Browsing by Subject "Topology"
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Item Advanced compulsator topologies and technologies(2002-05) Walls, William Alan; Driga, Mricea D.Increasing the compactness of compensated pulsed alternators (compulsators) has been an ongoing effort at since the mid-1980’s, when the U. S. Army interest in electric armaments began to emerge in a significant way. Much progress has been made from the early proof of concept machine built at UTCEM in the late 1970’s after its invention by Weldon, Driga, and Woodson. Today, the compulsator is the best approach for achieving compact energy storage and pulsed power generation for multi-MJ, multi-GW applications requiring voltages of up to 20 kV and discharge durations between 1 and 10 milliseconds. Electromagnetic railguns, coil launchers, and directed energy systems requiring high-power, high-energy pulsed-power in compact form are ideal candidate loads for compulsators. Since the initial validation of compulsator theory in the late 1970’s, a transition from iron-core to air-core magnetic circuits and the associated incorporation of composite materials for the rotor and self-excitation has provided substantial increases in both stored energy density and power density. While the advancement has been significant, further gains in compactness are possible by exploiting continued innovation in the topology of the machine, the mode in which it operates, and by developing new component technologies specifically for compulsator application. In addition, the use of this type of machine in very high voltage (>50 kV) pulsed power application, a concept not previously explored, is considered through the novel integration of resonant and cascaded transformer arrangements to boost the output voltage. In contributing to the continued compulsator performance improvement, this research effort has performed an in depth study of machine topologies and identified a new topology, the flywheel-compulsator. When combined with improvements in materials and switching technologies, this new machine configuration can improve energy and power density by factors of three and five, respectively, compared to the current state of the art. To allow rapid sizing of compulsator systems, a scaling algorithm was developed, validated against demonstrated machines and advanced designs, and used to design systems for advanced applications. Reduced to a set of linked spreadsheets, the scaling algorithms were also used to identify compulsator component technologies areas where improvements will provide the greatest overall performance impact. Another area of significant contribution to compulsator technology embodied in this research is the application of these machines to very high voltage systems. Two general concepts were conceived and developed. The first combines the energy storage and pulsed power generation of the compulsator with voltage increasing circuits, including both resonant and cascaded transformers. For compact high voltage systems, the concept of generating the high voltage directly within the compulsator was evaluated, and a high voltage compulsator using a helical winding was optimized for a specific set of requirements.Item Age-Dependent Evolution of the Yeast Protein Interaction Network Suggests a Limited Role of Gene Duplication and Divergence(Public Library of Science, 2008-11-28) Kim, Wan Kyu; Marcotte, Edward M.Proteins interact in complex protein–protein interaction (PPI) networks whose topological properties—such as scale-free topology, hierarchical modularity, and dissortativity—have suggested models of network evolution. Currently preferred models invoke preferential attachment or gene duplication and divergence to produce networks whose topology matches that observed for real PPIs, thus supporting these as likely models for network evolution. Here, we show that the interaction density and homodimeric frequency are highly protein age–dependent in real PPI networks in a manner which does not agree with these canonical models. In light of these results, we propose an alternative stochastic model, which adds each protein sequentially to a growing network in a manner analogous to protein crystal growth (CG) in solution. The key ideas are (1) interaction probability increases with availability of unoccupied interaction surface, thus following an anti-preferential attachment rule, (2) as a network grows, highly connected sub-networks emerge into protein modules or complexes, and (3) once a new protein is committed to a module, further connections tend to be localized within that module. The CG model produces PPI networks consistent in both topology and age distributions with real PPI networks and is well supported by the spatial arrangement of protein complexes of known 3-D structure, suggesting a plausible physical mechanism for network evolution.Item Course summary of geometry and topology(2010-08) Craig, Tara Theresa; Armendáriz, Efraim P.; Daniels, Mark L.The foundation of Luecke’s course M: 396 Geometry and Topology is that collaboration amongst mathematicians and biologists caused tremendous gains in DNA research. The field of topology has led to significant strides in understanding of the topological properties of the genetic molecule DNA. Through the integration of biological phenomena and knowledge of topology and Euclidean geometry, biologists can describe and quantize enzyme mechanisms and therefore determine enzyme mechanisms causing the changes. Understanding mathematical applications in contexts outside of mathematics on any level helps to explain why mathematics is a core content area in primary and secondary education. Requiring secondary educators to take such a course could result in mathematics taught with real world application on the secondary level as well as on the graduate level, as shown in Luecke’s course.Item Creating and utilizing symbolic representations of spatial knowledge using mobile robots(2008-08) Beeson, Patrick Foil, 1977-; Kuipers, BenjaminA map is a description of an environment allowing an agent--a human, or in our case a mobile robot--to plan and perform effective actions. From a single location, an agent’s sensors can not observe the whole structure of a complex, large environment. For this reason, the agent must build a map from observations gathered over time and space. We distinguish between large-scale space, with spatial structure larger than the agent’s sensory horizon, and small-scale space, with structure within the sensory horizon. We propose a factored approach to mobile robot map-building that handles qualitatively different types of uncertainty by combining the strengths of topological and metrical approaches. Our framework is based on a computational model of the human cognitive map; thus it allows robust navigation and communication within several different spatial ontologies. Our approach factors the mapping problem into natural sub-goals: building a metrical representation for local small-scale spaces; finding a topological map that represents the qualitative structure of large-scale space; and (when necessary) constructing a metrical representation for large-scale space using the skeleton provided by the topological map. The core contributions of this thesis are a formal description of the Hybrid Spatial Semantic Hierarchy (HSSH), a framework for both small-scale and large-scale representations of space, and an implementation of the HSSH that allows a robot to ground the largescale concepts of place and path in a metrical model of the local surround. Given metrical models of the robot’s local surround, we argue that places at decision points in the world can be grounded by the use of a primitive called a gateway. Gateways separate different regions in space and have a natural description at intersections and in doorways. We provide an algorithmic definition of gateways, a theory of how they contribute to the description of paths and places, and practical uses of gateways in spatial mapping and learning.Item Developing a qualitative geometry from the conceptions of young children(2010-05) Greenstein, Steven Baron; Stroup, Walter M.; Empson, Susan B.; Carmona, Guadalupe; Petrosino, Anthony; Starbird, MichaelMore than half a century ago, Piaget concluded from an investigation of children’s representational thinking about the nature of space that the development of children’s representational thought is topological before it is Euclidean. This conclusion, commonly referred to as the “topological primacy thesis,” has essentially been rejected. By giving emphasis to the ideas that develop rather than the order in which they develop, this work set out to develop a new form of non-metric geometry from young children’s early and intuitive topological, or at least non-metric, ideas. I conducted an eighteen-week teaching experiment with two children, ages six and seven. I developed a new dynamic geometry environment called Configure that I used in tandem with clinical interviews in each of the episodes of the experiment to elicit these children’s non-metric conceptions and subsequently support their development. I found that these children developed significant and authentic forms of geometric reasoning. It is these findings, which I refer to as qualitative geometry, that have implications for the teaching of geometry and for research into students’ mathematical reasoning.Item Exotic smoothings via large R⁴’s in Stein surfaces(2015-12) Bennett, Julia C.; Gompf, Robert E., 1957-; Etnyre, John; Gordon, Cameron; Luecke, John; Reid, AlanWe study the relationship between exotic R⁴'s and Stein surfaces as it applies to smoothing theory on more general open 4–manifolds. In particular, we construct the first known examples of large exotic R⁴'s that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4–manifolds. Under sufficiently nice conditions, we show that using these R⁴'s as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant. Additionally, we prove that a family of knots known to be topologically slice but not smoothly slice is rationally slice, realizing a combination of slice properties that seems to have been previously unknown to exist.Item Explorations in algebra and topology(2016-05) Gal, Itamar; Hadani, Ronny; Allcock, Daniel; Blumberg, Andrew; Vaaler, JeffreyThree independent investigations are expounded, two in the domain of algebra and one in the domain of topology. We first consider algebraic extensions generated by elements of bounded degree and consider the question of whether or not the finite sub-extensions of such fields can be bounded. We give partial results which will hopefully lead to a full classification in the future. These results are fundamentally group theoretic but have applications to number theory. Next we develop the notational system originated by Conway and Sloane for working with quadratic forms over the 2-adic integers and prove its correctness. This provides a proof which was missing from the literature. Finally we study distributions of persistent homology barcodes obtained by sampling finite point sets from metric measure spaces. The main result here is the derivation of robust statistics for topological data analysis.Item Exploring Invariants in Knot Theory(2024) Patel, Kaushal; Austin, JenniferThis thesis will explore the importance of invariants within the mathematical branch of Knot Theory with an emphasis on polynomial invariants and one interesting invariant within the subfield of tangle theory. The paper will begin by providing a basic foundation of mathematical knots and links that will be essential to understanding the discussion of invariants. The paper will then explain the importance of invariants and how they have and can be used to answer some of the most crucial questions in Knot Theory. This paper will largely focus on the Alexander Polynomial, the Conway Polynomial, the Jones Polynomial, and the HOMFLYPT Polynomial. For each polynomial, the paper will explain how it is calculated for both knots and links, discuss some important qualities, and analyze crucial limitations. At the end of this section, the paper will provide a discussion comparing and contrasting the relative values of each invariant over different types of knots. Furthermore, the paper will transition into Tangle Theory. I will begin by providing the fundamentals of tangles, emphasizing rational tangles, which is necessary for understanding the rest of the paper. Next, the paper will explain exactly how tangles can be transitioned into knots and links to tie the new concepts into the start of the paper. Finally, the paper will explore an interesting link invariant of tangle theory, specifically highlighting how it differs from the polynomial invariants in its bidirectionality. Lastly, the paper will give a short discussion on potential further explorations of invariants, highlighting goals, open questions, and struggles currently faced.Item Homogenization Design and Layered Manufacturing of a Lower Control Arm in Project MAXWELL(1993) Johanson, Roy; Kikuchi, Noboru; Papalambros, PanosWe briefly describe a new methodology for the design and manufacture of mechanical components and demonstrate the process for the design of an automobile suspension component. The methodology is a unique coupling between advanced technologies for product design and manufacture, which leads to the rapid realization of superior products. The concurrent design and analysis strategy yields information about the optimal structural layout, as well as details about the material composition. The fabrication of such designs requires unconventional manufacturing processes, such as layered manufacturing. Project MAXWELL, therefore, offers the possibility for the integration of layered manufacturing into the mainstream product development and fabrication process.Item Integrating topology into the standard high school geometry curriculum(2012-08) Kiker, William George; Odell, E. (Edward); Daniels, MarkThis report conveys some of the modern investigations surrounding the use of topology in a contextual setting. Topics discussed include applications of topology relating to the modeling of biological structures and common objects like sunshades, elementary knot theory, and the connection between the fields of topology and algebra. A brief overview and discussion of the incorporation of elementary topology into the standard Geometry curriculum of secondary schools is also examined.Item K-theoretic computation of the Verlinde ring(2018-05-04) Zakharevich, Valentin; Freed, Daniel S.; Neitzke, Andrew; Ben-Zvi, David; Perutz, Timothy; Teleman, ConstantinWe compute Verlinde rings of the groups SU(3) semidirect product Z/2Z and Spin(8) semdirect product Sym(3) at level 1. We use the K-theory formulation developed by Freed, Hopkins and Teleman. More precisely, we compute the twisted equivariant K-theory of G acting on itself by conjugation. The fusion product, corresponding to the Pontryagin product on the level of K-theory, is partially computed.Item Network-on-chip architectures for scalability and service guarantees(2011-08) Grot, Boris; Keckler, Stephen W.; Burger, Douglas C.; Mutlu, Onur; Witchel, Emmett; Zhang, YinRapidly increasing transistor densities have led to the emergence of richly-integrated substrates in the form of chip multiprocessors and systems-on-a-chip. These devices integrate a variety of discrete resources, such as processing cores and cache memories, on a single die with the degree of integration growing in accordance with Moore's law. In this dissertation, we address challenges of scalability and quality-of-service (QOS) in network architectures of highly-integrated chips. The proposed techniques address the principal sources of inefficiency in networks-on-chip (NOCs) in the form of performance, area, and energy overheads. We also present a comprehensive network architecture capable of interconnecting over a thousand discrete resources with high efficiency and strong guarantees. We first show that mesh networks, commonly employed in existing chips, fall significantly short of achieving their performance potential due to transient congestion effects that diminish network performance. Adaptive routing has the potential to improve performance through better load distribution. However, we find that existing approaches are myopic in that they only consider local congestion indicators and fail to take global network state into account. Our approach, called Regional Congestion Awareness (RCA), improves network visibility in adaptive routers via a light-weight mechanism for propagating and integrating congestion information. By leveraging both local and non-local congestion indicators, RCA improves network load balance and boosts throughput. Under a set of parallel workloads running on a 49-node substrate, RCA reduces on-chip network latency by 16%, on average, compared to a locally-adaptive router. Next, we target NOC latency and energy efficiency through a novel point-to-multipoint topology. Ring and mesh networks, favored in existing on-chip interconnects, often require packets to go through a number of intermediate routers between source and destination nodes, resulting in significant latency and energy overheads. Topologies that improve connectivity, such as fat tree and flattened butterfly, eliminate much of the router overhead, but require non-minimal channel lengths or large channel count, reducing energy-efficiency and/or performance as a result. We propose a new topology, called Multidrop Express Channels (MECS), that augments minimally-routed express channels with multi-drop capability. The resulting richly-connected NOC enjoys a low hop count with favorable delay and energy characteristics, while improving wire utilization over prior proposals. Applications such as virtualized servers-on-a-chip and real-time systems require chip-level quality-of-service (QOS) support to provide fairness, service differentiation, and guarantees. Existing network QOS approaches suffer from considerable performance and area overheads that limit their usefulness in a resource-limited on-die network. In this dissertation, we propose a new QOS scheme called Preemptive Virtual Clock (PVC). PVC uses a preemptive approach to provide hard guarantees and strong performance isolation while dramatically reducing queuing requirements that burden prior proposals. Finally, we introduce a comprehensive network architecture that overcomes the bottlenecks of earlier designs with respect to area, energy, and QOS in future highly-integrated chips. The proposed NOC uses a topology-centric QOS approach that restricts the extent of hardware QOS support to a fraction of the network without compromising guarantees. In doing so, network area and energy efficiency are significantly improved. Further improvements are derived through a novel flow-control mechanism, along with switch- and link-level optimizations. In concert, these techniques yield a network capable of interconnecting over a thousand terminals on a die while consuming 47% less area and 26% less power than a state-of-the-art QOS-enabled NOC. The mechanisms proposed in this dissertation are synergistic and enable efficient, high-performance interconnects for future chips integrating hundreds or thousands of on-die resources. They address deficiencies in routing, topologies, and flow control of existing architectures with respect to area, energy, and performance scalability. They also serve as a building block for cost-effective advanced services, such as QOS guarantees at the die level.Item On some residual and locally virtual properties of groups(2010-05) Katerman, Eric Michael; Reid, Alan W.; Gordon, Cameron; Luecke, John; Allcock, Daniel; G�l, AnnaWe define a strong form of subgroup separability, which we call RS separability, and we use this to combine LERF and Agol’s RFRS condition on groups into a property called LVRSS. We show that some infinite classes of groups that are known to be both subgroup separable and virtually RFRS are also LVRSS. We also provide evidence for the naturalness of RS separability and LVRSS by showing that they are preserved under various operations on groups.Item PipeSynth : automated topological and parametric design of fluid networks(2010-12) Patterson, William Rey; Campbell, Matthew I.; Seepersad, Carolyn C.PipeSynth is a design automation approach that combines various optimization research and artificial intelligence methods for synthesizing fluid networks. Starting with only the port locations, PipeSynth generates and optimizes the most effective network for a given application. This ideal network is found by not only optimizing the sizes of each pipe and orientation of fittings in the network (parameters), but also optimizing the layouts of how they are all connected (topology). Using Uniform-Cost-Search for topology optimization, and a combination of non-gradient based parametric optimization methods,PipeSynth demonstrates how advances in automated design can enable engineers to manage much more complex fluid network problems. PipeSynth uses a unique representation of fluid networks that synthesizes and optimizes networks one pipe at a time, in three-dimensional space. PipeSynth has successfully solved several problems containing multiple interlaced networks concurrently with multiple inputs and outputs. PipeSynth shows the power of automated design and optimization in producing solutions more effectively and efficiently than traditional design approaches.Item Pretzel knots of length three with unknotting number one(2012-05) Staron, Eric Joseph; Gordon, Cameron, 1945-; Gompf, Robert; Luecke, John; Namazi, Hossein; Ozsvath, Peter; Reid, AlanThis thesis provides a partial classification of all 3-stranded pretzel knots K=P(p,q,r) with unknotting number one. Scharlemann-Thompson, and independently Kobayashi, have completely classified those knots with unknotting number one when p, q, and r are all odd. In the case where p=2m, we use the signature obstruction to greatly limit the number of 3-stranded pretzel knots which may have unknotting number one. In Chapter 3 we use Greene's strengthening of Donaldson's Diagonalization theorem to determine precisely which pretzel knots of the form P(2m,k,-k-2) have unknotting number one, where m is an integer, m>0, and k>0, k odd. In Chapter 4 we use Donaldson's Diagonalization theorem as well as an unknotting obstruction due to Ozsv\'ath and Szab\'o to partially classify which pretzel knots P(2,k,-k) have unknotting number one, where k>0, odd. The Ozsv\'ath-Szab\'o obstruction is a consequence of Heegaard Floer homology. Finally in Chapter 5 we explain why the techniques used in this paper cannot be used on the remaining cases.Item RAxML and FastTree: Comparing Two Methods for Large-Scale Maximum Likelihood Phylogeny Estimation(Public Library of Science, 2011-11-21) Liu, Kevin; Linder, C. Randal; Warnow, TandyStatistical methods for phylogeny estimation, especially maximum likelihood (ML), offer high accuracy with excellent theoretical properties. However, RAxML, the current leading method for large-scale ML estimation, can require weeks or longer when used on datasets with thousands of molecular sequences. Faster methods for ML estimation, among them FastTree, have also been developed, but their relative performance to RAxML is not yet fully understood. In this study, we explore the performance with respect to ML score, running time, and topological accuracy, of FastTree and RAxML on thousands of alignments (based on both simulated and biological nucleotide datasets) with up to 27,634 sequences. We find that when RAxML and FastTree are constrained to the same running time, FastTree produces topologically much more accurate trees in almost all cases. We also find that when RAxML is allowed to run to completion, it provides an advantage over FastTree in terms of the ML score, but does not produce substantially more accurate tree topologies. Interestingly, the relative accuracy of trees computed using FastTree and RAxML depends in part on the accuracy of the sequence alignment and dataset size, so that FastTree can be more accurate than RAxML on large datasets with relatively inaccurate alignments. Finally, the running times of RAxML and FastTree are dramatically different, so that when run to completion, RAxML can take several orders of magnitude longer than FastTree to complete. Thus, our study shows that very large phylogenies can be estimated very quickly using FastTree, with little (and in some cases no) degradation in tree accuracy, as compared to RAxML.Item Smaller Gene Networks Permit Longer Persistence in Fast-Changing Environments(Public Library of Science, 2011-04-25) Malcom, Jacob W.The environments in which organisms live and reproduce are rarely static, and as the environment changes, populations must evolve so that phenotypes match the challenges presented. The quantitative traits that map to environmental variables are underlain by hundreds or thousands of interacting genes whose allele frequencies and epistatic relationships must change appropriately for adaptation to occur. Extending an earlier model in which individuals possess an ecologically-critical trait encoded by gene networks of 16 to 256 genes and random or scale-free topology, I test the hypothesis that smaller, scale-free networks permit longer persistence times in a constantly-changing environment. Genetic architecture interacting with the rate of environmental change accounts for 78% of the variance in trait heritability and 66% of the variance in population persistence times. When the rate of environmental change is high, the relationship between network size and heritability is apparent, with smaller and scale-free networks conferring a distinct advantage for persistence time. However, when the rate of environmental change is very slow, the relationship between network size and heritability disappears and populations persist the duration of the simulations, without regard to genetic architecture. These results provide a link between genes and population dynamics that may be tested as the -omics and bioinformatics fields mature, and as we are able to determine the genetic basis of ecologically-relevant quantitative traits.Item Smaller, Scale-Free Gene Networks Increase Quantitative Trait Heritability and Result in Faster Population Recovery(Public Library of Science, 2011-02-09) Malcom, Jacob W.One of the goals of biology is to bridge levels of organization. Recent technological advances are enabling us to span from genetic sequence to traits, and then from traits to ecological dynamics. The quantitative genetics parameter heritability describes how quickly a trait can evolve, and in turn describes how quickly a population can recover from an environmental change. Here I propose that we can link the details of the genetic architecture of a quantitative trait—i.e., the number of underlying genes and their relationships in a network—to population recovery rates by way of heritability. I test this hypothesis using a set of agent-based models in which individuals possess one of two network topologies or a linear genotype-phenotype map, 16–256 genes underlying the trait, and a variety of mutation and recombination rates and degrees of environmental change. I find that the network architectures introduce extensive directional epistasis that systematically hides and reveals additive genetic variance and affects heritability: network size, topology, and recombination explain 81% of the variance in average heritability in a stable environment. Network size and topology, the width of the fitness function, pre-change additive variance, and certain interactions account for ~75% of the variance in population recovery times after a sudden environmental change. These results suggest that not only the amount of additive variance, but importantly the number of loci across which it is distributed, is important in regulating the rate at which a trait can evolve and populations can recover. Taken in conjunction with previous research focused on differences in degree of network connectivity, these results provide a set of theoretical expectations and testable hypotheses for biologists working to span levels of organization from the genotype to the phenotype, and from the phenotype to the environment.Item State sums in two dimensional fully extended topological field theories(2011-05) Davidovich, Orit; Freed, Daniel S.; Ben-Zvi, David; Distler, Jacques; Reid, Alan; Rodriguez-Villegas, FernandoA state sum is an expression approximating the partition function of a d-dimensional field theory on a closed d-manifold from a triangulation of that manifold. To consider state sums in completely local 2-dimensional topological field theories (TFT's), we introduce a mechanism for incorporating triangulations of surfaces into the cobordism ([infinity],2)-category. This serves to produce a state sum formula for any fully extended 2-dimensional TFT possibly with extra structure. We then follow the Cobordism Hypothesis in classifying fully extended 2-dimensional G-equivariant TFT's for a finite group G. These are oriented theories in which bordisms are equipped with principal G-bundles. Combining the mechanism mentioned above with our classification results, we derive Turaev's state sum formula for such theories.Item Symmetries of knots, branched cyclic covers, and L-spaces(2021-08) Turner, Hannah Kathryn; Gordon, Cameron, 1945-; Boyer, Steven; Danciger, Jeffrey; Luecke, JohnThis dissertation studies the L-space conjecture among manifolds which are branched cyclic covers of links. We present three main results. First, we construct new families of knots all of whose branched cyclic covers are L-spaces. Then, we give an almost complete characterization of which cyclic branched covers of double-twist knots have left-orderable fundamental groups. Finally, we relate the notion of visibility of certain symmetries of an alternating knot to the Heegaard Floer homology of its cyclic branched covers.