Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1973. / Vita. / Bibliography: leaves 132-136. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (p. 183-190) and Index. / This thesis studies surfaces which minimize area, subject to a fixed boundary and to a free boundary with length constraint. Based on physical experiments, I make two conjectures. First, I conjecture that minimizers supported on generic wires have finitely many surface components. I approach this conjecture by proving that surface components of near-wire minimizers are Lipschitz graphs in wire Frenet coordinates, and appear near maxima of wire curvature. Second, I conjecture and prove that surface components of near-wire minimizers are C1 at corners where the thread touches the wire interior. Moreover, the limit of the surface normal field is the Frenet binormal of the wire at the corner point. This shows local wire geometry dominates global wire geometry in influencing the surface corner. Third, I show that these two conjectures are related: assuming additional regularity up to the corner, the finiteness conjecture follows. / by Benjamin K. Stephens. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (p. 109-111). / In this thesis, I report the results of a combined experimental and theoretical investigation of a journal bearing, specifically, a cylinder suspended in a viscous fluid housed within a cylindrical shell, rolling down an incline under the influence of gravity. Particular attention is given to rationalizing the distinct modes of motion observed. We performed a series of experiments in which the inner cylinder density and the fluid viscosity were varied. Three distinct types of behavior were observed. First, in what we shall call the "rocking" mode, after an initial settling period, the shell rocks back and forth without moving down the ramp. Second, we observed "slow, quasi-steady rolling"; this mode is characterized by the system proceeding down the hill at essentially a constant velocity. Finally, the cylinders roll down the incline with constant acceleration; we shall call this mode "unbounded acceleration." An accompanying theoretical model is developed and enables us to rationalize the rocking and accelerating modes. In the rocking solutions, potential and kinetic energy are dissipated in the fluid as the inner cylinder approaches the bottom of the outer cylinder. / (cont.) In the accelerating solutions, the whole system moves as a solid body so that no dissipation occurs and potential energy is continually converted into kinetic energy. In order to understand the quasi-steady motion, we analyze the motion of a similar system: a metal cylinder is placed inside a larger plastic cylinder filled with fluid and attached to a motor which fixes the larger cylinder's rotation rate. Our observations of this system, specifically, the differences between experiments and theory lead us to consider the effect of internal friction due to surface roughness. The resulting model's predictions are well supported by our observations. Finally, to rationalize the slow, quasi-steady rolling motion of the system, we incorporate surface roughness and cavitation into the theoretical model. These effects provide a restoring force on the inner cylinder; however, we find that surface roughness is the dominant effect. / by David Paul Vener. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 99-103). / In the first part of this thesis, we give a classification of all self-similar solutions to the curve shortening flow in the Euclidean plane R² and discuss basic properties of the curves. The problem of finding the curves is reduced to the study of a twodimensional system of ODEs with two parameters that determine the type of the self-similar motion. In the second part, we describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at their limiting behaviour as the pitch of the helicoidal motion goes to 0 and compare it with the limiting behaviour of the classical helicoidal minimal surfaces. Finally, we give a classification of the immersed cylinders in the family of constant mean curvature helicoidal surfaces. In the third part, we introduce the mean curvature flow of curves in the Minkowski plane R¹,¹ and give a classification of all the self-similar solutions. In addition, we demonstrate five non-self-similar exact solutions to the flow. / by Höskuldur Pétur Halldórsson. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 157-165). / Many real-world systems involve distributed and selfish agents who optimize their own objective function. In these systems, we need to design efficient mechanisms so that system-wide objective is optimized despite agents acting in their own self interest. In this thesis, we develop approximation algorithms and decentralized mechanisms for various combinatorial optimization problems in such systems. First, we investigate the distributed caching and a general set of assignment problems. We develop an almost tight LP-based ... approximation algorithm and a local search ... approximation algorithm for these problems. We also design efficient decentralized mechanisms for these problems and study the convergence of the corresponding games. In the following chapters, we study the speed of convergence to high quality solutions on (random) best-response paths of players. First, we study the average social value on best response paths in basic-utility, market sharing, and cut games. Then, we introduce the sink equilibrium as a new equilibrium concept. We argue that, unlike Nash equilibria, the selfish behavior of players converges to sink equilibria and all strategic games have a sink equilibrium. To illustrate the use of this new concept, we study the social value of sink equilibria in weighted selfish routing (or weighted congestion) games and valid-utility (or submodular-utility) games. In these games, we bound the average social value on random best-response paths for sink equilibria.. Finally, we study cross-monotonic cost sharings and group-strategyproof mechanisms. / (cont.) We study the limitations imposed by the cross-monotonicity property on cost-sharing schemes for several combinatorial optimization games including set cover and metric facility location. We develop a novel technique based on the probabilistic method for proving upper bounds on the budget-balance factor of cross-monotonic cost sharing schemes, deriving tight or nearly-tight bounds for these games. At the end, we extend some of these results to group-strategyproof mechanisms. / by Vahab S. Mirrokni. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 109-110). / Let C be a smooth projective curve over an algebraically closed field k of sufficiently large characteristic. Let G be a semisimple algebraic group over k and let GV be its Langlands dual group over k. Denote by BunG the moduli stack of G-bundles on C and LocSysGv the moduli stack of Gv-local systems on C. Let DBunG be the sheaf of crystalline differential operator algebra on BunG. In this thesis I construct an equivalence between the derived category D(QCoh(LocSys~v)) of quasi-coherent sheaves on some open subset LocSysov C LocSysGv and derived category D(DOunG mod) of modules over some localization DBunG of DBunG. This generalizes the work of Bezrukavnikov-Braverman in the GL, case. / by Tsao-Hsien Chen. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 149-164). / In machine learning, the choice of space in which to represent our data is of vital importance to their effective and efficient analysis. In this thesis, we develop approaches to address a number of problems in representation learning. We employ deep learning as means of sculpting our representations, and also develop improved representations for deep learning models. We present contributions that are based on five papers and make progress in several different research directions. First, we present techniques which leverage spatial and relational structure to achieve greater computational efficiency of model optimization and query retrieval. This allows us to train distance metric learning models 5-30 times faster; optimize convolutional neural networks 2-5 times faster; perform content-based image retrieval hundreds of times faster on codes hundreds of times longer than feasible before; and improve the complexity of Bayesian optimization to linear in the number of observations in contrast to the cubic dependence in its naive Gaussian process formulation. Furthermore, we introduce ideas to facilitate preservation of relevant information within the learned representations, and demonstrate this leads to improved supervision results. Our approaches achieve state-of-the-art classification and transfer learning performance on a number of well-known machine learning benchmarks. In addition, while deep learning models are able to discover structure in high dimensional input domains, they only offer implicit probabilistic descriptions. We develop an algorithm to enable probabilistic interpretability of deep representations. It constructs a transformation to a representation space under which the map of the distribution is approximately factorized and has known marginals. This allows tractable density estimation and.inference within this alternate domain. / by Oren Rippel. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 48-49). / This thesis deals with AS-regular algebras, first defined by Michael Artin and William Schelter in Graded Algebras of Global Dimension 3. All such algebras of dimension three have been classified, but the corresponding problem in higher dimensions remains open. We construct new examples of four dimensional AS-regular algebras, and provide some information about their module structure. Results are provided for proving the regularity of such algebras. In addition we classify the AS-regular algebras of dimension four satisfying certain conditions. / by Ian Caines. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (p. 44-45). / by C. Kenneth Fan. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 143-149). / In this thesis, I study two examples of obstructions to rational and integral points on varieties. The first concerns the S-unit equation, which asks for solutions to x + y = 1 with x and y both S-units, or units in Z = Z[1/S]. This is equivalent to finding the set of Z-points of P1 \ {0, 1, [infinity]}. We follow work of Dan-Cohen-Wewers and Brown in applying a motivic version of the non-abelian Chabauty's method of Minhyong Kim to find polynomials in p-adic polylogarithms that vanish on this set of integral points. More specifically, we extend the computations already done by Dan-Cohen-Wewers to the integer ring Z[1/3], and we provide some significant simplifications to a previous algorithm of Dan-Cohen, especially in the case of Z[1/S]. One of the reasons for doing this is to verify cases of Kim's conjecture, which states that these p-adic functions precisely cut out the set of integral points. This is joint work with Ishai Dan-Cohen. The second is about obstructions to the local-global principle. The étale Brauer-Manin obstruction of Skorobogatov can be used to explain the failure of the local-global principle for many algebraic varieties. In 2010, Poonen gave the first example of failure of the local-global principle that cannot be explained by the étale Brauer-Manin obstruction. Further obstructions such as the étale homotopy obstruction and the descent obstruction are unfortunately equivalent to the étale Brauer-Manin obstruction. However, Poonen's construction was not accompanied by a definition of a new, finer obstruction. Here, we present a possible definition for such an obstruction by applying the Brauer-Manin obstruction to each piece of every stratification of the variety. We prove that this obstruction is necessary and sufficient, over imaginary quadratic fields and totally real fields unconditionally, and over all number fields conditionally on the section conjecture. This is part of a joint project with Tomer Schlank. / by David Alexander Corwin. / Ph. D.