Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 80-81). Available also in a digital version from Dissertation Abstracts.

In this dissertation, a problem related to Hadwiger's conjecture has been studied. We first proved a conjecture of Jakobsen from 1983 which states that every simple graphs on $n$ vertices and at least (11n-35)/2 edges either has a minor isomorphic to K_8 with one edge deleted or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2} and/or K_7 by identifying cliques of size five. We then studied the extremal functions for complete minors. We proved that every simple graph on nge9 vertices and at least 7n-27 edges either has a minor, or is isomorphic to K_{2, 2, 2, 3, 3}, or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2, 2} by identifying cliques of size six. This result extends Mader's theorem on the extremal function for K_p minors, where ple7. We discussed the possibilities of extending our methods to K_{10} and K_{11} minors. We have also found the extremal function for K_7 plus a vertex minor.

The purpose of this thesis is to investigate a class of four left-invariant optimal control problems on the special orthogonal group SO(3). The set of all control-affine left-invariant control systems on SO(3) can, without loss, be reduced to a class of four typical controllable left-invariant control systems on SO(3) . The left-invariant optimal control problem on SO(3) involves finding a trajectory-control pair on SO (3), which minimizes a cost functional, and satisfies the given dynamical constraints and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*SO(3) = SO(3) x so (3)* using the optimal Hamiltonian on so(3)*, where the maximum principle yields the optimal control. In a contribution to the study of this class of optimal control problems on SO(3), the extremal equations on so(3)* (ident ified with JR3) are integrated via elliptic functions to obtain explicit expressions for the solution curves in each typical case. The energy-Casimir method is used to give sufficient conditions for non-linear stability of the equilibrium states. / KMBT_363 / Adobe Acrobat 9.54 Paper Capture Plug-in

The aim of this thesis is to study a class of left-invariant optimal control problems on the matrix Lie group SE(2). We classify, under detached feedback equivalence, all controllable (left-invariant) control affine systems on SE(2). This result produces six types of control affine systems on SE(2). Hence, we study six associated left-invariant optimal control problems on SE(2). A left-invariant optimal control problem consists of minimizing a cost functional over the trajectory-control pairs of a left-invariant control system subject to appropriate boundary conditions. Each control problem is lifted from SE(2) to T*SE(2) ≅ SE(2) x se (2)*and then reduced to a problem on se (2)*. The maximum principle is used to obtain the optimal control and Hamiltonian corresponding to the normal extremals. Then we derive the (reduced) extremal equations on se (2)*. These equations are explicitly integrated by trigonometric and Jacobi elliptic functions. Finally, we fully classify, under Lyapunov stability, the equilibrium states of the normal extremal equations for each of the six types under consideration.

In 1988, Owen introduced empirical likelihood as a nonparametric method for constructing confidence intervals and regions. Since then, empirical likelihood has been studied extensively in the literature due to its generality and effectiveness. It is well known that empirical likelihood has several attractive advantages comparing to its competitors such as bootstrap: determining the shape of confidence regions automatically using only the data; straightforwardly incorporating side information expressed through constraints; being Bartlett correctable. The main part of this thesis extends the empirical likelihood method to several interesting and important statistical inference situations. This thesis has four components. The first component (Chapter II) proposes a smoothed jackknife empirical likelihood method to construct confidence intervals for the receiver operating characteristic (ROC) curve in order to overcome the computational difficulty when we have nonlinear constrains in the maximization problem. The second component (Chapter III and IV) proposes smoothed empirical likelihood methods to obtain interval estimation for the conditional Value-at-Risk with the volatility model being an ARCH/GARCH model and a nonparametric regression respectively, which have applications in financial risk management. The third component(Chapter V) derives the empirical likelihood for the intermediate quantiles, which plays an important role in the statistics of extremes. Finally, the fourth component (Chapter VI and VII) presents two additional results: in Chapter VI, we present an interesting result by showing that, when the third moment is infinity, we may prefer the Student's t-statistic to the sample mean standardized by the true standard deviation; in Chapter VII, we present a method for testing a subset of parameters for a given parametric model of stationary processes.

Thesis (MSc)--University of Stellenbosch, 2004. / ENGLISH ABSTRACT: The United Nations Convention on the Law of the Sea (UNCLOS) provides an opportunity for qualifying coastal signatory states to claim extended maritime estate. The opportunity to claim rests on the precept that in certain cases a continental shelf extends beyond the traditionally demarcated two hundred nautical mile (200M) Exclusive Economic Zone (EEZ) mark. In these cases a successful claim results in states having sovereign rights to the living and non-living resources of the seabed and subsoil, as well as the sedentary species, of the area claimed. Where the continental shelf extends beyond the 200M mark, the Foot of the Continental Slope (FoS) has to be determined as one of the qualifying criteria. Article 76 of UNCLOS de nes the FoS as ". . . the point of maximum change in the gradient at its base." Currently Caris Lots is the most widely used software which incorporates public domain data to determine the FoS as a step towards defining the offshore extent of an extended continental shelf. In this software, existing methods to compute the FoS are often subjective, typically involving an operator choosing the best perceived foot point during consideration of a two dimensional profile of the continental slope. These foot points are then joined by straight lines to form the foot line to be used in the desk top study (feasibility study). The purpose of this thesis is to establish a semi-automated and mathematically based three dimensional method for determination of the FoS using South African data as a case study. Firstly, a general background of UNCLOS is given (with emphasis on Article 76), including a brief discussion of the geological factors that influence the characteristics of a continental shelf and thus factors that could influence the determination of the FoS. Secondly, a mathematical method for determination of the surfaces of extremal curvature (on three dimensional data), originally proposed by Vanicek and Ou in 1994, is detailed and applied to two smooth, hypothetical sample surfaces. A discussion of the bathymetric data to be used for application introduces the factors to be taken into account when using extensive survey data as well as methods to process the raw data for use. The method is then applied to two sets of gridded bathymetric data of differing resolution for four separate regions around the South African coast. The ridges formed on the resulting surfaces of maximum curvature are then traced in order to obtain a foot line definition for each region and each resolution. The results obtained from application of the method are compared with example foot points provided by the subjective two dimensional method of computation within the Caris Lots software suite. A comparison of the results for the different resolutions of data is included to provide insight as to the effectiveness of the method with differing spatial coarseness of data. Finally, an indication of further work is provided in the conclusion to this thesis, in the form of a number of recommendations for possible adaptations of the mathematical and tracing methods, and improvements thereof. / AFRIKAANSE OPSOMMING: Die Verenigde Nasies se Konvensie oor die Wet van die See (UNCLOS) bied 'n geleentheid aan kwalifiserende state wat ondertekenaars van die Konvensie is om aanspraak te maak op uitgebreide maritieme gebied. Die geleentheid om op uitgebreide gebied aanspraak te maak berus op die veronderstelling dat 'n kontinentale tafel in sekere gevalle tot buite die tradisioneel afgebakende 200 seemyl eksklusiewe ekonomiese zone (EEZ) strek. In sulke gevalle het 'n suksesvolle aanspraak die gevolg dat die staat soewereine reg oor die lewende en nie-lewende bronne van die seevloer en ondergrond verkry, sowel as die inwonende spesies van die gebied buite die EEZ waarop aanspraak gemaak word. Die voet van die kontinentale tafel (FoS) moet vasgestel word as een van die bepalende kriteria vir afbakening van die aanspraak waar die kontinentale tafel tot buite die EEZ strek. Artikel 76 van UNCLOS defineer die FoS as ". . . die punt van maksimale verandering in die helling by sy basis." Die mees algemeen gebruikte rekenaar sagteware wat openbare domein data aanwend om die voet van die helling te bepaal, is tans "Caris Lots." Die metodes wat in die program gebruik word om die voet van die helling te bepaal, is dikwels subjektief en berus tipies op 'n operateur se keuse van die beste afgeskatte punt van die voet van die helling uit 'n oorweging van 'n twee dimensionele profiel van die kontinentale tafel. Die berekende voet-punte word dan deur middel van reguit lyne verbind om 'n hellingsvoetlyn te vorm. Hierdie voetlyn kan dan in die Suid-Afrikaanse lessenaarstudie (doenlikheidstudie) oor die bepaling van die voet van die kontinentale tafel gebruik word. Die doel van hierdie verhandeling is om 'n semi-outomatiese en wiskundig gebaseerde drie-dimensionele metode te beskryf vir die vasstelling van die FoS, deur as 'n gevallestudie van Suid-Afrikaanse data gebruik te maak. 'n Algemene agtergrond van UNCLOS, met beklemtoning van Artikel 76, word eerstens gegee. 'n Kort bespreking van die geologiese faktore wat die kontinentale tafel beïnvloed en wat gevolglik 'n invloed kan hê op die vasstelling van die voet van die helling, is ingesluit. Tweedens word 'n wiskundige metode, wat oorspronklik in 1994 deur Vanicek en Ou voorgestel is, vir bepaling van die oppervlaktes van maksimale kromming (gebaseer op drie-dimensionele data) in detail bespreek en 'n voorbeeld van 'n toepassing op twee gladde, denkbeeldige oppervaktes word beskryf. Die faktore wat in ag geneem moet word wanneer omvattende dieptemeting data gebruik word, en die metodes wat gebruik word om die rou data te verwerk, word ingelei deur 'n bespreking van die aard van die dieptemeting data wat gebruik is. Die metode word dan toegepas op twee stelle geruite dieptemeting data van verskillende resolusies vir vier afsonderlike streke om die Suid-Afrikaanse kus. Die riwwe wat op die resulterende oppervlaktes van maksimale kromming gevorm word, word dan nagetrek ten einde 'n lyndefinisie van die voet van die kontinentale tafel vir elke streek teen elke resolusie te bepaal. Die resultate verkry uit toepassings van die metode word vergelyk met hellingsvoetpunte soos bepaal deur die subjektiewe twee dimensionele berekeningsmetode in die "Caris Lots" rekenaar-program. 'n Vergelyking van die resultate vir die verskillende data resolusies word ingesluit om die doeltreffendheid van die metode met betrekking tot die hantering van verskillende ruimtelike data resolusies te ondersoek. 'n Aanduiding van verdere werk, bestaande uit 'n aantal aanbevelings vir moontlike aanpassings en verbeterings van die wiskundige en natrek metodes, word ten slotte in die gevolgtrekking van die verhandeling verskaf.

This thesis is primarily focused on problems in extremal combinatorics, although we will also consider some questions of analytic and algorithmic nature. The d-dimensional hypercube is the graph with vertex set {0,1}^d where two vertices are adjacent if they differ in exactly one coordinate. In Chapter 2 we obtain an upper bound on the 'saturation number' of Q_m in Q_d. Specifically, we show that for m ≥ 2 fixed and d large there exists a subgraph G of Q_d of bounded average degree such that G does not contain a copy of Q_m but, for every G' such that G ⊊ G' ⊆ Q_d, the graph G' contains a copy of Q_m. This result answers a question of Johnson and Pinto and is best possible up to a factor of O(m). In Chapter 3, we show that there exists ε > 0 such that for all k and for n sufficiently large there is a collection of at most 2^{(1-ε)k} subsets of [n] which does not contain a chain of length k+1 under inclusion and is maximal subject to this property. This disproves a conjecture of Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós. We also prove that there exists a constant c ∈ (0,1) such that the smallest such collection is of cardinality 2^{(1+o(1))^ck} for all k. In Chapter 4, we obtain an exact expression for the 'weak saturation number' of Q_m in Q_d. That is, we determine the minimum number of edges in a spanning subgraph G of Q_d such that the edges of E(Q_d)\E(G) can be added to G, one edge at a time, such that each new edge completes a copy of Q_m. This answers another question of Johnson and Pinto. We also obtain a more general result for the weak saturation of 'axis aligned' copies of a multidimensional grid in a larger grid. In the r-neighbour bootstrap process, one begins with a set A₀ of 'infected' vertices in a graph G and, at each step, a 'healthy' vertex becomes infected if it has at least r infected neighbours. If every vertex of G is eventually infected, then we say that A₀ percolates. In Chapter 5, we apply ideas from weak saturation to prove that, for fixed r ≥ 2, every percolating set in Q_d has cardinality at least (1+o(1))(d choose r-1)/r. This confirms a conjecture of Balogh and Bollobás and is asymptotically best possible. In addition, we determine the minimum cardinality exactly in the case r=3 (the minimum cardinality in the case r=2 was already known). In Chapter 6, we provide a framework for proving lower bounds on the number of comparable pairs in a subset S of a partially ordered set (poset) of prescribed size. We apply this framework to obtain an explicit bound of this type for the poset 𝒱(q,n) consisting of all subspaces of 𝔽_qⁿordered by inclusion which is best possible when S is not too large. In Chapter 7, we apply the result from Chapter 6 along with the recently developed 'container method,' to obtain an upper bound on the number of antichains in 𝒱(q,n) and a bound on the size of the largest antichain in a p-random subset of 𝒱(q,n) which holds with high probability for p in a certain range. In Chapter 8, we construct a 'finitely forcible graphon' W for which there exists a sequence (ε_i)^∞_i=1 tending to zero such that, for all i ≥ 1, every weak ε_i-regular partition of W has at least exp(ε_i^-2/2^{5log∗ε_i^-2}) parts. This result shows that the structure of a finitely forcible graphon can be much more complex than was anticipated in a paper of Lovász and Szegedy. For positive integers p,q with p/q ❘≥ 2, a circular (p,q)-colouring of a graph G is a mapping V(G) → ℤ_p such that any two adjacent vertices are mapped to elements of ℤ_p at distance at least q from one another. The reconfiguration problem for circular colourings asks, given two (p,q)-colourings f and g of G, is it possible to transform f into g by recolouring one vertex at a time so that every intermediate mapping is a p,q-colouring? In Chapter 9, we show that this question can be answered in polynomial time for 2 ≤ p/q < 4 and is PSPACE-complete for p/q ≥ 4.