Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main focus of this work is to study the classical Calder\'n-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in $(\rn, d\mu)$, where $\mu$ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.

Pre-election polls are used to test the political landscape and predict election results. The relative weights for the state-level data from the 2006 U.S. senatorial races are considered based on the date on which the polls were conducted. Long- and short-memory weight functions are developed to specify the relative value of historical polling data. An optimal weight function is estimated by minimizing the discrepancy function between estimates from weighted polls and the election outcomes.

Components of turbines, combustion chambers, multi-layered electronic packaging structures and nuclear reactors are subjected to transient thermal loads during their service life. In the presence of a discontinuity like crack or dislocation, the thermal load creates high temperature gradient, which in turn causes the stress intensification at the crack tips. If proper attention is not paid in the design and maintenance of components on this high stress in the vicinity of crack tips, it may lead to instability in the system and decrease in the service life. The concepts of thermal fracture mechanics and its major parameter called transient thermal stress intensity factors can greatly help in the assessment of stability and residual life prediction of such structures. The evaluation of thermal stress intensity factors becomes computationally difficult when the body constitutes of two different materials or is non-homogenous or made of composites. Fracture at bi-material interface is different from its homogenous counterpart because of mixed mode stress condition that prevails at the crack tip even when the geometry is symmetric and loading unidirectional. Because of this, the mode 1 and mode 2 stress intensity factors can not be decoupled to represent tension and shear stress fields as can be done in the case of homogeneous materials. Mathematically, the stress intensity factors at bi-material interfaces are complex due to oscillatory singularity that exists at the crack tip. Although plenty of literature is available for bi-material systems subjected to mechanical loads, very little information is available on problems related to thermal loads. Besides, problems related to transient thermal loads need special attention, since no thermal weight functions are available and the existing methods are computationally expensive. Therefore, the present investigation has been undertaken to develop computational and analytical approaches for obtaining the Mode 1 and Mode 2 stress intensity factors for bi-material interface crack problems using conservation of energy principle in conjunction with the weight function approach for various kinds of thermal loads. In the beginning of the studies, a method to extract the Mode 1 and Mode 2 stress intensity factors for bi-material interface crack subjected to mechanical load is proposed using the concept of Jk integrals. This is extended to thermal loads using J2 line integral and J2 domain integral. Furthermore, weight functions are analytically derived for thermal bi-material stress intensity factors and a computational scheme is developed. These methods are validated for several benchmark problems with known solutions.

In this thesis, we deal with finite Gibbs point processes, especially the processes with densities with respect to a Poisson point process. The main aim of this work is to investigate a four-parametric marked point process of circular discs in three dimensions with two and three way point interactions. In the second chapter, our goal is to simulate such a process. For that purpose, the birth- death Metropolis-Hastings algorithm is presented including theoretical results. After that, the algorithm is applied on the disc process and numerical results for different choices of parameters are presented. The third chapter consists of two approaches for the estimation of parameters. First is the Takacs-Fiksel estimation procedure with a choice of weight functions as the derivatives of pseudolikelihood. The second one is the estimation procedure aiming for the optimal choice of weight functions for the estimation in order to provide better quality estimates. The theoretical background for both of these approaches is derived as well as detailed calculations for the disc process. The numerical results for both methods are presented as well as their comparison. 1

V současnosti je velmi aktuálním tématem generování elektrické energie z alternativních zdrojů, zejména z vibrací. Zařízení, která přeměňují mechanickou energii na elektrickou, využívají často ke své činnosti piezoelektrický jev. Pro optimální nastavení takového elektromechanického měniče pro danou aplikaci je třeba mít k dispozici výpočtový model, který bude schopný postihnout všechny klíčové aspekty jeho provozu. Tato práce se tedy zabývá vytvořením takovéhoto nástroje, který je schopen komplexně popsat elektromechanickou odezvu studovaného piezoelektrického měniče energie v podobě vetknutého, vícevrstvého keramického nosníku s piezoelektrickými vrstvami. Uvažovaná vícevrstvá konstrukce je během své činnosti vystavena kinematickému buzení a je rovněž zatížena tepelnou zbytkovou napjatostí vznikající při její výrobě. Vytvořený výpočtový model využívá klasickou laminátovou teorii k určení statické elektromechanické odezvy dané konstrukce. Elektromechanická odezva při kmitání uvažované konstrukce v ustáleném stavu je získána s využitím Hamiltonova variačního principu a teorie kmitání prutů. Vytvořený výpočtový model je dále schopen odhadnout zdánlivou lomovou houževnatost dané vícevrstvé konstrukce pomocí metody váhových funkcí. Výstupy vytvořeného výpočtového modelu jsou ověřeny s využitím numerických simulací na bázi MKP a dostupných experimentálních výsledků. V diplomové práci je následně vytvořený výpočtový model aplikován při hledání optimálního rozložení jednotlivých vrstev konkrétního vícevrstvého nosníku s cílem maximalizovat jeho elektrický výkon a odolnost vůči šíření povrchových trhlin, resp. vzniku křehkého lomu. Tohoto cíle je dosaženo pomocí vhodného rozložení tepelných zbytkových napětí v jednotlivých vrstvách uvažované konstrukce (řízeného použitými materiály a tloušťkami jednotlivých vrstev).

Compression fracture specimens are ideally suited for miniaturization down to tens of microns. Fracture testing of thermal barrier coatings, ceramics and glasses are also best accomplished under compression or indentation. Compression fracture specimen of finite size with constant form factor was not available in the literature. The finite-sized specimen of edge cracked semicircular disk (ECSD) is designed which has the property of constant form factor. The novel ECSD specimen is explored further using weight function concept. This thesis, therefore, is mainly concerned with the design, development and geometric optimization of compression fracture specimen vis a vis their characterization of form factors, weight functions and isotropic points in the uncracked geometry. Inspired by the Brazilian disk geometry, a novel compression fracture specimen is designed in the form of a semicircular disk with an edge crack which opens up due to the bending moment caused by the compressive load applied along its straight edge. This new design evolved from a set of photoelastic experiments conducted on the Brazilian disk and its two extreme cases. Surprisingly, normalized mode-I stress intensity factor of the semicircular specimen loaded under a particular Hertzian way, is found constant for a wide range of relative crack lengths. This property of constant form factor leads to the development of weight function for ECSD for deeper analysis of the specimen. The weight function of a cracked geometry does not depend on loading configuration and it relates stress intensity factor to the stress distribution in the corresponding uncracked geometry through a weighted integral. The weight function for the disk specimen is synthesized in two different ways: using the conventional approach which requires crack opening displacement and the dual form factor method which is newly developed. Since stress distribution in the uncracked specimen is required in order to use weight function concept, analytical solution is attempted using linear elasticity theory. Since closed form solution for stresses in the uncracked semicircular disk is seldom possible with the available techniques, a new semi-analytical method called partial boundary collocation (PBC), is developed which may be used for solving any 2-D elasticity problem involving a semi-geometry. In the new method, part of the boundary conditions are identically satisfied and remaining conditions are satisfied at discrete boundary points. The classical stress concentration factor for a semi-in finite plate with a semicircular edge notch re-derived using PBC is found to be accurate to the eighth decimal. To enhance the form factor in order to test high-toughness materials, edge cracked semicircular ring (ECSR) specimen is designed in which bending moment at the crack-tip is increased significantly due to the ring geometry. ECSR is analyzed using nite element method and the corresponding uncracked problem is analyzed by PBC. Constant form factor is found possible for the ring specimen with tiny notch. In order to avoid varying semi-Hertzian angle during practice and thereby ensure consistent loading conditions, the designs are further modified by chopping at the loading zones and analyzed. Photoelastic isotropic points (IPs) which are a special case of zeroth order fringe (ZOF) are often found in uncracked and cracked specimens. An analytical technique based on Flamant solution is developed for solving any problem involving circular domain loaded at its boundary. Formation of IPs in a circular disk is studied. The coefficients of static friction between the surfaces of disk and loading fixtures, in photoelastic experiments of three-point and four-point loadings, are explored analytically to confirm with experimental results. The disk under multiple radial loads uniformly spaced on its periphery is found to give rise to one isolated IP at the center. Splitting of this IP into a number of IPs can be observed when the symmetry of normal loading is perturbed. Tangential loading is introduced along with normal loading to capture the effect of the composition on formation of IPs. Bernoulli's lemniscate is found to fit fringe order topology local to multiple IPs. Isotropic points along with other low fringe order zones including ZOF are ideal locations for material removal for weight reduction. Making a small hole in the prospective crack path at the IP location in the uncracked geometry might provide dual benefits: 1. Form factor enhancement; 2. Crack arrestor. Thus, this thesis describes experimental, theoretical and computational investigations for the design, development and calibration of novel compact compression fracture specimens.