<p>Extremal quantile index is a concept that the quantile index will drift to zero (or one)</p><p>as the sample size increases. The three chapters of my dissertation consists of three</p><p>applications of this concept in three distinct econometric problems. In Chapter 2, I</p><p>use the concept of extremal quantile index to derive new asymptotic properties and</p><p>inference method for quantile treatment effect estimators when the quantile index</p><p>of interest is close to zero. In Chapter 3, I rely on the concept of extremal quantile</p><p>index to achieve identification at infinity of the sample selection models and propose</p><p>a new inference method. Last, in Chapter 4, I use the concept of extremal quantile</p><p>index to define an asymptotic trimming scheme which can be used to control the</p><p>convergence rate of the estimator of the intercept of binary response models.</p> / Dissertation

This thesis considers three related structures on finite sets and outstanding conjectures on two of them. Several new problems and conjectures are stated.A union-closed collection of sets is a collection of sets which contains the union of each pair of sets in the collection. A completely separating system of sets is a collection of sets in which for each pair of elements of the universal set, there exists a set in the collection which contains the first element but not the second, and another set which contains the second element but not the first. An antichain (Sperner Family) is a collection of distinct sets in which no set is a subset of another set in the collection. The size of an antichain is the number of sets in the collection. The volume of an antichain is the sum of the cardinalities of the sets in the collection. A flat antichain is an antichain in which the difference in cardinality between any two sets in the antichain is at most one.The two outstanding conjectures considered are:The union-closed sets conjecture - In any union-closed collection of non-empty sets there is an element of the universal set in at least half of the sets in the collection;The flat antichain conjecture - Given an antichain with size s and volume V, there is a flat antichain with the same size and volume.Union-closed collections are considered in two ways. Improvements are made to the previously known bounds concerning the minimum size of a counterexample to the union-closed sets conjecture. Results are derived on the minimum size of a union-closed collection generated by a given number of k-sets. An ordering on sets is described, called order R and it is conjectured that choosing a collection of m k-sets in order R will always minimise the size of the union-closed collection generated by m k-sets.Several variants on completely separating systems of sets are considered. A ++ / determination is made of the minimum size of such collections, subject to various constraints on the collections. In particular, for each n and k, exact values or bounds are determined for the minimum size of completely separating systems on a n-set in which each set has cardinality k.Antichains are considered in their relationship to completely separating systems and the flat antichain conjecture is shown to be true in certain cases.

A graph is called star extremal if its fractional chromatic number is equal to its circular chromatic number. Given integers n,k,k' such that 1<=k<=k'<=n/2,the circulant graph G(n,S_k,k') has vertex set [n]={0,1,2,...,n-1} in which i~j if k<=|i-j|<=k' or n-k'<= |i-j|<=n-k. It was known that for n=q(k+k')+r,where 0<=r <k+k', if k'>=5/4k,then G(n,S_k,k') is star extremal. In the thesis, we prove that if k'>=7/6k and q>=4, then G(n,S_k,k') is star extremal.

Let f be an analytic and one-to-one function on the unit disk such that f(0)=0. Let Q(w)dw^2 be a quadratic differential. Suppose that f maps into the complex plane or the unit disk minus analytic arcs w(t) satisfying Q(w(t))(dw/dt)^2<0. We are interested in the question: if Q is unknown but of a specified form, does f determine the quadratic differential Q uniquely? Our main result is that for functions mapping into the unit disk and quadratic differentials with a pole of order 4 at the origin, the quadratic differential is uniquely determined up to exceptional cases. This question arises in the study of extremal functions for functionals over classes of analytic one-to-one maps.

Let f be an analytic and one-to-one function on the unit disk such that f(0)=0. Let Q(w)dw^2 be a quadratic differential. Suppose that f maps into the complex plane or the unit disk minus analytic arcs w(t) satisfying Q(w(t))(dw/dt)^2<0. We are interested in the question: if Q is unknown but of a specified form, does f determine the quadratic differential Q uniquely? Our main result is that for functions mapping into the unit disk and quadratic differentials with a pole of order 4 at the origin, the quadratic differential is uniquely determined up to exceptional cases. This question arises in the study of extremal functions for functionals over classes of analytic one-to-one maps.

In this thesis we investigate the quantum aspects of black holes near extremality. In particular we seek evidence that a near-extremal black hole has a microscopic description in terms of a two dimensional conformal field theory (CFT). We first demonstrate how the low temperature expansion of the first law of thermodynamics leads to an expression for the entropy of extremal black holes which can be recast as the Cardy formula for the entropy of a chiral two dimensional CFT, in agreement with the Extremal Black Hole/CFT correspondence. We apply Sen’s entropy function formalism to fortify this result by reproducing it in a gravitational setup. We extend our first law analysis to a class of near-Extremal Vanishing Horizon (near-EVH) black holes. These black holes have low entropy and temperature, and their geometries contain locally asymptotically AdS3 throats in the near horizon region. The low temperature expansion of the first law is compatible with the first law for a three dimensional BTZ black hole. As the BTZ black hole has an AdS/CFT description in terms of a non-chiral two dimensional CFT, our result can be viewed as thermodynamic evidence for the EVH/CFT correspondence, which states that gravity on the near horizon EVH geometry is described by a 2d CFT. A near-EVH black hole, or low energy excitation around an EVH black hole, is described by excitations of the dual 2d CFT. As case studies of our first law analysis and the EVH/CFT correspondence, we focus on two asymptotically AdS5× S5 classes of near-EVH black holes. The two cases have interesting individual properties and, by the AdS/CFT correspondence, dual descriptions as states in N = 4 super Yang-Mills theory . We can compare these (UV) pictures to the two dimensional descriptions that emerge from the near horizon, or low energy, dynamics. All EVH near horizon geometries have local AdS3 factors which become BTZ black holes when the configurations are excited from EVH to near-EVH. In the study of static black holes with three R-charges, we examine the non-BPS and near- BPS regimes separately. While the non-BPS near horizon limit is locally regular, in the near- BPS case the near horizon procedure requires focussing geometrically on a strip of the horizon, and the degrees of freedom of the dual CFT2 can be associated with stretched strings between giant gravitons in the transverse five-sphere. The near-EVH limit of non-BPS stationary charged black holes is obtained by taking the vanishing limit of one or both of the angular momenta. When one of the momenta is small, the AdS3 angle is a combination of azimuthal angles in the AdS5 and S5 regions of the geometry. Taking the vanishing limit of both of the angular momenta leads to a near horizon limit which contains a BTZ black hole that is non-trivially fibred by a three-sphere. For each of the case studies we use the AdS3/CFT2 dictionary to specify dual IR CFT2 descriptions of the black holes. We outline a map between the UV and IR near-EVH excitations and demonstrate how the first law reduces in the near-EVH limit to the first law of a BTZ black hole. As a consistency check we compare our results with those of the Kerr/CFT correspondence.

Extremal combinatorics is concerned with how large or small a combinatorial structure can be if we insist it satis es certain properties. In this thesis we investigate four different problems in extremal combinatorics, each with its own unique flavour. We begin by examining a graph saturation problem. We say a graph G is H-saturated if G contains no copy of H as a subgraph, but the addition of any new edge to G creates a copy of H. We look at how few edges a Kp- saturated graph can have when we place certain conditions on its minimum degree. We look at a problem in Ramsey Theory. The k-colour Ramsey number Rk(H) of a graph H is de ned as the least integer n such that every k- colouring of Kn contains a monochromatic copy of H. For an integer r > 3 let Cr denote the cycle on r vertices. By studying a problem related to colourings without short odd cycles, we prove new lower bounds for Rk(Cr) when r is odd. Bootstrap percolation is a process in graphs that can be used to model how infection spreads through a community. We say a set of vertices in a graph percolates if, when this set of vertices start off as infected, the whole graph ends up infected. We study minimal percolating sets, that is, percolating sets with no proper percolating subsets. In particular, we investigate if there is any relation between the smallest and the largest minimal percolating sets in bounded degree graph sequences. A tournament is a complete graph where every edge has been given an orientation. We look at the maximum number of directed k-cycles a tournament can have and investigate when there exist tournaments with many more k-cycles than expected in a random tournament.

This work provides first and second-order expressions to approximate neighboring solutions to the m-point boundary value problem. Multi-point problems arise in optimal control because of interior constraints or switching dynamics. Many problems have this form, and so this work fills a void in the study of extremal fields and neighboring optimal control of constrained systems. Only first and second-order terms are written down, but the approach is systematic and higher order expressions can be found similarly. The constraints and their parameters define an extremal field because any solution to the problem must satisfy the constraints. The approach is to build a Taylor series using constraint differentials, state differentials, and state variations. The differential is key to these developments, and it is a unifying element in the optimization of points, optimal control, and neighboring optimal control. The method is demonstrated on several types of problems including lunar descent, which has nonlinear dynamics, bounded thrust, and free final time. The control structure is bang-off-bang, and the method successfully approximates the unknown initial conditions, switch times, and final time. Compared to indirect shooting, computation time decreases by about three orders of magnitude.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 95-96). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 83-84).