Matching Items (11)
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- All Subjects: Mathematics
- Creators: Kawski, Matthias
Description
In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen.
ContributorsPatani, Nura (Author) / Kaliszewski, Steven (Thesis advisor) / Quigg, John (Thesis advisor) / Bremner, Andrew (Committee member) / Kawski, Matthias (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2011
Description
Bacteriophage (phage) are viruses that infect bacteria. Typical laboratory experiments show that in a chemostat containing phage and susceptible bacteria species, a mutant bacteria species will evolve. This mutant species is usually resistant to the phage infection and less competitive compared to the susceptible bacteria species. In some experiments, both susceptible and resistant bacteria species, as well as phage, can coexist at an equilibrium for hundreds of hours. The current research is inspired by these observations, and the goal is to establish a mathematical model and explore sufficient and necessary conditions for the coexistence. In this dissertation a model with infinite distributed delay terms based on some existing work is established. A rigorous analysis of the well-posedness of this model is provided, and it is proved that the susceptible bacteria persist. To study the persistence of phage species, a "Phage Reproduction Number" (PRN) is defined. The mathematical analysis shows phage persist if PRN > 1 and vanish if PRN < 1. A sufficient condition and a necessary condition for persistence of resistant bacteria are given. The persistence of the phage is essential for the persistence of resistant bacteria. Also, the resistant bacteria persist if its fitness is the same as the susceptible bacteria and if PRN > 1. A special case of the general model leads to a system of ordinary differential equations, for which numerical simulation results are presented.
ContributorsHan, Zhun (Author) / Smith, Hal (Thesis advisor) / Armbruster, Dieter (Committee member) / Kawski, Matthias (Committee member) / Kuang, Yang (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2012
Description
The Jordan curve theorem states that any homeomorphic copy of a circle into R2 divides the plane into two distinct regions. This paper reconstructs one proof of the Jordan curve theorem before turning its attention toward generalizations of the theorem and their proofs and counterexamples. We begin with an introduction to elementary topology and the different notions of the connectedness of a space before constructing the first proof of the Jordan curve theorem. We then turn our attention to algebraic topology which we utilize in our discussion of the Jordan curve theorem’s generalizations. We end with a proof of the Jordan-Brouwer theorems, extensions of the Jordan curve theorem to higher dimensions.
ContributorsClark, Kacey (Author) / Kawski, Matthias (Thesis director) / Paupert, Julien (Committee member) / Department of Physics (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Recursive Bayesian Estimation on Projective Spaces: Theoretical Foundations and Practical Algorithms
Description
This thesis develops geometrically and statistically rigorous foundations for multivariate analysis and bayesian inference posed on grassmannian manifolds. Requisite to the development of key elements of statistical theory in a geometric realm are closed-form, analytic expressions for many differential geometric objects, e.g., tangent vectors, metrics, geodesics, volume forms. The first part of this thesis is devoted to a mathematical exposition of these. In particular, it leverages the classical work of Alan James to derive the exterior calculus of differential forms on special grassmannians for invariant measures with respect to which integration is permissible. Motivated by various multi-sensor remote sensing applications, the second part of this thesis describes the problem of recursively estimating the state of a dynamical system propagating on the Grassmann manifold. Fundamental to the bayesian treatment of this problem is the choice of a suitable probability distribution to a priori model the state. Using the Method of Maximum Entropy, a derivation of maximum-entropy probability distributions on the state space that uses the developed geometric theory is characterized. Statistical analyses of these distributions, including parameter estimation, are also presented. These probability distributions and the statistical analysis thereof are original contributions. Using the bayesian framework, two recursive estimation algorithms, both of which rely on noisy measurements on (special cases of) the Grassmann manifold, are the devised and implemented numerically. The first is applied to an idealized scenario, the second to a more practically motivated scenario. The novelty of both of these algorithms lies in the use of thederived maximumentropy probability measures as models for the priors. Numerical simulations demonstrate that, under mild assumptions, both estimation algorithms produce accurate and statistically meaningful outputs. This thesis aims to chart the interface between differential geometry and statistical signal processing. It is my deepest hope that the geometric-statistical approach underlying this work facilitates and encourages the development of new theories and new computational methods in geometry. Application of these, in turn, will bring new insights and bettersolutions to a number of extant and emerging problems in signal processing.
ContributorsCrider, Lauren N (Author) / Cochran, Douglas (Thesis advisor) / Kotschwar, Brett (Committee member) / Scharf, Louis (Committee member) / Taylor, Thomas (Committee member) / Turaga, Pavan (Committee member) / Arizona State University (Publisher)
Created2021
Description
In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique. This thesis explores one possible construction (originally due to Hunt) in depth and uses it to produce arithmetic lattices, non-arithmetic lattices, and thin subgroups in SU(2,1).
ContributorsWells, Joseph (Author) / Paupert, Julien (Thesis advisor) / Kotschwar, Brett (Committee member) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kawski, Matthias (Committee member) / Arizona State University (Publisher)
Created2019
Description
The main part of this work establishes existence, uniqueness and regularity properties of measure-valued solutions of a nonlinear hyperbolic conservation law with non-local velocities. Major challenges stem from in- and out-fluxes containing nonzero pure-point parts which cause discontinuities of the velocities. This part is preceded, and motivated, by an extended study which proves that an associated optimal control problem has no optimal $L^1$-solutions that are supported on short time intervals.
The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system.
More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls.
The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such.
In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm.
The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system.
More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls.
The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such.
In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm.
ContributorsGong, Xiaoqian, Ph.D (Author) / Kawski, Matthias (Thesis advisor) / Kaliszewski, Steven (Committee member) / Motsch, Sebastien (Committee member) / Smith, Hal (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2019
Description
Robotic technology is advancing to the point where it will soon be feasible to deploy massive populations, or swarms, of low-cost autonomous robots to collectively perform tasks over large domains and time scales. Many of these tasks will require the robots to allocate themselves around the boundaries of regions or features of interest and achieve target objectives that derive from their resulting spatial configurations, such as forming a connected communication network or acquiring sensor data around the entire boundary. We refer to this spatial allocation problem as boundary coverage. Possible swarm tasks that will involve boundary coverage include cooperative load manipulation for applications in construction, manufacturing, and disaster response.
In this work, I address the challenges of controlling a swarm of resource-constrained robots to achieve boundary coverage, which I refer to as the problem of stochastic boundary coverage. I first examined an instance of this behavior in the biological phenomenon of group food retrieval by desert ants, and developed a hybrid dynamical system model of this process from experimental data. Subsequently, with the aid of collaborators, I used a continuum abstraction of swarm population dynamics, adapted from a modeling framework used in chemical kinetics, to derive stochastic robot control policies that drive a swarm to target steady-state allocations around multiple boundaries in a way that is robust to environmental variations.
Next, I determined the statistical properties of the random graph that is formed by a group of robots, each with the same capabilities, that have attached to a boundary at random locations. I also computed the probability density functions (pdfs) of the robot positions and inter-robot distances for this case.
I then extended this analysis to cases in which the robots have heterogeneous communication/sensing radii and attach to a boundary according to non-uniform, non-identical pdfs. I proved that these more general coverage strategies generate random graphs whose probability of connectivity is Sharp-P Hard to compute. Finally, I investigated possible approaches to validating our boundary coverage strategies in multi-robot simulations with realistic Wi-fi communication.
In this work, I address the challenges of controlling a swarm of resource-constrained robots to achieve boundary coverage, which I refer to as the problem of stochastic boundary coverage. I first examined an instance of this behavior in the biological phenomenon of group food retrieval by desert ants, and developed a hybrid dynamical system model of this process from experimental data. Subsequently, with the aid of collaborators, I used a continuum abstraction of swarm population dynamics, adapted from a modeling framework used in chemical kinetics, to derive stochastic robot control policies that drive a swarm to target steady-state allocations around multiple boundaries in a way that is robust to environmental variations.
Next, I determined the statistical properties of the random graph that is formed by a group of robots, each with the same capabilities, that have attached to a boundary at random locations. I also computed the probability density functions (pdfs) of the robot positions and inter-robot distances for this case.
I then extended this analysis to cases in which the robots have heterogeneous communication/sensing radii and attach to a boundary according to non-uniform, non-identical pdfs. I proved that these more general coverage strategies generate random graphs whose probability of connectivity is Sharp-P Hard to compute. Finally, I investigated possible approaches to validating our boundary coverage strategies in multi-robot simulations with realistic Wi-fi communication.
ContributorsPeruvemba Kumar, Ganesh (Author) / Berman, Spring M (Thesis advisor) / Fainekos, Georgios (Thesis advisor) / Bazzi, Rida (Committee member) / Syrotiuk, Violet (Committee member) / Taylor, Thomas (Committee member) / Arizona State University (Publisher)
Created2016
Description
The data explosion in the past decade is in part due to the widespread use of rich sensors that measure various physical phenomenon -- gyroscopes that measure orientation in phones and fitness devices, the Microsoft Kinect which measures depth information, etc. A typical application requires inferring the underlying physical phenomenon from data, which is done using machine learning. A fundamental assumption in training models is that the data is Euclidean, i.e. the metric is the standard Euclidean distance governed by the L-2 norm. However in many cases this assumption is violated, when the data lies on non Euclidean spaces such as Riemannian manifolds. While the underlying geometry accounts for the non-linearity, accurate analysis of human activity also requires temporal information to be taken into account. Human movement has a natural interpretation as a trajectory on the underlying feature manifold, as it evolves smoothly in time. A commonly occurring theme in many emerging problems is the need to \emph{represent, compare, and manipulate} such trajectories in a manner that respects the geometric constraints. This dissertation is a comprehensive treatise on modeling Riemannian trajectories to understand and exploit their statistical and dynamical properties. Such properties allow us to formulate novel representations for Riemannian trajectories. For example, the physical constraints on human movement are rarely considered, which results in an unnecessarily large space of features, making search, classification and other applications more complicated. Exploiting statistical properties can help us understand the \emph{true} space of such trajectories. In applications such as stroke rehabilitation where there is a need to differentiate between very similar kinds of movement, dynamical properties can be much more effective. In this regard, we propose a generalization to the Lyapunov exponent to Riemannian manifolds and show its effectiveness for human activity analysis. The theory developed in this thesis naturally leads to several benefits in areas such as data mining, compression, dimensionality reduction, classification, and regression.
ContributorsAnirudh, Rushil (Author) / Turaga, Pavan (Thesis advisor) / Cochran, Douglas (Committee member) / Runger, George C. (Committee member) / Taylor, Thomas (Committee member) / Arizona State University (Publisher)
Created2016
Description
This dissertation contains three main results. First, a generalization of Ionescu's theorem is proven. Ionescu's theorem describes an unexpected connection between graph C*-algebras and fractal geometry. In this work, this theorem is extended from ordinary directed graphs to higher-rank graphs. Second, a characterization is given of the Cuntz-Pimsner algebra associated to a tensor product of C*-correspondences. This is a generalization of a result by Kumjian about graphs algebras. This second result is applied to several important special cases of Cuntz-Pimsner algebras including topological graph algebras, crossed products by the integers and crossed products by completely positive maps. The result has meaningful interpretations in each context. The third result is an extension of the second result from an ordinary tensor product to a special case of Woronowicz's twisted tensor product. This result simultaneously characterizes Cuntz-Pimsner algebras of ordinary and graded tensor products and Cuntz-Pimsner algebras of crossed products by actions and coactions of discrete groups, the latter partially recovering earlier results of Hao and Ng and of Kaliszewski, Quigg and Robertson.
ContributorsMorgan, Adam (Author) / Kaliszewski, Steven (Thesis advisor) / Quigg, John (Thesis advisor) / Spielberg, Jack (Committee member) / Kawski, Matthias (Committee member) / Kotschwar, Brett (Committee member) / Arizona State University (Publisher)
Created2016
Description
This thesis explores several questions concerning the preservation of geometric structure under the Ricci flow, an evolution equation for Riemannian metrics. Within the class of complete solutions with bounded curvature, short-time existence and uniqueness of solutions guarantee that symmetries and many other geometric features are preserved along the flow. However, much less is known about the analytic and geometric properties of solutions of potentially unbounded curvature. The first part of this thesis contains a proof that the full holonomy group is preserved, up to isomorphism, forward and backward in time. The argument reduces the problem to the preservation of reduced holonomy via an analysis of the equation satisfied by parallel translation around a loop with respect to the evolving metric. The subsequent chapter examines solutions satisfying a certain instantaneous, but nonuniform, curvature bound, and shows that when such solutions split as a product initially, they will continue to split for all time. This problem is encoded as one of uniqueness for an auxiliary system constructed from a family of time-dependent, orthogonal distributions of the tangent bundle. The final section presents some details of an ongoing project concerning the uniqueness of asymptotically product gradient shrinking Ricci solitons, including the construction of a certain system of mixed differential inequalities which measures the extent to which such a soliton fails to split.
ContributorsCook, Mary (Author) / Kotschwar, Brett (Thesis advisor) / Paupert, Julien (Committee member) / Kawski, Matthias (Committee member) / Kaliszewski, Steven (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2021