<p>Paper 1 : A geometrical version of Hardy's inequality for W_0^{1,p}(D).</p><p>The aim of this article is to prove a Hardy-type inequality, concerning functions in W_0^{1,p}(D) for some domain D in R^n, involving the volume of D and the distance to the boundary of D. The inequality is a generalization of a previously proved inequality by M. and T. Hoffmann-Ostenhof and A. Laptev, which dealt with the special case p=2.</p><p>Paper 2 : A Hardy inequality in the Half-space.</p><p>Here we prove a Hardy-type inequality in the half-space which generalize an inequality originally proved by V. Maz'ya to the so-called L^p case. This inequality had previously been conjectured by the mentioned author. We will also improve the constant appearing in front of the reminder term in the original inequality (which is the first improved Hardy inequality appearing in the litterature).</p><p>Paper 3 : Hardy type inequalities for Many-Particle systems.</p><p>In this article we prove some results about the constants appearing in Hardy inequalities related to many particle systems. We show that the problem of estimating the best constants there is related to some interesting questions from Geometrical combinatorics. The asymptotical behaviour, when the number of particles approaches infinity, of a certain quantity directly related to this, is also investigated.</p><p>Paper 4 : Various results in the theory of Hardy inequalities and personal thoughts.</p><p>In this article we give some further results concerning improved Hardy inequalities in Half-spaces and other conic domains. Also, some examples of applications of improved Hardy inequalities in the theory of viscous incompressible flow will be given.</p>

The spectral radius of a matrix is a widely used concept in linear algebra. It expresses the asymptotic growth rate of successive powers of the matrix. This concept can be extended to sets of matrices, leading to the notion of "joint spectral radius". The joint spectral radius of a set of matrices was defined in the 1960's, but has only been used extensively since the 1990's. This concept is useful to study the behavior of multi-agent systems, to determine the continuity of wavelet basis functions or for expressing the capacity of binary codes. Although the joint spectral radius shares some properties with the usual spectral radius, it is much harder to compute, and the problem of approximating it is NP-hard. In this thesis, we first review theoretical results that lead to basic approximations of the joint spectral radius. Then, we list various specific cases where it is effectively computable, before presenting a specific type of sets of matrices, for which we solve the problem of computing it with a polynomial computational cost.

Thesis (Ph. D.)--Oregon State University, 2007. / Printout. Includes bibliographical references (leaves 125-128). Also available on the World Wide Web.

Successful methods for a large class of nonlinear convex optimization problems have recently been developed. This class, known as self-scaled optimization problems, has been defined by Nesterov and Todd in 1994. As noticed by Guler in 1996, this class is best described using an algebraic structure known as Euclidean Jordan algebra, which provides an elegant and powerful unifying framework for its study. Euclidean Jordan algebras are now a popular setting for the analysis of algorithms designed for self-scaled optimization problems : dozens of research papers in optimization deal explicitely with them. This thesis proposes an extensive and self-contained description of Euclidean Jordan algebras, and shows how they can be used to design and analyze new algorithms for self-scaled optimization. Our work focuses on the so-called spectral functions on Euclidean Jordan algebras, a natural generalization of spectral functions of symmetric matrices. Based on an original variational analysis technique for Euclidean Jordan algebras, we discuss their most important properties, such as differentiability and convexity. We show how these results can be applied in order to extend several algorithms existing for linear or second-order programming to the general class of self-scaled problems. In particular, our methods allowed us to extend to some nonlinear convex problems the powerful smoothing techniques of Nesterov, and to demonstrate its excellent theoretical and practical efficiency.

Let {Tsub(p) : q1 ≤ p ≤ q2} be a family of consistent Csub(0) semigroups on Lφ(Ω) with q1, q2 ∈ [1, ∞)and Ω ⊆ IRn open. We show that certain commutator conditions on Tφ and on the resolvent of its generator Aφ ensure the φ independence of the spectrum of Aφ for φ ∈ [q1, q2]. Applications include the case of Petrovskij correct systems with Hölder continuous coeffcients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coeffcients.

We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed.

Paper 1 : A geometrical version of Hardy's inequality for W_0^{1,p}(D). The aim of this article is to prove a Hardy-type inequality, concerning functions in W_0^{1,p}(D) for some domain D in R^n, involving the volume of D and the distance to the boundary of D. The inequality is a generalization of a previously proved inequality by M. and T. Hoffmann-Ostenhof and A. Laptev, which dealt with the special case p=2. Paper 2 : A Hardy inequality in the Half-space. Here we prove a Hardy-type inequality in the half-space which generalize an inequality originally proved by V. Maz'ya to the so-called L^p case. This inequality had previously been conjectured by the mentioned author. We will also improve the constant appearing in front of the reminder term in the original inequality (which is the first improved Hardy inequality appearing in the litterature). Paper 3 : Hardy type inequalities for Many-Particle systems. In this article we prove some results about the constants appearing in Hardy inequalities related to many particle systems. We show that the problem of estimating the best constants there is related to some interesting questions from Geometrical combinatorics. The asymptotical behaviour, when the number of particles approaches infinity, of a certain quantity directly related to this, is also investigated. Paper 4 : Various results in the theory of Hardy inequalities and personal thoughts. In this article we give some further results concerning improved Hardy inequalities in Half-spaces and other conic domains. Also, some examples of applications of improved Hardy inequalities in the theory of viscous incompressible flow will be given.

Spectral decomposition is a modern tool that utilizes seismic data to generate additional useful information in seismic exploration for hydrocarbon detection, lithology identification, stratigraphic interpretation, filtering and others. Different spectral decomposition methods with applications to seismic data were reported and investigated in past years. Many methods usually do not consider the non-stationary features of seismic data and, therefore, are not likely to give satisfactory results. S-transform developed in recent years is able to provide time-dependent frequency analysis while maintaining a direct relationship with the Fourier spectrum, a unique property that other methods of spectral decomposition may not have. In this thesis, I investigated the feasibility and efficiency of using S-transform for hydrocarbon detection and time-varying surface wave filtering. S-transform was first applied to two seismic data sets from a clastic reservoir in the North Sea and a deep carbonate reservoir in the Sichuan Basin, China. Results from both cases demonstrated that S-transform decomposition technique can detect hydrocarbon zones effectively and helps to build the relationships between lithology changes and high frequency variation and between hydrocarbon occurrence and low-frequency anomaly. However, its time resolution needs to be improved. In the second part of my thesis, I used S-transform to develop a novel Time-frequency-wave-number-domain (T-F-K) filtering method to separate surface wave from reflected waves in seismic records. The S-T-F-K filtering proposed here can be used to analyze surface waves on separate f-k panels at different times. The method was tested using hydrophone records of four-component seismic data acquired in the shallow-water Persian Gulf where the average water depth is about 10m and Scholte waves and other surfaces wave persistently strong. Results showed that this new S-T-F-K method is able to separate and sttenuate surface waves and to improve greatly the quality of seismic reflection signals that are otherwise completely concealed by the aliased surface waves.

Mining operations can produce highly localized, low intensity earthquakes that are referred to as microseismic events. Monitoring of microseismic events is useful in predicting and comprehending hazards, and in evaluating the overall performance of a mine design. <p>A robust localization algorithm is used to estimate the source position of the microseismic event by selecting the hypothesized source location that maximizes an energy function generated from the sum of the time--aligned sensor signals. The accuracy of localization for the algorithm characterized by the variance depends in part upon the configuration of sensors. Two algorithms, MAXSRC and MINMAX, are presented that use the variance of localization error, in a particular direction, as a performance measure for a given sensor configuration.<p>The variance of localization error depends, in part, upon the energy spectral density of the microseismic event. The energy spectral density characterization of sensor signals received in two potash mines are presented and compared using two spectral estimation techniques: multitaper estimation and combined time and lag weighting. It is shown that the difference between the the two estimation techniques is negligible. However, the differences between the two mine characterizations, though not large, is significant. An example uses the characterized energy spectral densities to determine the variance of error for a single step localization algorithm.<p>The MAXSRC and MINMAX algorithms are explained. The MAXSRC sensor placement algorithm places a sensor as close as possible to the source position with the maximum variance. The MINMAX sensor placement algorithm minimizes the variance of the source position with the maximum variance after the sensor has been placed. The MAXSRC algorithm is simple and can be solved using an exhaustive search while the MINMAX algorithm uses a genetic algorithm to find a solution. These algorithms are then used in three examples, two of which are simple and synthetic. The other example is from Lanigan Potash Mine. The results show that both sensor placement algorithms produce similar results, with the MINMAX algorithm consistently doing better. The MAXSRC algorithm places a single sensor approximately 100 times faster than the MINMAX algorithm. The example shows that the MAXSRC algorithm has the potential to be an efficient and intuitively simple sensor placement algorithm for mine microseismic event monitoring. The MINMAX algorithm provides, at an increase in computational time, a more robust placement criterion which can be solved adequately using a genetic algorithm.

The importance of understanding non-metallic material behaviour at high strain rates is becoming ever more important as new materials are being developed and used in shock loading applications. Applying conventional methods for high strain rate testing to non-metallic materials proved ineffective due to impedance mismatch between the specimen and apparatus and so a new test method was developed. A polymeric Hopkinson bar was developed enabling non-metallic materials, such as polycarbonate and rubber, to be tested at strain rates from 500 s^-1 to 4000 s^-1. Conventional Hopkinson bar analysis is invalid due to the viscoelastic nature of the polymeric bar material. As waves propagate along the bar length, the inherent material behaviour causes the waves to undergo a degree of attenuation and dispersion. Through the use of spectral analysis, a comparison of the dispersive relationships for 6061 T-6 aluminium, extruded acrylic and low density polyethylene is presented. The application of spectral methods to viscoelastic wave analysis was validated by three separate methods. A comparison of predicted and measured waves along the bar length allowed the dispersive relationship to be substantiated. The use of an enhanced laser velocity system further verified the predicted wave magnitude. A comparison of results for polycarbonate and ballistic gelatin to published results showed good agreement.