Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 85-86). / Abstract In polynomial regression ... . In this thesis, I developed a residual based test, the turning point test for residuals, which tests the hypothesis that the kth order polynomial regression holds with ... while the alternative can simply be the negation or be more specific, e.g., polynomial regression with order higher than k. This test extends the rather well known turning point test and requires approximation of residuals by errors for large n. The simple linear regression model, namely k = 1, will be studied in most detail. It is proved that the expected absolute difference of numbers of turning points in the errors and residuals cannot become large and under mild conditions becomes small at given rates for large n. The power of the test is then compared with another residual based test, the convexity point test, using simulations. The turning point test is shown to be more powerful against quadratic alternatives. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 63-64). / We study affine permutation diagrams and their labellings with positive integers. Balanced labellings of a Rothe diagram of a finite permutation were defined by Fomin- Greene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column-strict balanced labellings is the affine Stanley symmetric function which plays an important role in the geometry of the affine Grassmannian. Furthermore, we define set-valued balanced labellings in which the labels are sets of positive integers, and we investigate the relations between set-valued balanced labellings and nilHecke words in the nilHecke algebra. A signed generating function of column-strict set-valued balanced labellings is shown to coincide with the affine stable Grothendieck polynomial which is related to the K-theory of the affine Grassmannian. Moreover, for finite permutations, we show that the usual Grothendieck polynomial of Lascoux-Schiitzenberger can be obtained by flagged column-strict set-valued balanced labellings. Using the theory of balanced labellings, we give a necessary and sufficient condition for a diagram to be a permutation diagram. An affine diagram is an affine permutation diagram if and only if it is North-West and admits a special content map. We also characterize and enumerate the patterns of permutation diagrams. / by Taedong Yun. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 63). / In this thesis I study the Seiberg-Witten equations on the product of a genus g surface [Sigma] and a circle. I exploit S1 invariance to reduce to the vortex equations on [Sigma] and thus completely describe the Seiberg-Witten monopoles. In the case where the monopoles are not Morse-Bott regular, I explicitly perturb the equations to obtain such a situation and thus find a candidate for the chain complex that calculates the Seiberg-Witten Floer homology groups. / by William Manuel Lopes. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 191-197). / The world is filled with objects that appear to follow some perfect model. A sleeping baby might look still and a house's roof .should be straight. However, both the baby and the roof can deviate subtly from their ideal models of perfect stillness and perfect straightness. These deviations can reveal important information like whether the baby is breathing normally or whether the house's roof is sagging. In this dissertation, we make the observation that these subtle deviations produce a visual signal that while invisible to the naked eye can be extracted from ordinary and ubiquitous images and videos. We propose new computational techniques to reveal these subtle deviations by producing new images and videos, in which the tiny deviations have been magnified. We focus on magnifying deviations from two ideal models: perfect stillness and perfect geometries in space. In the first case, we leverage the complex steerable pyramid, a localized version of the Fourier transform, whose notion of local phase can be used to process and manipulate small motions or changes from stillness in videos. In the second case, we find hidden geometric deformations in images by localizing edges to sub-pixel precision. In both cases, we experimentally validate that the tiny deviations we magnify are indeed real, comparing them to alternative ways of measuring tiny motions and subtle geometric deformations in the world. We also give a careful analysis of how noise in videos impacts our ability to see tiny motions. Additionally, we show the utility of revealing hidden deviations in a wide variety of fields, such as biology, physics and structural analysis. / by Neal Wadhwa. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Cataloged from student-submitted PDF version of thesis. / Includes bibliographical references (pages 297-302). / In 1983, Ira Gessel introduced the ring of quasisymmetric functions (QSym), an extension of the ring of symmetric functions and nowadays one of the standard examples of a combinatorial Hopf algebra. In this thesis, I elucidate three aspects of its theory: 1) Gessel's P-partition enumerators are quasisymmetric functions that generalize, and share many properties of, the Schur functions; their Hopf-algebraic antipode satisfies a simple and explicit formula. Malvenuto and Reutenauer have generalized this formula to quasisymmetric functions "associated to a set of equalities and inequalities". I reformulate their generalization in the handier terminology of double posets, and present a new proof and an even further generalization in which a group acts on the double poset. 2) There is a second bialgebra structure on QSym, with its own "internal" comultiplication. I show how this bialgebra can be constructed using the Aguiar-Bergeron- Sottile universal property of QSym by extending the base ring; the same approach also constructs the so-called "Bernstein homomorphism", which makes any connected graded commutative Hopf algebra into a comodule over this second bialgebra QSym. 3) I prove a recursive formula for the "dual immaculate quasisymmetric functions" (a certain special case of P-partition enumerators) conjectured by Mike Zabrocki. The proof introduces a dendriform algebra structure on QSym. Two further results appearing in this thesis, but not directly connected to QSym, are: 4) generalizations of Whitney's formula for the chromatic polynomial of a graph in terms of broken circuits. One of these generalizations involves weights assigned to the broken circuits. A formula for the chromatic symmetric function is also obtained. 5) a proof of a conjecture by Bergeron, Ceballos and Labbé on reduced-word graphs in Coxeter groups (joint work with Alexander Postnikov). Given an element of a Coxeter group, we can form a graph whose vertices are the reduced expressions of this element, and whose edges connect two reduced expressions which are "a single braid move apart". The simplest part of the conjecture says that this graph is bipartite; we show finer claims about its cycles. / by Darij Grinberg. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 71-72). / In the first part of this dissertation, we provide the first example of a singular energy minimizing free boundary. This singular solution occurs in dimension 7 and higher, and in fact it is conjectured that there are no singular minimizers in dimension lower than 7. Our example is the analogue of the 8-dimensional Simons cone in the theory of minimal surfaces. The minimality of the Simons cone is closely related to the existence of a complete minimal graph in dimension 9, which is not a hyperplane. The first step toward solving the analogous problem in the free boundary context, consists in developing a local existence and regularity theory for monotone solutions to a free boundary problem. This is the objective of the second part of our thesis. We also provide a partial result in the global context.. / by Daniela De Silva. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (p. 148-150). / by Yuan Ma. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1990. / Includes bibliographical references (leaves 78-80). / by José M. González-Barrios. / Ph.D.

Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1951. / Vita. / Bibliography: leaves 170-174. / by Richard Horace Battin. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 74-78). / Stanley's Cauchy identity for Jack symmetric functions defines a Jack measure, a model random partitions for every analytic real function v(w) on the unit circle and parameters E2 < 0 < E1. Jacks are eigenfunctions of the Hamiltonian ... of the quantum Benjamin-Ono equation with periodic boundary conditions, dispersion and quantization corresponding E1 + E2 and -E1E2 respectively. From this point of view, Jack measures are the random energy distribution of a coherent state around a classical configuration v(w). Taking E2 --> 0 <-- E1 at a comparable rate [beta]/2, we prove that the slopes of the profiles of the random partition concentrate on a limit shape independent of [beta], the push-forward of the uniform measure on the circle along v. This is the conserved density of the classical inviscid Hopf hierarchy on the circle, following Dubrovin (2014). At the quantum Hopf hierarchy ([beta] = 2), we recover Okounkov's limit shape for Schur measures (2003) as a verification of the correspondence principle. Our main result is the computation of macroscopic fluctuations of random profiles around the limit shape: they converge to the push-forward along v of the restriction to the circle of a Gaussian free field on the upper half-plane whose covariance is independent of [beta]. At [beta] = 2, our result matches Breuer-Duits' central limit theorem (2013) for Borodin's biorthogonal ensembles. Our limit theorems follow from a diagrammatic all-order convergent expansion of joint cumulants of linear statistics over "ribbon paths". This expansion has the same form as the 1/N refined topological asymptotic expansion over ribbon graphs on surfaces for [beta]-ensembles on the line in one-cut potentials V due to Chekhov-Eynard (2006) and Borot-Guionnet (2012). Our analysis relies on the Lax operator L for the quantum Benjamin-Ono hierarchy introduced by Nazarov-Sklyanin (2013). L is expressed through Toeplitz operators whose symbols are affine Kac-Moody currents for gl1. We use the spectral shift function of L to construct a generating function y(u) of local Hamiltonians commuting with y3. This explicit y(u) is a special case of the y-operator defined implicitly by functional calculus in Nekrasov (2016). / by Alexander Moll. / Ph. D.