The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method [26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that are not coupled, which has never been done to our knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS.

Directional wave-field decomposition for heterogeneousanisotropic media with in-stantaneous response is establishedfor both the acoustic and the electromagnetic equations. We derive a sufficient condition for ellipticity of thesystem's matrix in the Laplace domain and show that theconstruction of the splitting matrix via a Dunford-Taylorintegral over the resolvent of the non-compact, non-normalsystem's matrix is well de ned. The splitting matrix also hasproperties that make it possible to construct the decompositionwith a generalized eigenvector procedure. The classical way ofobtaining the decomposition is equivalent to solving analgebraic Riccati operator equation. Hence the proceduredescribed above also provides a solution to the algebraicRiccati operator equation. The solution to the wave-field decomposition for theisotropic wave equation is expressed in terms of theDirichlet-to-Neumann map for a plane. The equivalence of thisDirichlet-to-Neumann map is the acoustic admittance, i.e. themapping between the pressure and the particle velocity. Theacoustic admittance, as well as the related impedance aresolutions to algebraic Riccati operator equations and are keyelements in the decomposition. In the electromagnetic case thecorresponding impedance and admittance mappings solve therespective algebraic Riccati operator equations and henceprovide solutions to the decomposition problem. The present research shows that it is advantageous toutilize the freedom implied by the generalized eigenvectorprocedure to obtain the solution to the decomposition problemin more general terms than the admittance/impedancemappings. The time-reversal approach to steer an acoustic wave eld inthe cavity and half space geometries are analyzed from aboundary control perspective. For the cavity it is shown thatwe can steer the field to a desired final configuration, withthe assumption of local energy decay. It is also shown that thetime-reversal algorithm minimizes a least square error forfinite times when the data are obtained by measurements. Forthe half space geometry, the boundary condition is expressedwith help of the wave-field decomposition. In the homogeneousmaterial case, the response of the time-reversal algorithm iscalculated analytically. This procedure uses the one-wayequations together with the decomposition operator.

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This work details an O(n^2) algorithm for computing the spectra of discrete Schroedinger operators with periodic potentials. Spectra of these objects enhance our understanding of fundamental aperiodic physical systems and contain rich theoretical structure of interest to the mathematical community. Previous work on the Harper model led to an O(n^2) algorithm relying on properties not satisfied by other aperiodic operators. Physicists working with the Fibonacci Hamiltonian, a popular quasicrystal model, have instead used a problematic dynamical map approach or a sluggish O(n^3) procedure for their calculations. The algorithm presented in this work, a blend of well-established eigenvalue/vector algorithms, provides researchers with a more robust computational tool of general utility. Application to the Fibonacci Hamiltonian in the sparsely studied intermediate coupling regime reveals structure in canonical coverings of the spectrum that will prove useful in motivating conjectures regarding band combinatorics and fractal dimensions.

This master thesis has been carried out on behalf of Volvo Construction Equipment. A front frame of an articulated hauler should be analysed according to the Finite Element Method and vibration tests should be made. The results from the experimental tests should be correlated with the analytical test results here using MAC-values. These values will show if the FE-model represents the physical structure correctly. Visualisations are made on both the experimental and analytical results to get a better understanding about the eigenmodes of the frame. The final results showed that the FE-model was not a match to the physical structure which the experimental tests were made on. It should be noted that the final result only states the present situation. The CAD-model had not been completed when this thesis was performed and therefore some deviation occurred in the results. Some actions are needed in order to reach a better result and they are stated in this report. When they are made, the results can be improved by following the work that has been done in this master thesis.

Radio Communication technologies are undergoing drastic demand over the past two decades. The precious radio resource, electromagnetic radio spectrum, is in vain as technology advances. It is required to come up with a solution to improve its wise uses. Cognitive Radio enabled by Software-Defined Radio brings an intelligent solution to efficiently use the Radio Spectrum. It is a method to aware the radio communication system to be able to adapt to its radio environment like signal power and free spectrum holes. The approach will pose a question on how to efficiently detect a signal. In this thesis different spectrum sensing algorithm will be explained and a special concentration will be on new sensing algorithm based on the Eigenvalues of received signal. The proposed method adapts blind signal detection approach for applications that lacks knowledge about signal, noise and channel property. There are two methods, one is ratio of the Maximum Eigenvalue to Minimum Eigenvalue and the second is ratio of Signal Power to Minimum Eigenvalue. Random Matrix theory (RMT) is a branch of mathematics and it is capable in analyzing large set of data or in a conclusive approach it provides a correlation points in signals or waveforms. In the context of this thesis, RMT is used to overcome both noise and channel uncertainties that are common in wireless communication. Simulations in MATLAB and real-time measurements in LabVIEW are implemented to test the proposed detection algorithms. The measurements were performed based on received signal from an IF-5641R Transceiver obtained from National Instruments.

We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

The thesis explores the new expansions of eigenvalues for -£Gu =£f£lu in S with the Dirichlet boundary condition u=0 on $partial S$ by two conforming elements: the linear element $P_1$ and the bilinear element $Q_1$, and three non-conforming elements: the rotated bilinear element (denoted $Q_1^{rot}$), the extension of $Q_1^{rot}$ (denoted $EQ_1^{rot}$) and Wilson's element. The expansions indicate that $P_1$, $Q_1$ and $Q_1^{rot}$ provide the upper bounds of the eigenvalues, and $EQ_1^{rot}$ and Wilson's elements provide the lower bounds of the eigenvalues. Comparing the five finite elements, the $Q_1^{rot}$ element is more accurate. By the extrapolation, the superconvergence $O(h^4)$ can be obtained where $h$ is the boundary length of uniform squares. Numerical experiment are carried to verify the theoretical analysis made. (°Ñ·Ó¹q¤lÀÉp.4)

For Laplace's eigenvalue problems, this thesis presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use a iteration process to yield approximate eigenvalues and eigenfunctions. The new iteration method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.