We study two sets of models: independent percolation models in half spaces Zᵈ⁻¹ x Z₊, and Ising/Potts models as well as the Fortuin-Kasteleyn (FK) random cluster models on branching planes T x Z, where Z is the one-dimensional lattice, Z₊ = {0,1,2,...} and T is a Bethe lattice. We prove that for independent percolation in half spaces, the infinite cluster is unique whenever it exists. For the Ising/Potts models on branching planes, there are (at least) two phase transitions; that is, there exist(s) a unique Gibbs state, tree-like nonunique Gibbs states or plane-like nonunique Gibbs states corresponding to high temperature, intermediate temperature or low temperature. In the low temperature plus phase, the plus infinite cluster is unique and it "traps" the space T x Z and prevents co-existence of the minus infinite cluster. For the FK random cluster models (which are dependent percolation models) on T x Z, the number of infinite (open) clusters may be zero, infinity or one depending on the value of p--the probability of each bond being open. This is an extension of Grimmett and Newman's results for independent percolation on T x Z. We also prove that both the independent percolation model and the FK random cluster models satisfy a finite island property when p is close to 1. Chapter 1 is an introduction. Chapter 2 contains the proof of the uniqueness theorem for independent percolation in half spaces. The proof utilizes only a large deviation estimate and translation invariance of the models along the hyperplane Zᵈ⁻¹ x {0}. The Ising/Potts models and the FK random cluster models on the branching planes are studied in Chapter 3. The methods are to use the FK representation of Ising/Potts systems as dependent percolation models to carry over Grimmett and Newman's results for independent percolation to the Ising/Potts models. However, in order to prove the plane-like behavior of the Ising/Potts models, the corresponding results for independent percolation are not sufficient and this led us to investigate independent percolation again and prove a new finite island property. Chapters 2 and 3 are independent. Readers with basic knowledge of percolation and Ising models can omit chapter 1 and read chapters 2 and 3 directly.

This work is concerned with some finiteness statements and explicit computations in the arithmetic 0/ algebraic curves. For an excellent introduction to this topic see Barry Mazur's article A1'ithmetic on Curves ([Mz]). We begin with an example: Mazur's article discusses Gerd Faltings' 1983 proof of the Mordell Conjecture, which says that any curve of genus at least 2 over a number field J( has only finitely many J( -rational points. This implies, for instance, that the Fermat curves FN given in projective coordinates by Xᴺ + Yᴺ = Zᴺ, N ≥ 4 defined over the rational numbers Q have only finitely many Q-rational points, and only finitely many J( -rational points for any number field K. The famous conjecture of Fermat (Fermat's Last Theorem) states that in fact these curves possess only the trivial Q-rational points, ie. those for which one of the coordinates is O. (This conjecture is now known to follow from the conjecture of Taniyama on elliptic curves thanks to work of Ribet and Frey, and it appears to be, as of June 1993, a Theorem of Andrew Wiles.) Now for number fields J( (finite extensions of Q) one can also ask for an explicit description of the set of J( -rational points F(N)(K). The most natural way to proceed is to start with fields of low degree d = [K : Q]. We will say that F(N)(K) is non-trivial if it contains at least one point which is not Q-rational. One finds the somewhat unexpected fact (which remains true for any smooth plane curve of degree N ≥ 7): Theorem(Debarre-Klassen, [DK]) For N ≠ 16, there are only finitely many number fields K with degree d = [K : Q] ≤ N - 2 such that F(N)(K) is non-trivial. This implies that for each N there is a finite list of number fields of degree at most N - 2, and a finite set of non-trivial points on F(N) with coordinates in these fields. If we take N ≥ 5 a prime number, the only known examples of such points are given by the intersection of F(N) with the line X + Y = Z, which contains the three rational points (see section 2.4). In Theorem 12 we extend the known results for N = 5. First we show that the Jacobian of F₅ has exactly 25 or 125 rational points. We then produce a group of 25 rational points, and show that if these are all, then there are no number fields K of degree 3 over Q with F(N)(K) non-trivial. The above Theorem is an application of a result of Faltings (see [Fa2]) , combined with other work of Coppens and Abramovich (see [Cp], and [Ab].) These results and background material are the subject of chapters 1 and 2. In the first chapter we explain how finiteness of points of low degree is reduced to finiteness of rational points on certain symmetric product varieties. We also give an example due to Faddeev on the Fermat quartic. In chapter two, we describe results of Abramovich, Coppens, and Frey, and give a proof of the above result for F(N) , N prime, in Theorem 7. We also give a description of all points of degree ≤ 6 on the Fermat quintic. In chapter 3, we begin a study of Coleman's p-adic Abelian integrals applied to the symmetric products of curves. The goal is to mimic Coleman's paper Effective Chabauty [Co2], where he uses these integrals to produce effective bounds on the numbers of rational points on certain curves with low rank. The inspiration to pursue these topics was supplied by the papers of McCallum ([Mcl] and [Mc2]) where the techniques of [Co2] are applied to Fermat curves in such a way as to elucidate some of the very deep connections between the class groups of cyclotomic fields, and the Mordell-Weil rank of the Jacobian, and numbers of rational points on these curves. In this chapter we prove a finiteness result for rational points away from a closed subvariety (a canonical divisor), under certain rank assumptions. Then we also give a criterion for when there is at most one rational point in a given residue class. Although this does not yield specific bounds on numbers of points of low degree, we hope to apply the results in this chapter to explicit examples in future papers.

Given a triangulated piecewise-flat surface and a function on the vertices we can define the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. To find the Delaunay triangulation of the piecewise flat surface, we modify the triangulation by a sequence of edge flips, called the flip algorithm, which transform an edge which is not Delaunay into one that is Delaunay. It is known that the flip algorithm works in the plane as well as for a piecewise-flat surface, where we have to ensure that only finitely many triangulations are possible.When the vertices of a piecewise-flat surface have weights, we want to find the weighted Delaunay triangulation using a flip algorithm. In this dissertation, we prove that the maximum edge length during the algorithm is bounded, which guarantees that there are finitely many triangulations. Thus the flip algorithm terminates and the resulting triangulation is weighted Delaunay.Additionally, we give a new way to find what we call the relaxed weighted Delaunay on a flat surface.

By a theorem of Mnëv, every possible algebraic singularity occurs in certain incidence subspaces of (P²)ⁿ x (P²)ᵐ for some n, m. These incidence subspaces are defined by conditions that include (1) certain points must lie on certain lines, (2) i of the n points coincide, and (3) j of the m lines coincide. To capture these incidence spaces, we define a configuration space Xᵒ(n) ⊂ (P²)n x (P²)^(n/2) along with its closure X(n), which is singular for n ≥ 3. The open strata Xᵒ(n) parameterizes configurations of n points in P² that are in general linear position. When n = 3, a smooth compactification of Xn was discovered by Schubert and refined by Semple in 1954. In order to desingularize X₃, we add the net of conics through the three points. This is equivalent to blowing up at one of three strata θᵢ (Theorem 2.12). Some of the possible blowups that would desingularize X₃ fit into the framework of the Atiyah flop. There are forgetful morphisms Fᵢ : X₄ → X₃ that omit the ith point and the three lines incident on the ith point. The space X₄ is homogeneously covered by open affines. Each open affine has a coordinate ring generated by coordinates pulled back from a flag variety and edge coordinates indexed by the edges of a graph Γ (Proposition 4.33). The relations are given by certain linear relations, quadratic relations that come from the triangular subconfigurations, and cubic relations indexed by hexagons in the graph Γ (Proposition 4.37). These generators and relations are used to prove that blowing up X₄ at F(j)⁻¹(ε); F(k)⁻¹(ε); F(l)⁻¹(ε); and Fᵢ⁻¹(τ), where ε and τ are strata in X₃, results in a smooth space (Theorem 5.24) whose boundary consists of smooth divisorial components (Theorem 5.29).

<p>Square principles are statements about an important class of infinitary combinatorial objects. They may hold or fail to hold at singular cardinals depending on our large cardinal assumptions, but their precise consistency strengths are not yet known. </p><p> In this paper I present two theorems which greatly lower the known upper bounds of the consistency strengths of the failure of several square principles at singular cardinals. I do this using forcing constructions. First, using a quasicompact* cardinal I construct a model of the failure of &not;&square;([special characters omitted], &lt; &omega;). Second, using a cardinal which is both subcompact and measurable, I construct a model of &square;_&kappa;,2 + &not;&square;_&kappa; in which &kappa; is singular. This paves the way for several natural extensions of these results.</p>

This dissertation studies two different techniques for analyzing control systems whose dynamics include impulses, or more specifically, are measure-driven. In such systems, the state trajectories will have discontinuities corresponding to the atoms of the Borel measure driving the dynamics, and these discontinuities require further definition in order for the control system to be treated with the broad range of results available to non-impulsive systems. Both techniques considered involve a reparameterization of the system variables including state, time, and controls. The first method is that of the graph completion, which provides an explicit reparameterization of the time and state variables. The reparameterization is continuous, which allows for the analysis of the system within classical control theory, yet it retains enough information about the discontinuous, or impulsive, trajectories that the results of such analyses may be interpreted for the original impulsive system. We utilize this reparameterization to formulate equivalent solution concepts between impulsive differential inclusions and impulsive differential equations. We also demonstrate that the graph completion is generally equivalent to a solution concept established for a neural spiking model, and make use of a specific such model as a numerical example. The second method considered is similar to the graph completion but differs in that it utilizes implicit reparameterizations of all variables considered as families of functions which meet continuity and other requirements. This is particularly beneficial to optimal control problems as the choices of controls, impulsive and non-impulsive, may be varied within the optimization problem and analysis thereof. Necessary conditions for optimal control problems of Mayer form with fixed end time have been established under this reparameterization technique, and we extend these necessary conditions in a general context to a Mayer problem with free end time. Corollary to this, we deduce necessary conditions for a Bolza problem and a minimum time problem for impulsive control systems. Much of these results are obtained through reformulation techniques.

We propose an Adaptive Stochastic Conjugate Gradient (ASCG) optimization algorithm for temporal medical image registration. This method combines the advantages of Conjugate Gradient (CG) method and Adaptive Stochastic Gradient Descent (ASGD) method. The main idea is that the search direction of ASGD is replaced by stochastic approximations of the conjugate gradient of the cost function. In addition, the step size of ASCG is based on the approximation of the Lipschitz constant of the stochastic gradient function. Thus, this algorithm could maintain the good properties of the conjugate gradient method, meanwhile it uses less gradient computation time per iteration and adjusts the step size adaptively as the ASGD method. As a result, this algorithm takes less CPU time than the previous ASGD method. We demonstrate the efficiency of our algorithm on the public available 4D Lung CT data and our clinical Lung/Tumor CT data using the general 4D image registration model. We compare the ASCG with several existing iterative optimization strategies: steepest gradient descent method, conjugate gradient method, Quasi-Newton method (LBFGS) and adaptive stochastic gradient descent method. Our preliminary results indicate that our ASCG algorithm achieves 22% higher accuracy on the POPI dataset and it also performs better than existing methods on other datasets(DIR-Lab dataset and our clinical dataset). Furthermore, we demonstrate that compared with other methods, our ASCG algorithm is more robust to image noises.

This dissertation begins with an introduction to matroids and graphs. In the first chapter, we develop matroid and graph theory definitions and preliminary results sufficient to discuss the problems presented in the later chapters. These topics include duality, connectivity, matroid minors, and Cunningham and Edmonds's tree decomposition for connected matroids. One of the most well-known excluded-minor results in matroid theory is Tutte's characterization of binary matroids. The class of binary matroids is one of the most widely studied classes of matroids, and its members have many attractive qualities. This motivates the study of matroid classes that are close to being binary. One very natural such minor-closed class Z consists of those matroids M such that the deletion or the contraction of e is binary for all elements e of M. Chapter 2 is devoted to determining the set of excluded minors for Z. Duality plays a central role in the study of matroids. It is therefore natural to ask the following question: which matroids guarantee that, when present as minors, their duals are present as minors? We answer this question in Chapter 3. We also consider this problem with additional constraints regarding the connectivity and representability of the matroids in question. The main results of Chapter 3 deal with 3-connected matroids.

In this thesis we study the problem of secure key establishment, motivated by the construction of secure channels protocols to protect information transmitted over an open network. In the past, the purported security of a key establishment protocol was justified if it could be shown to withstand popular attack scenarios by heuristic analysis. Since this approach does not account for all possible attacks, the security guarantees are limited and often insufficient. This thesis examines the provable security approach to the analysis of key establishment protocols. We present the security models and definitions developed in 2001 and 2002 by Canetti and Krawczyk, critique the appropriateness of the models, and provide several security proofs under the definitions. In addition, we consider the importance of the key compromise impersonation resilience property in the context of these models. We list some open problems that were encountered in the study.

The edge space of a finite graph <em>G</em> = (<em>V</em>, <em>E</em>) over a field F is simply an assignment of field elements to the edges of the graph. The edge space can equally be thought of us an |<em>E</em>|-dimensional vector space over F. The cycle space and bond space are the subspaces of the edge space generated by the cycle and bonds of the graph respectively. It is easy to prove that the cycle space and bond space are orthogonal complements. <br /><br /> Unfortunately many of the basic results in finite dimensional vector spaces no longer hold in infinite dimensions. Therefore extending the cycle and bond spaces to infinite graphs is not at all a trivial exercise. <br /><br /> This thesis is mainly concerned with the algebraic properties of the cycle and bond spaces of a locally finite, infinite graph. Our approach is to first topologize and then compactify the graph. This allows us to enrich the set of cycles to include infinite cycles. We introduce two cycle spaces and three bond spaces of a locally finite graph and determine the orthogonality relations between them. We also determine the sum of two of these spaces, and derive a version of the Edge Tripartition Theorem.