Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. 71-74). / by Yue Lei. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1970. / Vita. / Bibliography: leaves 61-63. / by Edmund Francis Kelly. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 143-154). / In this thesis we develop a new family of fast and scalable algorithms to solve the 2D high-frequency Helmholtz equation in heterogeneous medium. The algorithms rely on a layered domain decomposition and a coupling between subdomains using the Green's representation formula, which reduces the problem to a boundary integral system at the interfaces between subdomains. Simultaneously, we introduce a polarization of the waves in up- and down-going components using incomplete Green's integrals, which induces another equivalent boundary integral formulation that is easy to precondition. The computation is divided in two stages: an offline stage, a computationally expensive but embarrassingly parallel precomputation performed only once; and an online stage, a highly parallel computation with low complexity performed for each right-hand side. The computational efficiency of the algorithms is achieved by shifting most of the computational burden to an offline precomputation, and by reducing the sequential bottleneck in the online stage using an efficient preconditioner, based on the polarized decomposition, coupled with compressed linear algebra. The resulting algorithms have online runtime O(N/P), where N is the number of unknowns, and P is the number of nodes in a distributed memory environment; provided that P = O (N [alpha]). Typically [alpha] = 1/5 or 1/8. / by Leonardo Andrés Zepeda-Núñ̃ez. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (leaves 195-202 / by András A. Benczúr. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 57-58). / by Grigore Raul Tataru. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1991. / Includes bibliographical references (p. 98-99). / by Yi Hu. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1941. / Vita. / by Kenneth James Arnold. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (p. 112-113). / by Mark S. Joshi. / Ph.D.

Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1946. / Vita. / Bibliography: leaf 27. / by Warren Simms Loud. / Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 29). / In their paper Metaplectic Forms, D. A. Kazhdan and S. J. Patterson developed a generalization of automorphic forms that are defined on metaplectic groups. These groups are non-trivial covering groups of usual algebraic groups, and the forms defined on them are representations that respect the covering. As in the case for automorphic forms, these representations fall into a principle series, indexed by characters on a torus of the metaplectic group, and there is an associated an L-function. In the final section of their paper, an equivalence is shown in the rank one case between this -function and an Dirichlet series defined using Gauss sums, in order to demonstrate the arithmetic content. In this paper we reexamine this connection in the particular case that was discussed in Metaplectic Forms. By looking through the scope of twisted multiplicativity, a property of L-series, the computation is simplified and more easily generalized. / by Sawyer Tabony. / S.M.