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- All Subjects: Mathematics
- All Subjects: Resource allocation
- Creators: Sen, Arunabha
- Creators: Spielberg, John
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
The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. It is shown that the quantum mechanical uncertainty principle is a special case of an inequality from J-holomorphic map theory, that is, J-holomorphic curves minimize the difference between the quantum covariance matrix determinant and a symplectic area. An immediate consequence is that a minimal determinant is a topological invariant, within a fixed homology class of the curve. Various choices of quantum operators are studied with reference to the implications of the J-holomorphic condition. The mean curvature vector field and Maslov class are calculated for a lagrangian torus of an integrable quantum system. The mean curvature one-form is simply related to the canonical connection which determines the geometric phases and polarization linear response. Adiabatic deformations of a quantum system are analyzed in terms of vector bundle classifying maps and related to the mean curvature flow of quantum states. The dielectric response function for a periodic solid is calculated to be the curvature of a connection on a vector bundle.
ContributorsSanborn, Barbara (Author) / Suslov, Sergei K (Thesis advisor) / Suslov, Sergei (Committee member) / Spielberg, John (Committee member) / Quigg, John (Committee member) / Menéndez, Jose (Committee member) / Jones, Donald (Committee member) / Arizona State University (Publisher)
Created2011
Description
Sparse learning is a technique in machine learning for feature selection and dimensionality reduction, to find a sparse set of the most relevant features. In any machine learning problem, there is a considerable amount of irrelevant information, and separating relevant information from the irrelevant information has been a topic of focus. In supervised learning like regression, the data consists of many features and only a subset of the features may be responsible for the result. Also, the features might require special structural requirements, which introduces additional complexity for feature selection. The sparse learning package, provides a set of algorithms for learning a sparse set of the most relevant features for both regression and classification problems. Structural dependencies among features which introduce additional requirements are also provided as part of the package. The features may be grouped together, and there may exist hierarchies and over- lapping groups among these, and there may be requirements for selecting the most relevant groups among them. In spite of getting sparse solutions, the solutions are not guaranteed to be robust. For the selection to be robust, there are certain techniques which provide theoretical justification of why certain features are selected. The stability selection, is a method for feature selection which allows the use of existing sparse learning methods to select the stable set of features for a given training sample. This is done by assigning probabilities for the features: by sub-sampling the training data and using a specific sparse learning technique to learn the relevant features, and repeating this a large number of times, and counting the probability as the number of times a feature is selected. Cross-validation which is used to determine the best parameter value over a range of values, further allows to select the best parameter value. This is done by selecting the parameter value which gives the maximum accuracy score. With such a combination of algorithms, with good convergence guarantees, stable feature selection properties and the inclusion of various structural dependencies among features, the sparse learning package will be a powerful tool for machine learning research. Modular structure, C implementation, ATLAS integration for fast linear algebraic subroutines, make it one of the best tool for a large sparse setting. The varied collection of algorithms, support for group sparsity, batch algorithms, are a few of the notable functionality of the SLEP package, and these features can be used in a variety of fields to infer relevant elements. The Alzheimer Disease(AD) is a neurodegenerative disease, which gradually leads to dementia. The SLEP package is used for feature selection for getting the most relevant biomarkers from the available AD dataset, and the results show that, indeed, only a subset of the features are required to gain valuable insights.
ContributorsThulasiram, Ramesh (Author) / Ye, Jieping (Thesis advisor) / Xue, Guoliang (Committee member) / Sen, Arunabha (Committee member) / Arizona State University (Publisher)
Created2011
Description
Data mining is increasing in importance in solving a variety of industry problems. Our initiative involves the estimation of resource requirements by skill set for future projects by mining and analyzing actual resource consumption data from past projects in the semiconductor industry. To achieve this goal we face difficulties like data with relevant consumption information but stored in different format and insufficient data about project attributes to interpret consumption data. Our first goal is to clean the historical data and organize it into meaningful structures for analysis. Once the preprocessing on data is completed, different data mining techniques like clustering is applied to find projects which involve resources of similar skillsets and which involve similar complexities and size. This results in "resource utilization templates" for groups of related projects from a resource consumption perspective. Then project characteristics are identified which generate this diversity in headcounts and skillsets. These characteristics are not currently contained in the data base and are elicited from the managers of historical projects. This represents an opportunity to improve the usefulness of the data collection system for the future. The ultimate goal is to match the product technical features with the resource requirement for projects in the past as a model to forecast resource requirements by skill set for future projects. The forecasting model is developed using linear regression with cross validation of the training data as the past project execution are relatively few in number. Acceptable levels of forecast accuracy are achieved relative to human experts' results and the tool is applied to forecast some future projects' resource demand.
ContributorsBhattacharya, Indrani (Author) / Sen, Arunabha (Thesis advisor) / Kempf, Karl G. (Thesis advisor) / Liu, Huan (Committee member) / Arizona State University (Publisher)
Created2013
Description
Communication networks, both wired and wireless, are expected to have a certain level of fault-tolerance capability.These networks are also expected to ensure a graceful degradation in performance when some of the network components fail. Traditional studies on fault tolerance in communication networks, for the most part, make no assumptions regarding the location of node/link faults, i.e., the faulty nodes and links may be close to each other or far from each other. However, in many real life scenarios, there exists a strong spatial correlation among the faulty nodes and links. Such failures are often encountered in disaster situations, e.g., natural calamities or enemy attacks. In presence of such region-based faults, many of traditional network analysis and fault-tolerant metrics, that are valid under non-spatially correlated faults, are no longer applicable. To this effect, the main thrust of this research is design and analysis of robust networks in presence of such region-based faults. One important finding of this research is that if some prior knowledge is available on the maximum size of the region that might be affected due to a region-based fault, this piece of knowledge can be effectively utilized for resource efficient design of networks. It has been shown in this dissertation that in some scenarios, effective utilization of this knowledge may result in substantial saving is transmission power in wireless networks. In this dissertation, the impact of region-based faults on the connectivity of wireless networks has been studied and a new metric, region-based connectivity, is proposed to measure the fault-tolerance capability of a network. In addition, novel metrics, such as the region-based component decomposition number(RBCDN) and region-based largest component size(RBLCS) have been proposed to capture the network state, when a region-based fault disconnects the network. Finally, this dissertation presents efficient resource allocation techniques that ensure tolerance against region-based faults, in distributed file storage networks and data center networks.
ContributorsBanerjee, Sujogya (Author) / Sen, Arunabha (Thesis advisor) / Xue, Guoliang (Committee member) / Richa, Andrea (Committee member) / Hurlbert, Glenn (Committee member) / Arizona State University (Publisher)
Created2013
Description
Gray codes are perhaps the best known structures for listing sequences of combinatorial objects, such as binary strings. Simply defined as a minimal change listing, Gray codes vary greatly both in structure and in the types of objects that they list. More specific types of Gray codes are universal cycles and overlap sequences. Universal cycles are Gray codes on a set of strings of length n in which the first n-1 letters of one object are the same as the last n-1 letters of its predecessor in the listing. Overlap sequences allow this overlap to vary between 1 and n-1. Some of our main contributions to the areas of Gray codes and universal cycles include a new Gray code algorithm for fixed weight m-ary words, and results on the existence of universal cycles for weak orders on [n]. Overlap cycles are a relatively new structure with very few published results. We prove the existence of s-overlap cycles for k-permutations of [n], which has been an open research problem for several years, as well as constructing 1- overlap cycles for Steiner triple and quadruple systems of every order. Also included are various other results of a similar nature covering other structures such as binary strings, m-ary strings, subsets, permutations, weak orders, partitions, and designs. These listing structures lend themselves readily to some classes of combinatorial objects, such as binary n-tuples and m-ary n-tuples. Others require more work to find an appropriate structure, such as k-subsets of an n-set, weak orders, and designs. Still more require a modification in the representation of the objects to fit these structures, such as partitions. Determining when and how we can fit these sets of objects into our three listing structures is the focus of this dissertation.
ContributorsHoran, Victoria E (Author) / Hurlbert, Glenn H. (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Colbourn, Charles (Committee member) / Sen, Arunabha (Committee member) / Arizona State University (Publisher)
Created2012
DescriptionCantor sets are totally disconnected, compact, metrizable, and contain no isolated points. All Cantor sets are homeomorphic to each other, but the addition of the metric yields new properties which can be detected by their correspondence with the boundaries of infinite rooted trees.
ContributorsAmes, Robert (Author) / Spielberg, John (Thesis advisor) / Kaliszewski, Steven (Committee member) / Quigg, John (Committee member) / Arizona State University (Publisher)
Created2022
Description
Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
ContributorsYie, Jangwon (Author) / Czygrinow, Andrzej (Thesis advisor) / Kierstead, Henry (Committee member) / Colbourn, Charles (Committee member) / Fishel, Susanna (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2018
Description
Emerging from years of research and development, the Internet-of-Things (IoT) has finally paved its way into our daily lives. From smart home to Industry 4.0, IoT has been fundamentally transforming numerous domains with its unique superpower of interconnecting world-wide devices. However, the capability of IoT is largely constrained by the limited resources it can employ in various application scenarios, including computing power, network resource, dedicated hardware, etc. The situation is further exacerbated by the stringent quality-of-service (QoS) requirements of many IoT applications, such as delay, bandwidth, security, reliability, and more. This mismatch in resources and demands has greatly hindered the deployment and utilization of IoT services in many resource-intense and QoS-sensitive scenarios like autonomous driving and virtual reality.
I believe that the resource issue in IoT will persist in the near future due to technological, economic and environmental factors. In this dissertation, I seek to address this issue by means of smart resource allocation. I propose mathematical models to formally describe various resource constraints and application scenarios in IoT. Based on these, I design smart resource allocation algorithms and protocols to maximize the system performance in face of resource restrictions. Different aspects are tackled, including networking, security, and economics of the entire IoT ecosystem. For different problems, different algorithmic solutions are devised, including optimal algorithms, provable approximation algorithms, and distributed protocols. The solutions are validated with rigorous theoretical analysis and/or extensive simulation experiments.
I believe that the resource issue in IoT will persist in the near future due to technological, economic and environmental factors. In this dissertation, I seek to address this issue by means of smart resource allocation. I propose mathematical models to formally describe various resource constraints and application scenarios in IoT. Based on these, I design smart resource allocation algorithms and protocols to maximize the system performance in face of resource restrictions. Different aspects are tackled, including networking, security, and economics of the entire IoT ecosystem. For different problems, different algorithmic solutions are devised, including optimal algorithms, provable approximation algorithms, and distributed protocols. The solutions are validated with rigorous theoretical analysis and/or extensive simulation experiments.
ContributorsYu, Ruozhou, Ph.D (Author) / Xue, Guoliang (Thesis advisor) / Huang, Dijiang (Committee member) / Sen, Arunabha (Committee member) / Zhang, Yanchao (Committee member) / Arizona State University (Publisher)
Created2019
Description
We live in a networked world with a multitude of networks, such as communication networks, electric power grid, transportation networks and water distribution networks, all around us. In addition to such physical (infrastructure) networks, recent years have seen tremendous proliferation of social networks, such as Facebook, Twitter, LinkedIn, Instagram, Google+ and others. These powerful social networks are not only used for harnessing revenue from the infrastructure networks, but are also increasingly being used as “non-conventional sensors” for monitoring the infrastructure networks. Accordingly, nowadays, analyses of social and infrastructure networks go hand-in-hand. This dissertation studies resource allocation problems encountered in this set of diverse, heterogeneous, and interdependent networks. Three problems studied in this dissertation are encountered in the physical network domain while the three other problems studied are encountered in the social network domain.
The first problem from the infrastructure network domain relates to distributed files storage scheme with a goal of enhancing robustness of data storage by making it tolerant against large scale geographically-correlated failures. The second problem relates to placement of relay nodes in a deployment area with multiple sensor nodes with a goal of augmenting connectivity of the resulting network, while staying within the budget specifying the maximum number of relay nodes that can be deployed. The third problem studied in this dissertation relates to complex interdependencies that exist between infrastructure networks, such as power grid and communication network. The progressive recovery problem in an interdependent network is studied whose goal is to maximize system utility over the time when recovery process of failed entities takes place in a sequential manner.
The three problems studied from the social network domain relate to influence propagation in adversarial environment and political sentiment assessment in various states in a country with a goal of creation of a “political heat map” of the country. In the first problem of the influence propagation domain, the goal of the second player is to restrict the influence of the first player, while in the second problem the goal of the second player is to have a larger market share with least amount of initial investment.
The first problem from the infrastructure network domain relates to distributed files storage scheme with a goal of enhancing robustness of data storage by making it tolerant against large scale geographically-correlated failures. The second problem relates to placement of relay nodes in a deployment area with multiple sensor nodes with a goal of augmenting connectivity of the resulting network, while staying within the budget specifying the maximum number of relay nodes that can be deployed. The third problem studied in this dissertation relates to complex interdependencies that exist between infrastructure networks, such as power grid and communication network. The progressive recovery problem in an interdependent network is studied whose goal is to maximize system utility over the time when recovery process of failed entities takes place in a sequential manner.
The three problems studied from the social network domain relate to influence propagation in adversarial environment and political sentiment assessment in various states in a country with a goal of creation of a “political heat map” of the country. In the first problem of the influence propagation domain, the goal of the second player is to restrict the influence of the first player, while in the second problem the goal of the second player is to have a larger market share with least amount of initial investment.
ContributorsMazumder, Anisha (Author) / Sen, Arunabha (Thesis advisor) / Richa, Andrea (Committee member) / Xue, Guoliang (Committee member) / Reisslein, Martin (Committee member) / Arizona State University (Publisher)
Created2016