation of Maxwell\u27s equations. The precise method of coupling allows us to solve the system at the cost of the individual parts, with the same convergence rates under the same regularity assumptions and the same CFL conditions as for an uncoupled examination. Numerical experiments illustrate and expand on the theoretical results and demonstrate the applicability of the methods. For the formulation of the boundary integral equations, the study of the Laplace transform is inevitable. We collect and extend the properties of the Laplace transform from literature. In the suitable functional analytic setting, we give extensive proofs in a self contained way of all the required properties

Chromatic roots as algebraic integers

A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane

PynPoint: a modular pipeline architecture for processing and analysis of high-contrast imaging data

The direct detection and characterization of planetary and substellar companions at small angular separations is a rapidly advancing field. Dedicated high-contrast imaging instruments deliver unprecedented sensitivity, enabling detailed insights into the atmospheres of young low-mass companions. In addition, improvements in data reduction and PSF subtraction algorithms are equally relevant for maximizing the scientific yield, both from new and archival data sets. We aim at developing a generic and modular data reduction pipeline for processing and analysis of high-contrast imaging data obtained with pupil-stabilized observations. The package should be scalable and robust for future implementations and in particular well suitable for the 3-5 micron wavelength range where typically (ten) thousands of frames have to be processed and an accurate subtraction of the thermal background emission is critical. PynPoint is written in Python 2.7 and applies various image processing techniques, as well as statistical tools for analyzing the data, building on open-source Python packages. The current version of PynPoint has evolved from an earlier version that was developed as a PSF subtraction tool based on PCA. The architecture of PynPoint has been redesigned with the core functionalities decoupled from the pipeline modules. Modules have been implemented for dedicated processing and analysis steps, including background subtraction, frame registration, PSF subtraction, photometric and astrometric measurements, and estimation of detection limits. The pipeline package enables end-to-end data reduction of pupil-stabilized data and supports classical dithering and coronagraphic data sets. As an example, we processed archival VLT/NACO L' and M' data of beta Pic b and reassessed the planet's brightness and position with an MCMC analysis, and we provide a derivation of the photometric error budget.Comment: 16 pages, 9 figures, accepted for publication in A&A, PynPoint is available at https://github.com/PynPoint/PynPoin

Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let \bv = \{v_e\}_{e\in E}, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over K(\bv), and give three self-contained proofs that the Galois group of $\hat Z_M$ over K(\bv) is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.Comment: 8 pages, final version, to appear in J. Alg. Comb. Substantial revisions, including the addition of two alternative proofs of the main resul

Algebraic number-theoretic properties of graph and matroid polynomials

PhDThis thesis is an investigation into the algebraic number-theoretical properties of certain polynomial invariants of graphs and matroids. The bulk of the work concerns chromatic polynomials of graphs, and was motivated by two conjectures proposed during a 2008 Newton Institute workshop on combinatorics and statistical mechanics. The first of these predicts that, given any algebraic integer, there is some natural number such that the sum of the two is the zero of a chromatic polynomial (chromatic root); the second that every positive integer multiple of a chromatic root is also a chromatic root. We compute general formulae for the chromatic polynomials of two large families of graphs, and use these to provide partial proofs of each of these conjectures. We also investigate certain correspondences between the abstract structure of graphs and the splitting fields of their chromatic polynomials. The final chapter concerns the much more general multivariate Tutte polynomials—or Potts model partition functions—of matroids. We give three separate proofs that the Galois group of every such polynomial is a direct product of symmetric groups, and conjecture that an analogous result holds for the classical bivariate Tutte polynomial

How to sustain entrepreneurial performance during the current financial crisis

In a debt-ridden society that badly needs to grow economically, policies controlling the flows of economic accounts (revenues and expenditures) should be consistent with an efficient “asset and liability management”. The extra money obtained from immediate sales of idle or low-productive government properties can boost economic growth if lent to innovative entrepreneurial firms