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<p> We study pedestrian dynamics using Kalman filtering methods. Kalman filters apply the Bayesian approach to a time series and incorporate the uncertainty in both the measurement and the mathematical model to create a better state estimate of a system, and are commonly used to reduce error in state estimates for applications ranging from radar systems to GPS. </p><p> Our goal is to combine a Kalman filter and the Kuhn-Munkres algorithm to not only predict the locations of the pedestrians but also reconstruct their walking path trajectories. We first worked on the linear system of stochastic differential equations to model undisturbed pedestrian dynamics, which is motivated by the work from Corbetta et al. The model uses Langevin equations to describe the Brownian motion-like perturbations in pedestrian trajectories. We utilize the data collected over one year from the Eindhoven University of Technology MetaForum building. While our measurement data are recorded with relatively high frequency and low noise, we also investigate the effect of reduced data quality by downsampling our data set. </p><p> In the next step, we extend our study to a pedestrian model similar to the social force model described by Helbing et al. which includes pedestrian avoidance behavior on a second data set, the Eindhoven train station. The data set contains pedestrian data for different crowd densities and includes more interactions among pedestrians. </p><p> We find that using a Kalman filter improved results for both data sets, especially for connecting trajectories when more pedestrians are present. Using the pedestrian model with avoidance versus without avoidance showed no improvement.</p><p>

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1940. / Vita. / Includes bibliographical references (leaf 63). / by Claude Elwood Shannon. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (leaves 151-[152]). / by Michael John Hawrylycz. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (leaves 52-53). / by Debra Kay Borkovitz. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 253-254). / For a Lagrangian submanifold, we define a moduli space of trees of holomorphic disk maps with Morse flow lines as edges, and construct an ambient space around it which we call the quotient space of disk trees. We show that this ambient space is an M-polyfold with boundary and corners by combining the infinite dimensional analysis in sc-Banach space with the finite dimensional analysis in Deligne-Mumford space. We then show that the Cauchy-Riemann section is sc-Fredholm, and by applying the polyfold perturbation we construct an A[infinity]. algebra over Z₂ coefficients. Under certain assumptions, we prove the invariance of this algebra with respect to choices of almost-complex structures. / by Jiayong Li. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 129-132). / The first portion of this work focuses on leaky modes in the atmospheric sciences. Leaky modes (related to quasi-modes, scattering resonances, and the singularity expansion method) are discrete, oscillatory and decaying modes that arise in conservative systems where waves are partially trapped. By replacing the infinite domain with a finite domain and appropriate boundary conditions it is possible in many cases to construct a complete basis for the solution in terms of these modes. Formulating such effective boundary conditions requires a notion of the direction of propagation of the waves. For this purpose we introduce a generalization of the concept of group speed for exponentially decaying but conservative waves. This is found via an extended modulation argument and a generalization of Whitham's Average Lagrangian theory. The theory also shows that a close relationship exists between the branch cuts of the dispersion relation and the propagation direction, and is used to create spectral decompositions for simple problems in internal gravity waves. The last chapter considers deep-sea nodule mining operations, which potentially involve plans for discharge plumes to be released into the water column by surface operation vessels. We consider the effects of non-uniform, realistic stratifications with vertical shear on forced compressible plumes. The plume model is developed to account for the influence of thermal conduction through the discharge pipe and an initial adjustment phase. We investigate the substantial role of compressibility, for which a dimensionless number is introduced to determine its importance compared to that of the background stratification. Our results show that (i) small-scale stratification features can have a significant impact, (ii) in a static ambient there exists a discharge flow rate that minimizes the plume vertical extent, (iii) the ambient velocity profile plays an important role in determining the final plume scale and dilution factor, and (iv) for a typical plume the dilution factor is expected to be several hundred to a thousand. / by Andrew Joseph Rzeznik. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / 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 are common in everyday experience, but have long-resisted a complete theoretical description. Here, we consider the regime of slow, dense granular flow, for which there is no general model, representing a considerable hurdle to industry, where grains and powders must frequently be manipulated. Much of the complexity of modeling granular materials stems from the discreteness of the constituent particles, and a key theme of this work has been the connection of the microscopic particle motion to a bulk continuum description. This led to development of the "spot model", which provides a microscopic mechanism for particle rearrangement in dense granular flow, by breaking down the motion into correlated group displacements on a mesoscopic length scale. The spot model can be used as the basis of a multiscale simulation technique which can accurately reproduce the flow in a large-scale discrete element simulation of granular drainage, at a fraction of the computational cost. In addition, the simulation can also successfully track microscopic packing signatures, making it one of the first models of a flowing random packing. To extend to situations other than drainage ultimately requires a treatment of material properties, such as stress and strain-rate, but these quantities are difficult to define in a granular packing, due to strong heterogeneities at the level of a single particle. However, they can be successfully interpreted at the mesoscopic spot scale, and this information can be used to directly test some commonly-used hypotheses in modeling granular materials, providing insight into formulating a general theory. / by Christopher Harley Rycroft. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 107-108). / by Michael G. Shefter. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 83-86). / This thesis begins with the study of a class of symmetric functions ... Which are generating functions for ribbon tableaux (hereon called ribbon functions), first defined by Lascoux, Leclerc and Thibon. Following work of Fomin and Greene, I introduce a set of operators called ribbon Schur operators on the space of partitions. I develop the theory of ribbon functions using these operators in an elementary manner. In particular, I deduce their symmetry and recover a theorem of Kashiwara, Miwa and Stern concerning the Fock space F of the quantum affine algebras ... Using these results, I study the functions ... in analogy with Schur functions, giving: * a Pieri and dual-Pieri formula for ribbon functions, * a ribbon Murnaghan-Nakayama formula, * ribbon Cauchy and dual Cauchy identities, * and a C-algebra isomorphism ... The study of the functions ... will be connected to the Fock space representation F of ...via a linear map [Iota]: F ... which sends the standard basis of F to the ribbon functions. Kashiwara, Miwa and Stern [29] have shown that a copy of the Heisenberg algebra H acts on F commuting with the action of ... Identifying the Fock Space of H with the ring of symmetric functions A(q) I will show that · is in fact a map of H-modules with remarkable properties. In the second part of the thesis, I give a combinatorial generalisation of the classical Boson-Fermion correspondence and explain how the map [phi] is an example of this more general phenomena. I show how certain properties of many families of symmetric functions arise naturally from representations of Heisenberg algebras. The main properties I consider are a tableaux-like definition, a Pieri-style rule and a Cauchy-style identity. / (cont.) Families of symmetric functions which can be viewed in this manner include Schur functions, Hall- Littlewood functions, Macdonald polynomials and the ribbon functions. Using work of Kashiwara, Miwa, Petersen and Yung, I define generalised ribbon functions for certain affine root systems 1 of classical type. I prove a theorem relating these generalised ribbon functions to a speculative global basis of level 1 q-deformed Fock spaces. / by Thomas F. Lam. / Ph.D.