Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 245-254). / Granular materials such as sand or gravel surround us everyday and yet remain poorly understood. In this thesis, two models are developed for dense granular flow, each capable of predicting flows with accuracy in multiple environments. The models are based on differing perspectives of grain-level dynamics, with one deriving flow from a stochastic mechanism and the other from a deterministic deformation law. The Stochastic Flow Rule (SFR): This work models granular flow as a sequence of localized collective grain displacements. As in the Spot Model for drainage (Bazant 2001), grain clusters move as dictated by "spots" which travel through the material as biased random-walkers. The SFR derives spot motion directly from the material stresses, thus generalizing and extending the Spot Model beyond drainage to any quasi-2D geometry with a computable stress field. Limit-State Mohr-Coulomb Plasticity is used to approximate the stress profile in a slow flowing granular assembly. The SFR then describes quantitatively how to convert the slip-line field and stresses into the necessary parameters to fully define a spot's random trajectory through the material and generate a steady flow profile. Results are compared to known flow data. Nonlinear Granular Elasto-Plasticity: This work models granular deformation at the meso-scale as a deterministic consequence of the local stresses and state parameters. Recently proposed models for granular elasticity (Jiang and Liu 2003) and plastic flow (Jop et al. 2006) are combined into one universal granular continuum law, capable of predicting both flowing regions and stagnant zones simultaneously in any arbitrary 3D flow geometry. / (cont.) The unification is performed by first motivating physically, and then implementing a Kroner-Lee elasto-plastic decomposition. The model is then numerically solved in multiple geometries and results are compared to experiments and discrete simulations. / by Kenneth Norman Kamrin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993. / Includes bibliographical references (leaves 138-142). / by Jöns Richard Gustaf Ehrenborg. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 211-227). / Recent experiments show that active fluids stirred by swimming bacteria or ATPpowered microtubule networks can exhibit complex flow dynamics and emergent pattern scale selection. Here, I will investigate a simplified phenomenological approach to 'active turbulence', a chaotic non-equilibrium steady-state in which the solvent flow develops a dominant vortex size. This approach generalizes the incompressible Navier-Stokes equations by accounting for active stresses through a linear instability mechanism, in contrast to externally driven classical turbulence. This minimal model can reproduce experimentally observed velocity statistics and is analytically tractable in planar and curved geometry. Exact stationary bulk solutions include Abrikosovtype vortex lattices in 2D and chiral Beltrami fields in 3D. Numerical simulations for a plane Couette shear geometry predict a low viscosity phase mediated by stress defects, in qualitative agreement with recent experiments on bacterial suspensions. Considering the active analog of Stokes' second problem, our numerical analysis predicts that a periodically rotating ring will oscillate at a higher frequency in an active fluid than in a passive fluid, due to an activity-induced reduction of the fluid inertia. The model readily generalizes to curved geometries. On a two-sphere, we present exact stationary solutions and predict a new type of upward energy transfer mechanism realized through the formation of vortex chains, rather than the merging of vortices, as expected from classical 2D turbulence. In 3D simulations on periodic domains, we observe spontaneous mirror-symmetry breaking realized through Beltrami-like flows, which give rise to upward energy transfer, in contrast to the classical direct Richardson cascade. Our analysis of triadic interactions supports this numerical prediction by establishing an analogy with forced rigid body dynamics and reveals a previously unknown triad invariant for classical turbulence. / by Jonasz Słomka. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 87-88). / The algebra of symmetric functions, the representation theory of the symmetric group, and the geometry of the Grassmannian are related to each other via Schur functions, Specht modules, and Schubert varieties, all of which are indexed by partitions and their Young diagrams. We will generalize these objects to allow for not just Young diagrams but arbitrary collections of boxes or, equally well, bipartite graphs. We will then provide evidence for a conjecture that the relation between the areas described above can be extended to these general diagrams. In particular, we will prove the conjecture for forests. Along the way, we will use a novel geometric approach to show that the dimension of the Specht module of a forest is the same as the normalized volume of its matching polytope. We will also demonstrate a new Littlewood-Richardson rule and provide combinatorial, algebraic, and geometric interpretations of it. / by Ricky Ini Liu. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / 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 (p. 67-69). / We consider a variety of questions related to pattern avoidance in alternating permutations and generalizations thereof. We give bijective enumerations of alternating permutations avoiding patterns of length 3 and 4, of permutations that are the reading words of a "thickened staircase" shape (or equivalently of permutations with descent set {k, 2k, 3k, . . .}) avoiding a monotone pattern, and of the reading words of Young tableaux of any skew shape avoiding any of the patterns 132, 213, 312, or 231. Our bijections include a simple bijection involving binary trees, variations on the Robinson-Schensted-Knuth correspondence, and recursive bijections established via isomorphisms of generating trees. / by Joel Brewster Lewis. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 119-125). / by Dmitri Betaneli. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1988. / Includes bibliographical references. / by Francesco Brenti. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (leaves 59-60). / Given a degenerate (n + 1)-simplex in a n-dimensional Euclidean space Rn, which is embedded in a (n + 1)-dimensional Euclidean space Rn+l. We allow all its vertices to have continuous motion in the space, either in Rn+l or restricted in Rn. For a given k, based on certain rules, we separate all its k-faces into 2 groups. During the motion, we give the following restriction: the volume of the k-faces in the 1st group can not increase (these faces are called "k-cables"); the volume of the k-faces in the 2nd group can not decrease ("k-struts"). We will prove that, under more conditions, all the volumes of the k-faces will be preserved for any sufficiently small motion. We also partially generalize the above result to spherical space Sn and hyperbolic space Hn. / by Lizhao Zhang. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1969. / Vita. / Bibliography: leaf 102. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1983. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 57-58. / by Rajiv Gupta. / Ph.D.