This thesis studies the so-called “criticality problem”, an important generalised eigenvalue problem arising in neutron transport theory. The smallest positive real eigenvalue of the problem contains valuable information about the status of the fission chain reaction in the nuclear reactor (i.e. the criticality of the reactor), and thus plays an important role in the design and safety of nuclear power stations. Because of the practical importance, efficient numerical methods to solve the criticality problem are needed, and these are the focus of this thesis. In the theory we consider the time-independent neutron transport equation in the monoenergetic homogeneous case with isotropic scattering and vacuum boundary conditions. This is an unsymmetric integro-differential equation in 5 independent variables, modelling transport, scattering, and fission, where the dependent variable is the neutron angular flux. We show that, before discretisation, the nonsymmetric eigenproblem for the angular flux is equivalent to a related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator(in space only). Furthermore, we prove the existence of a simple smallest positive real eigenvalue with a corresponding eigenfunction that is strictly positive in the interior of the reactor. We discuss approaches to discretise the problem and present discretisations that preserve the underlying symmetry in the finite dimensional form. The thesis then describes methods for computing the criticality in nuclear reactors, i.e. the smallest positive real eigenvalue, which are applicable for quite general geometries and physics. In engineering practice the criticality problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inneriterations. This leads to inexact iterative methods for criticality computations, for which there appears to be no rigorous convergence theory. The fact that, under appropriate assumptions, the integro-differential eigenvalue problem possesses an underlying symmetry (in a space of reduced dimension) allows us to perform a systematic convergence analysis for inexact inverse iteration and related methods. In particular, this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. The theory is illustrated with numerical examples on several test problems of physical relevance, using GMRES as the inner solver. We also illustrate the use of Monte Carlo methods for the solution of neutron transport source problems as well as for the criticality problem. Links between the steps in the Monte Carlo process and the underlying mathematics are emphasised and numerical examples are given. Finally, we introduce an iterative scheme (the so-called “method of perturbation”) that is based on computing the difference between the solution of the problem of interest and the known solution of a base problem. This situation is very common in the design stages for nuclear reactors when different materials are tested, or the material properties change due to the burn-up of fissile material. We explore the relation ofthe method of perturbation to some variants of inverse iteration, which allows us to give convergence results for the method of perturbation. The theory shows that the method is guaranteed to converge if the perturbations are not too large and the inner problems are solved with sufficiently small tolerances. This helps to explain the divergence of the method of perturbation in some situations which we give numerical examples of. We also identify situations, and present examples, in which the method of perturbation achieves the same convergence rate as standard shifted inverse iteration. Throughout the thesis further numerical results are provided to support the theory.

The frame tube buildings have been the most efficient structural system used for building which is in the range of 40-100storey. The soaring heights and the demanding structural efficiency have led to them having smaller reserves of stiffness and consequently stability. In this thesis a Non-linear analysis and stability check of frame-tube building is done. Nonlinear analysis offers several options for addressing problems of nonlinearity and in this work focus is on Geometric Non-linearity. The main sources can be identified as P-Æ’´ effect of gravity loading acting on a transversely displaced structure due to lateral loading and can also be due to member imperfections, such as member camber and out of plumb erection of the frame. During analysis the element response keep continuously changing as a function of the applied load so simple step computing methods have been employed instead of direct analytical methods. The problem here is dealt in a piece wise linear way and solved. In this thesis a program using the matrix approach has been developed. The program developed can calculate the buckling load and can do Linear and Non-linear analysis using the Mat-lab as the computing platform. Numerical results obtained from the program have been compared with the Finite Element software Mastan2. The comparative solutions presented later on in the report clearly prove the accuracy of the program and go on to show, how exploiting simple matrix equation can help solve the most complex structures in fraction of seconds. The program is modular in structure. It provides opportunity for user to make minor manipulation or can append his own module to make it work for his specific needs and will get reliable results.

Este trabalho trata de estimativas inferiores para o primeiro autovalor de Dirichlet para dom nios multiplamente conexos contidos em variedades riemannianas. Essas estimativas consideram o supremo da curvatura seccional da variedade e a curvatura do bordo do domínio. Para obter os resultados, usa-se uma estimativa C0 para solucões da equação de Poisson. / Lower bounds for the rst Dirichlet eigenvalue are presented. We consider multiply connected domains in riemannian manifolds. The estimates are obtained using hypothesis on the supremum of the manifold's sectional curvature and on the domain's boundary curvature. C0 estimates for solutions of Poissons equation are used to prove the results.

In this work, we demonstrate that the Leray-Schauder topological degree theory can be used for the development of a topological degree theory for maximal monotone perturbations of demicontinuous operators of type (S+) in separable reflexive Banach spaces. This is an extension of Berkovits’ degree development for operators as the perturbations above. Berkovits has developed a topological degree for demicontinuous mappings of type (S+), and has shown that the degree mapping is unique under the assumption that it satisfies certain general properties. He proved that if f is a bounded demicontinous mapping of type (S+), G is an open bounded subset of X, and 0 ∈/ f(∂G), then there exists ε0 > 0 such that for every ε ∈ (0, ε0) we have 0 ∈/ (I+ (1/ε)QQ∗ (f))(∂G). Here, Q is a compact linear injection from a Hilbert space H into X, such that Q(H) is dense in X, and Q∗ its adjoint. The map I+ 1 εQQ∗ (f) is a compact displacement of the identity, for which the Leray-Schauder degree is well defined. The Berkovits degree is obtained as the limit of this Leray-Schauder degree as ε tends to zero. We utilize a demicontinuous (S+)-approximation of the form Tt + f, where Tt is the Yosida approximant of T. Namely, we show that if G is an open bounded set in X and 0 ∈/ (T + f)(∂G), then there exist ε0 > 0, t0 > 0, such that for every ε ∈ (0, ε0), t ∈ (0, t0), we have 0 ∈/ (I + (1/ε)QQ∗ (Tt + f))(∂G). Our degree is the limit of the Leray-Schauder degree of the compact displacement of the identity I + (1/ε)QQ∗ (Tt + f) as ε, t → 0. Various extension of the degree has been considered. Finally some properties and applications in invariance of domain, eigenvalue and surjectivity results have also been discussed.

Very fine discretizations of differential operators often lead to large, sparse matrices A, where the condition number of A is large. Such ill-conditioning has well known effects on both solving linear systems and eigenvalue computations, and, in general, computing solutions with relative accuracy independent of the condition number is highly desirable. This dissertation is divided into two parts. In the first part, we discuss a method of preconditioning, developed by Ye, which allows solutions of Ax=b to be computed accurately. This, in turn, allows for accurate eigenvalue computations. We then use this method to develop discretizations that yield accurate computations of the smallest eigenvalue of the biharmonic operator across several domains. Numerical results from the various schemes are provided to demonstrate the performance of the methods. In the second part we address the role of the condition number of A in the context of multigrid algorithms. Under various assumptions, we use rigorous Fourier analysis on 2- and 3-grid iteration operators to analyze round off errors in floating point arithmetic. For better understanding of general results, we provide detailed bounds for a particular algorithm applied to the 1-dimensional Poisson equation. Numerical results are provided and compared with those obtained by the schemes discussed in part 1.

The primary arithmetic information attached to a Maass cusp form is its Laplace eigenvalue. However, in the case of cuspidal Maass forms, the range that these eigenvalues can take is not well-understood. In particular it is unknown if, given a real number r, one can prove that there exists a primitive Maass cusp form with Laplace eigenvalue 1/4 + r2. Conversely, given the Fourier coefficients of a primitive Maass cusp form f on Γ0(D), it is not clear whether or not one can determine its Laplace eigenvalue. In this paper we show that given only a finite number of Fourier coefficients one can first determine the level D, and then compute the Laplace eigenvalue to arbitrarily high precision. The key to our results will be understanding the resonance and rapid decay properties of Maass cusp forms. Let f be a primitive Maass cusp form with Fourier coefficients λf (n). The resonance sum for f is given by SX(f;α;β) = Εn≥1λf(n)‑Φ(n/X) e(αnβ) where φ ∈ Cc∞((1, 2)) is a Schwartz function and α ∈ R and β, X > 0 are real numbers. Sums of this form have been studied for many different classes of functions f, including holomorphic modular forms for SL(2, Z), and Maass cusp forms for SL(n,Z). In this paper we take f to be a primitive Maass cusp form for a congruence subgroup Γ0(D) ⊂ SL(2, Z). Thus our result extends the family of automorphic forms for which their resonance properties are understood. Similar analysis and algorithms can be easily implemented for holomorphic cusp forms for Γ0(D). Our techniques include Voronoi summation, weighted exponential sums, and asymptotics expansions of Bessel functions. We then use these estimates in a new application of resonance sums. In particular we show that given only limited information about a Maass cusp form f (in particular a finite list of high Fourier coefficients), one can determine its level and estimate its spectral parameter, and thus its Laplace eigenvalue. This is done using a large parallel computing cluster running MATLAB and Mathematica

Longitudinal data analysis is common in biomedical research area. Generalized estimating equations (GEE) approach is widely used for longitudinal marginal models. The GEE method is known to provide consistent regression parameter estimates regardless of the choice of working correlation structure, provided the square root of n consistent nuisance parameters are used. However, it is important to use the appropriate working correlation structure in small samples, since it improves the statistical efficiency of β estimate. Several working correlation selection criteria have been proposed (Rotnitzky and Jewell, 1990; Pan, 2001; Hin and Wang, 2009; Shults et. al, 2009). However, these selection criteria have the same limitation in that they perform poorly when over-parameterized structures are considered as candidates. In this dissertation, new working correlation selection criteria are developed based on generalized eigenvalues. A set of generalized eigenvalues is used to measure the disparity between the bias-corrected sandwich variance estimator under the hypothesized working correlation matrix and the model-based variance estimator under a working independence assumption. A summary measure based on the set of the generalized eigenvalues provides an indication of the disparity between the true correlation structure and the misspecified working correlation structure. Motivated by the test statistics in MANOVA, three working correlation selection criteria are proposed: PT (Pillai's trace type criterion),WR (Wilks' ratio type criterion) and RMR (Roy's Maximum Root type criterion). The relationship between these generalized eigenvalues and the CIC measure is revealed. In addition, this dissertation proposes a method to penalize for the over-parameterized working correlation structures. The over-parameterized structure converges to the true correlation structure, using extra parameters. Thus, the true correlation structure and the over-parameterized structure tend to provide similar variance estimate of the estimated β and similar working correlation selection criterion values. However, the over-parameterized structure is more likely to be chosen as the best working correlation structure by "the smaller the better" rule for criterion values. This is because the over-parameterization leads to the negatively biased sandwich variance estimator, hence smaller selection criterion value. In this dissertation, the over-parameterized structure is penalized through cluster detection and an optimization function. In order to find the group ("cluster") of the working correlation structures that are similar to each other, a cluster detection method is developed, based on spacings of the order statistics of the selection criterion measures. Once a cluster is found, the optimization function considering the trade-off between bias and variability provides the choice of the "best" approximating working correlation structure. The performance of our proposed criterion measures relative to other relevant criteria (QIC, RJ and CIC) is examined in a series of simulation studies.

Exact solution to the vibration of rectangular plates is available only for plates with two opposite edges subject to simply supported conditions. Otherwise, they are analysed by using approximate methods. There are several approximate methods to conduct a vibration analysis, such as the Rayleigh-Ritz method, the Finite Element Method, the Finite Difference Method, and the Superposition Method. The Rayleigh-Ritz method and the finite element method give upper bound results for the natural frequencies of plates. However, there is a disadvantage in using this method in that the error due to discretisation cannot be calculated easily. Therefore, it would be good to find a suitable method that gives lower bound results for the natural frequencies to complement the results from the Rayleigh-Ritz method. The superposition method is also a convenient and efficient method but it gives lower bound solution only in some cases. Whether it gives upper bound or lower bound results for the natural frequencies depends on the boundary conditions. It is also known that the finite difference method always gives lower bound results. This thesis presents bounded eigenvalues, which are dimensionless form of natural frequencies, calculated using the superposition method and the finite difference method. All computations were done using the MATLAB software package. The convergence tests show that the superposition method gives a lower bound for the eigenvalues of fully clamped plates, and an upper bound for the completely free plates. It is also shown that the finite difference method gives a lower bound for the eigenvalues of completely free plates. Finally, the upper bounds and lower bounds for the eigenvalues of fully clamped and completely free plates are given.

Objects such as documents, people, words or utilities, that are related in some way, for instance by citations, friendship, appearance in definitions or physical connections, may be conveniently represented using graphs or networks. An increasing number of such relational databases, as for instance the World Wide Web, digital libraries, social networking web sites or phone calls logs, are available. Relevant information may be hidden in these networks. A user may for instance need to get authority web pages on a particular topic or a list of similar documents from a digital library, or to determine communities of friends from a social networking site or a phone calls log. Unfortunately, extracting this information may not be easy. This thesis is devoted to the study of problems related to information extraction in large graphs with the help of dominant vectors of nonnegative matrices. The graph structure is indeed very useful to retrieve information from a relational database. The correspondence between nonnegative matrices and graphs makes Perron--Frobenius methods a powerful tool for the analysis of networks. In a first part, we analyze the fixed points of a normalized affine iteration used by a database matching algorithm. Then, we consider questions related to PageRank, a ranking method of the web pages based on a random surfer model and used by the well known web search engine Google. In a second part, we study optimal linkage strategies for a web master who wants to maximize the average PageRank score of a web site. Finally, the third part is devoted to the study of a nonlinear variant of PageRank. The simple model that we propose takes into account the mutual influence between web ranking and web surfing.

In this thesis, a mixture-model cluster analysis technique under different covariance structures of the component densities is developed and presented, to capture the compactness, orientation, shape, and the volume of component clusters in one expert system to handle Gaussian high dimensional heterogeneous data sets to achieve flexibility in currently practiced cluster analysis techniques. Two approaches to parameter estimation are considered and compared; one using the Expectation-Maximization (EM) algorithm and another following a Bayesian framework using the Gibbs sampler. We develop and score several forms of the ICOMP criterion of Bozdogan (1994, 2004) as our fitness function; to choose the number of component clusters, to choose the correct component covariance matrix structure among nine candidate covariance structures, and to select the optimal parameters and the best fitting mixture-model. We demonstrate our approach on simulated datasets and a real large data set, focusing on early detection of breast cancer. We show that our approach improves the probability of classification error over the existing methods.