Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. 161-165). / Solar activity, as indexed by sunspots, has a cycle of length varying from about 9 to 13 years. Statisticians have fitted several models to predict sunspot numbers one year ahead. We instead focus on predicting the magnitudes of the next cycle maximum, the next cycle minimum, the time from the initial minimum of a cycle to its maximum, called the rise time, and the time from a cycle maximum to the next cycle minimum, called the fall time. The predictions are based on sunspot numbers just far enough into a new cycle to establish that a cycle has started. We propose parsimonious regression models for the maximum and rise time. For the fall time and minimum we propose crude models, based on the sample means and variances for all past cycles. We compare our models to simulation results for many models we found in the literature including autoregressive (AR) models, subset AR (SAR) models, and related nonlinear models including a threshold AR model, transformed by squaring (TTAR model), a bilinear model, a so-called ASTAR model, and a neural network (CNAR) model. We also consider a model proposed by solar physicists, based on a functional form for cycles. / (cont.) Numerical results show that among all the models considered for sunspot numbers, our regression models give the smallest MSEs (mean-square errors) for the maxima, and our crude models give the smallest MSEs for the minima. For fall times our crude model has the second-smallest MSE after CNAR. ASTAR does very well for modeling the rise times. Observations of the sun have become progressively more accurate, but we found that giving higher weight to more recent observations gave worse results. Among the model selection criteria we tried for the autoregressive and regression models, the two that worked best as judged by cross-validations were the well-known Akaike Information Criterion and a modification of the G. Schwarz BIC criterion called BIC* due to D. and J. Haughton and A. Izenman. / by Li He. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. 81-83). / Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian covering can be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitary representation of G. W. Graham and D. Vogan propose such a construction, relying on the notions of orbital varieties and admissible orbit data. The first part of the thesis seeks to understand the set of orbital varieties contained in a given nipotent orbit. Starting from N. Spaltenstein's parameterization of the irreducible components of the variety of flags fixed by a unipotent, we produce a parameterization of the orbital varieties lying in the corresponding fiber of the Steinberg map. The parameter set is the family of standard Young or domino tableau of a given shape. The key to the proof is understanding certain closed cycles as defined by D. Garfinkle. This parameterization is particularly useful; it provides a method of determining the r-invariant of each orbital variety, as well as a way of relating an orbital variety in any classical group to one lying in type A. / (cont.) The second part of the thesis addresses the representations V(V, ir) constructed by Graham and Vogan. A natural question is how well the V(V, 7r) approximate the set of unipotent representations that ought to be attached to the nilpotent orbit 0. The answer is promising in the setting of spherical orbits. When it is possible to carry out the Graham-Vogan construction, the corresponding infinitesimal character lies in the set of characters suggested by W. M. McGovern. Furthermore, we show that it is possible to carry out the Graham-Vogan construction for a suffient number of orbital varieties to account for all the infinitesimal characters attached to 0 by McGovern. / by Thomas Pietraho. / Ph.D.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D. / Bibliography: leaves 102-103. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaves 63-64). / by Jianke Yang. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 153-155). / Consider an instance h of the Gaussian free field on a simply connected domain ... We study several properties of the level lines: continuity, monotonicity, reversibility and target-independence ... In the second part, we study Liouville quantum gravity(LQG). / by Menglu Wang. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaves 139-141). / by Gábor Hetyei. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 133-135). / by Dimitri Vaynblat. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / In title on title page, "N" of GLN appears as subscript of upper case letter N. Cataloged from PDF version of thesis. / Includes bibliographical references (p. 73). / Let C be a smooth connected projective curve of genus > 1 over an algebraically closed field k of characteristic p > 0, and c [epsilon] k \ Fp. Let BunN be the stack of rank N vector bundles on C and Ldet the line bundle on BunN given by determinant of derived global sections. In this thesis, we construct an equivalence of derived categories of modules for certain localizations of the twisted crystalline differential operator algebras DBunNet and DBunN, L-1/cdet The first step of the argument is the same as that of [BB] for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier-Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of p-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de Rham local systems on C. / by Roman Travkin. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 99-101). / We study optimal and nearly-optimal quantum strategies for non-local XOR games. First, we prove the following general result: for every non-local XOR game, there exists a set of relations with the properties: (1) a quantum strategy is optimal for the game if and only if it satisfies the relations, and (2) a quantum strategy is nearly optimal for the game if and only if it approximately satisfies the relations. Next, we focus attention on a specific infinite family of XOR games: the CHSH(n) games. This family generalizes the well-known CHSH game. We describe the general form of CHSH(n) optimal strategies. Then, we adapt the concept of intertwining operator from representation theory and use that to characterize nearly-optimal CHSH(n) strategies / by Dimiter Ostrev. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 69-70). / The purpose of this work is give some field notes on exploring the idea that a generalized Tate construction tk reduces chromatic level in stable homotopy theory. The first parts introduce the construction and discuss chromatic reduction. The next section makes a computation and gives the duals of L(n) = L(n)1. The last part looks ahead, mentioning how this computation could be extended to finding the duals of Steinberg summands in corresponding Thom spectra of negative representations, L(n)-q, and presents an equivariant loopspace machine. Finally, observations made are pulled together and brought back to compute the base case of the generalized Tate construction, evaluated on a sphere. Results parallel work of A. Cathcart, B. Guillou and P. May, and N. Stapleton, among others. / by Olga Stroilova. / Ph.D.