The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on n nodes and m edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest Laplacian eigenvalues exceeds that of the k largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the k largest Laplacian eigenvalues.

We give a strongly polynomial time combinatorial algorithm to minimise the largest eigenvalue of the weighted Laplacian of a bipartite graph. This is accomplished by solving the dual graph embedding problem which arises from a semidefinite programming formulation. In particular, the problem for trees can be solved in time cubic in the number of vertices.

The aim of this work is to derive mathematical descriptions of gravitation. Postulating gravitation as a force field, Newton's law of gravitation is heuristically derived by considering linear differential operators invariant under euclidean isometries and by finding the fundamental solution to Helmholtz equation in three dimensions. Thereafter, the theory of differential geometry is introduced, providing a framework for the subsequent review of gravitation as curvature. Lastly, in the light of Einstein's postulates and equivalence principle, Lovelock's proof of uniqueness of Einstein's field equations is presented.

As an orbiter cruising to Jupiter will encounter different plasma environments, variety of spacecraft surface charging is expected. This surface potential can lead to inaccurate and wrong in-situ plasma measurements of on-board sensors, which explain the interest in simulating the charging.In this thesis the spacecraft-plasma interactions for a future mission to Jupiter are modelled with the help of the Spacecraft Plasma Interaction System, taking the case of a Jupiter Ganymede Orbiter (JGO) and a Jupiter Europa Orbiter (JEO) as an archetype for a future mission.It is shown that in solar wind at Earth and Jupiter, spacecraft potentials of about 8 V for the JEO, and 10 V to 11 V for the JGO are expected. Furthermore, at a distance of 15 Jupiter radii from Jupiter, the JGO is expected to charge to an electric potential of 2 V, except in the planetary shadow, where it will charge to a high negative potential of -40 V. Moreover, close to the orbit of Callisto, JGO will charge to 12 V in the sun and to 4.6 V in eclipse, due to a high secondary electron emission yield. / <p>Validerat; 20120115 (anonymous)</p>

A model is developed to describe solute transport and retention in fractured rocks. It accounts for the fact that solutes not only can diffuse directly from the flowing channel into the adjacent rock matrix composed of different geological layers but can also at first diffuse into the stagnant water zone occupied in part of the fracture and then from there into the rock matrix adjacent to it. Moreover, the effect of radioactive decay-chain has also been studied in the presence of matrix comprising different geological layers. In spite of the complexities of the system, the analytical solution obtained for the Laplace-transformed concentration at the outlet of the flowing channel can conveniently be transformed back to the time domainby use of e.g. De Hoog algorithm. This allows one to readily include it into a fracture network modelorachannelnetwork model to predictnuclide transport through channels in heterogeneous fracturedmedia consisting of an arbitrary number of rock units withpiecewise constant properties. Simulations made in this study indicate that, in addition to the intact wall rock adjacent to the flowing channel, the stagnant water zone and the rock matrix adjacent to it may also lead to a considerable retardation of solute in cases with a narrow channel. The results further suggest that it is necessary to account for decay-chain and also rock matrix comprising at least two different geological layers in safety and performance assessment of the repositories for spent nuclear fuel. The altered zone may cause a great decrease of the nuclide concentration at the outlet of the flowing channel. The radionuclide decay, when accounted for, will drastically decrease the concentration of nuclides, while neglecting radioactive ingrowth would underestimate the concentration of daughter nuclides. / <p>QC 20140224</p>

The Birnbaum-Saunders (BS) distribution is a positively skewed distribution and is a popular model for analyzing lifetime data. In this thesis, we first develop an improved method of estimation for the BS distribution and the corresponding inference. Compared to the maximum likelihood estimators (MLEs) and the modified moment estimators (MMEs), the proposed method results in estimators with smaller bias, but having the same mean squared errors (MSEs) as these two estimators. Next, the existence and uniqueness of the MLEs of the parameters of BS distribution are discussed based on Type-I, Type-II and hybrid censored samples. In the case of five-parameter bivariate Birnbaum-Saunders (BVBS) distribution, we use the distributional relationship between the bivariate normal and BVBS distributions to propose a simple and efficient method of estimation based on Type-II censored samples. Regression analysis is commonly used in the analysis of life-test data when some covariates are involved. For this reason, we consider the regression problem based on BS and BVBS distributions and develop the associated inferential methods. One may generalize the BS distribution by using Laplace kernel in place of the normal kernel, referred to as the Laplace BS (LBS) distribution, and it is one of the generalized Birnbaum-Saunders (GBS) distributions. Since the LBS distribution has a close relationship with the Laplace distribution, it becomes necessary to first carry out a detailed study of inference for the Laplace distribution before studying the LBS distribution. Several inferential results have been developed in the literature for the Laplace distribution based on complete samples. However, research on Type-II censored samples is somewhat scarce and in fact there is no work on Type-I censoring. For this reason, we first start with MLEs of the location and scale parameters of Laplace distribution based on Type-II and Type-I censored samples. In the case of Type-II censoring, we derive the exact joint and marginal moment generating functions (MGF) of the MLEs. Then, using these expressions, we derive the exact conditional marginal and joint density functions of the MLEs and utilize them to develop exact confidence intervals (CIs) for some life parameters of interest. In the case of Type-I censoring, we first derive explicit expressions for the MLEs of the parameters, and then derive the exact conditional joint and marginal MGFs and use them to derive the exact conditional marginal and joint density functions of the MLEs. These densities are used in turn to develop marginal and joint CIs for some quantities of interest. Finally, we consider the LBS distribution and formally show the different kinds of shapes of the probability density function (PDF) and the hazard function. We then derive the MLEs of the parameters and prove that they always exist and are unique. Next, we propose the MMEs, which can be used as initial values in the numerical computation of the MLEs. We also discuss the interval estimation of parameters. / Thesis / Doctor of Science (PhD)

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