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2014

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A Greedy Method To Simulate Drainage In Cross Sections, Long Li Dec 2014

A Greedy Method To Simulate Drainage In Cross Sections, Long Li

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Porous media, such as rocks and soil, make up the largest proportion of mass in nature. Pore-network model is a widely used model to describe the tiny and complicated inner structure of porous media. Assisted by the development of X-ray computed microtomography in recent years, researchers are able to extract the geometric characters of inner pores and throats from porous media, then further explore two-phase flow in porous media more exactly. Drainage, a process that a non-wetting phase fluid replaces the other wetting fluid, is the research topic of this thesis. The boundaries between two different phases, named as menisci …

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Multi-Armed Bandits With Applications To Markov Decision Processes And Scheduling Problems, Isa Mithqal Muqattash Dec 2014

Multi-Armed Bandits With Applications To Markov Decision Processes And Scheduling Problems, Isa Mithqal Muqattash

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

The focus of this work is on practical applications of stochastic multi-armed bandits (MABs) in two distinctive settings. First, we develop and present REGA, a novel adaptive sampling-based algorithm for control of finite-horizon Markov decision processes (MDPs) with very large state spaces and small action spaces. We apply a variant of the epsilon-greedy multi-armed bandit algorithm to each stage of the MDP in a recursive manner, thus computing an estimation of the " reward-to-go" value at each stage of the MDP. We provide a finite-time analysis of REGA. In particular, we provide a bound on the probability that the approximation …

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Mechanical Properties Of 3-D Printed Cellular Foams With Triangular Cells, Pratap Kumar Bunga Dec 2014

Mechanical Properties Of 3-D Printed Cellular Foams With Triangular Cells, Pratap Kumar Bunga

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

In the present work, poly lactic acid (PLA) is used as a model system to investigate the mechanical behavior of 3-D printed foams with triangular cells. Solid PLA tension and compression specimens and foams made of PLA were fabricated using fused deposition 3-D printing technique. The solid PLA tension specimens were characterized for their densities and found to be about 10% lower in density as compared to their bulk counter parts. The triangular foams had a relative density of about 64%. The relationships between the structure of the foams and its deformation behavior under compression along two in-plane directions were …

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Modeling Of Parachute Dynamics With Gpu Enhanced Continuum Fabric Model And Front Tracking Method, Qiangqiang Shi Dec 2014

Modeling Of Parachute Dynamics With Gpu Enhanced Continuum Fabric Model And Front Tracking Method, Qiangqiang Shi

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

An advanced mesoscale spring-mass model is used to mimic fabric surface motion. The fabric surface is represented by a high-quality triangular surface mesh. Both the tensile stiffness and the angular stiffness of each spring are determined by the material's Young's modulus and Poisson ratio, as well as the geometrical characteristics of the surface mesh. The spring-mass system is a nonlinear Ordinary Differential Equation (ODE) system solved by fourth order Runge-Kutta method. The model is shown to be numerically convergent under the constraint that the summation of points masses is constant. Through coupling with an incompressible fluid solver and the front …

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Rheological Characterization Of Nano Composite Hydrogels, Jack Lombardi Dec 2014

Rheological Characterization Of Nano Composite Hydrogels, Jack Lombardi

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Engineered Polymer hydrogels and hydrogels from Bio macromolecules have visco-elastic properties that can be measured using Oscillatory Shear Rheology. Manipulation and measurement of physical properties in gels including F-127 Pluronic Block Co-Polymer and Poly(N-isopropylacrylamide)-Clay are shown through OSR by addition of salts, clays and glucose at physiological levels. Rheological analysis of f-127 illustrates changes in G' reduction with phase transition temperature. Measurements also indicate physical changes due to the aforementioned additives vary as a function of the gel physical and chemical structure. In particular, non-enzymatic glycation is shown to change the modulus of elasticity in both of the gels tested. …

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Molecular Modeling Of The Interaction Between Chitosan And Silver And Palladium, Wenbo Wang Dec 2014

Molecular Modeling Of The Interaction Between Chitosan And Silver And Palladium, Wenbo Wang

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

The purpose of this research was to study the interaction between chitosan and metal surfaces at the atomic level in order to better understand the mechanism of formation of catalytic noble metal nanoparticle (where chitosan serves as a shaping or encapsulating agent). In this study, Molecular dynamics (MD) simulation was employed to study chitosan adsorption on different Silver and Palladium crystallographic planes at the atomic level. Chitosan were formulated by molecular dynamics simulation named Materials Studio 6.0 using COMPASS force field. By calculated result of the solubility parameter δ , the length of chitosan was chosen 20 repeated units. Through …

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Space Of Kã¤Hler Potentials On Singular And Non-Compact Manifolds | Space Of Kähler Potentials On Singular And Non-Compact Manifolds, Seyed Ali Aleyasin Dec 2014

Space Of Kã¤Hler Potentials On Singular And Non-Compact Manifolds | Space Of Kähler Potentials On Singular And Non-Compact Manifolds, Seyed Ali Aleyasin

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Let H be the space of Kähler metrics in a fixed cohomology class. This space may be endowed with a Weil-Petersson-type metric, referred to as the Mabuchi metric, which allows one to study the geometry of H. It is now well-known that the geometry of the space of Kähler potentials, in particular, the geodesics in H, may be used for studying `canonical metrics' on the base manifold. In order to be interpreted as the potential of a Kähler metric, however, one needs to prove certain regularity for such solutions. In the first part, I shall discuss deriving of weighted estimates …

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Applications Of The Seiberg-Witten Equations To The Differential Geometry Of Non-Compact Kahler Manifolds, Ilya Elson Dec 2014

Applications Of The Seiberg-Witten Equations To The Differential Geometry Of Non-Compact Kahler Manifolds, Ilya Elson

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Soon after the introduction of the Seiberg-Witten equations, and their magnificent application to the differential topology of 4-manifolds, LeBrun [LeB95] used these equations to study differential geometry and prove a rigidity theorem for compact complex hyperbolic manifolds. Biquard [Biq97] extended these results to non-compact, finite volume complex hyperbolic manifolds, and Rollin [Rol04] extended these techniques to CH2. Finally, Di Cerbo[DC12b, DC11] applied Biquard's techniques to the product of two negatively curved Riemann surfaces. The main tool that allows one to use the Seiberg-Witten equations to study differential geometry is an integral scalar curvature estimate The principle difficulty in …

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Braiding Non-Ribbon Surfaces And Constructing Broken Fibrations On Smooth 4-Manifolds, Mark Clifford Hughes Dec 2014

Braiding Non-Ribbon Surfaces And Constructing Broken Fibrations On Smooth 4-Manifolds, Mark Clifford Hughes

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

In this thesis we study various notions of surface braidings in 4-space, and their applications to the construction of singular fibrations on smooth oriented 4-manifolds. We define the notion of braided link cobordisms in S3 × [0,1], which generalize Viro's closed 2-braids in S4. We prove that via isotopy any properly embedded oriented surface W in S3 × [0,1] can be brought to this special position, and that the isotopy can be taken rel boundary when the boundary already consists of closed braids. These surfaces are closely related to another notion of surface braiding in D2 × D2, called braided …

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On The Uniqueness Of Singular Kaehler-Einstein Metrics, Long Li Dec 2014

On The Uniqueness Of Singular Kaehler-Einstein Metrics, Long Li

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Bando and Mabuchi proved the uniqueness of Kaehler-Einstein metrics on Fano manifolds up to a holomorphic automorphism in 1987. Then recently Berndtsson generalized the uniqueness result of Kaehler-Einstein metrics to bounded potentials. We give a new proof of the Bando-Mabuch-Berndtsson uniqueness theorem in a different aspect, based on a new technique developed from Chen's C^{1,1} geodesic and Futaki's spectral formula. Finally, the uniqueness of the conical Kaehler-Einstein metrics will be discussed under the assumption of properness of twisted Ding-functional. | 61 pages

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Uniqueness Of Ricci Flow Solution On Non-Compact Manifolds And Integral Scalar Curvature Bound, Xiaojie Wang Dec 2014

Uniqueness Of Ricci Flow Solution On Non-Compact Manifolds And Integral Scalar Curvature Bound, Xiaojie Wang

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

In this dissertation, we prove two results. The first is about the uniqueness of Ricci flow solution. B.-L. Chen and X.-P. Zhu first proved the uniqueness of Ricci flow solution to the initial value problem by assuming bilaterally bounded curvature over the space-time. Here we show that, when the initial data has bounded curvature and is non-collapsing, the complex sectional curvature bounded from below over the space-time guarantees the short-time uniqueness of solution. The second is about the integral scalar curvature bound. A. Petrunin proved that for any complete boundary free Riemannian manifold, if the sectional curvature is bounded from …

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Shintani Zeta Functions And Weyl Group Multiple Dirichlet Series, Jun Wen Dec 2014

Shintani Zeta Functions And Weyl Group Multiple Dirichlet Series, Jun Wen

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

In recent years, substantial progresses have been made towards the development of a general theory of multiple Dirichlet series with functional equations. In this dissertation, we investigate the Shintani zeta function associated to a prehomogeneous vector space and identify it with a Weyl group multiple Dirichlet series. The example under consideration is the set of 2 by 2 by 2 integer cubes, that is the integral lattice in a certain pre-homogeneous vector space acted on by three copies of GL(2). One of M. Bhargava's achievements is the determination of the corresponding integral orbits and the discovery of an extension of …

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Numerical Modeling And Combustion Studies Of Scramjet Problem, Ying Xu Dec 2014

Numerical Modeling And Combustion Studies Of Scramjet Problem, Ying Xu

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Physical and numerical modeling of turbulent mixing and turbulent combustion in a Scramjet combustion chamber is studied. A high-order finite volume scheme is applied to the flow, accompanied with dynamic subgrid-scale model and local averaging procedure to account for the effect of unresolved small length scales turbulent fluid behavior. The Arrhenius law kinetic mechanism is applied to describe the nonlinearity of chemical reactions. To model combustion, the finite rate chemistry is developed to characterize the combustion process inside the combustion chamber. Results are compared among the finite rate model, the flamelet model/progress variable approach adopted by Stanford PSAAP center and …

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On Geometric And Motivic Realizations Of Variations Of Hodge Structure Over Hermitian Symmetric Domains, Zheng Zhang Dec 2014

On Geometric And Motivic Realizations Of Variations Of Hodge Structure Over Hermitian Symmetric Domains, Zheng Zhang

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Based on the work of Gross and Sheng and Zuo, Friedman and Laza show that over every irreducible Hermitian symmetric domain there exists a canonical variation of real Hodge structure of Calabi-Yau type. The first part of the thesis concerns motivic realizations of the canonical Calabi-Yau variations over irreducible Hermitian symmetric domains of tube type. In particular, we show that certain rational descents of the canonical variations of Calabi-Yau type over irreducible tube domains of type $A$ can be realized as sub-variations of Hodge structure of certain variations which are naturally associated to families of abelian varieties of Weil type. …

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Effects Of Marine Environments And Cyclic Loading On Carbon Fiber-Vinyl Ester Composites, Arash Afshar Dec 2014

Effects Of Marine Environments And Cyclic Loading On Carbon Fiber-Vinyl Ester Composites, Arash Afshar

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Fiber-reinforced composites are increasingly being employed in advanced composite based military marine crafts as they can assist in achieving cost savings, weight reductions and increased stealth capabilities. In these crafts, composites are considered for a wide range of applications, such as for superstructures, decks, bulkheads, masts and propellers, among others. However, in marine applications, composites are often exposed to aggressive environments, which include: ultraviolet radiation, salt water and cyclic loadings, to name a few. With time, such harsh elements can individually and cooperatively degrade the mechanical properties of fiber-reinforced polymer composites, diminishing their structural integrity and durability. Accordingly, ensuring the …

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On The Dynamics Of Non-Linear Systems And Actuation Devices, Dake Feng Dec 2014

On The Dynamics Of Non-Linear Systems And Actuation Devices, Dake Feng

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

The research results being presented in this thesis is divided into two areas. The first area is related to the development of a systematic method for model parameter identification of a large class of fully and not fully controlled nonlinear dynamics systems such as robot manipulators. The developed method is based on Trajectory Pattern Method (TPM). The developed method uses trajectory patterns with feed-forward controls to identify the system model parameters. The developed method ensures full system stability; does not require close initial estimated values for the system parameters to be identified; and provides a systematic method of emphasizing on …

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Finding Place: The Art Of Wangechi Mutu, Maria Magdalena Campos-Pons And Ingridmwangiroberthutter, Eva Susanne Hericks-Bares Dec 2014

Finding Place: The Art Of Wangechi Mutu, Maria Magdalena Campos-Pons And Ingridmwangiroberthutter, Eva Susanne Hericks-Bares

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

This dissertation examines the issue of " place" in the lives and works of three contemporary women artists of the African Diaspora: Maria Magdalena Campos-Pons, IngridMwangiRobertHutter, and Wangechi Mutu. Place as a location is a core challenge of globalism: though fundamental to our existence, it has ceased to be permanent or definite. However, place is more than location--it is cultural, gendered, historical, interactive, personal and social, among other things--and can only be fully understood in considering several of these different facets. Each of the artists in this dissertation has had to navigate through instances of bias (such as racism and …

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Enhanced Backbone Dihedral Parameters For Protein Simulations, Carmenza Martinez Dec 2014

Enhanced Backbone Dihedral Parameters For Protein Simulations, Carmenza Martinez

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Molecular Dynamics (MD) and Monte Carlo (MC) simulations of biomolecules have become a supportive tool to experimental methods for structure refinement, protein folding, drug discovery and many more applications. This is due to the increase in computational power and necessity of finding a method capable of describing the forces at work in chemical systems without the need to do extensive quantum calculations. All of these are possible because of the classical definition of the potential force field function. The force field function is essential to perform MD simulations, since this equation carries the physical description of the molecules being studied. …

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Improved Generalized Born Solvent Model Parameters For Protein And Nucleic Acid Simulations, Hai Minh Nguyen Dec 2014

Improved Generalized Born Solvent Model Parameters For Protein And Nucleic Acid Simulations, Hai Minh Nguyen

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Generalized Born solvent model offers inexpensive approach for solvation energy calculation as compared to explicit solvent and Poisson-Boltzmann (PB) method. Thanks to its speed, GB models have been widely used in molecular dynamics (MD) simulations. However the speed comes with tradeoffs. Literatures have pointed out the weaknesses of GB models in inaccurately calculating solvation energy that leads to helical bias in protein simulation or unstable DNA/RNA duplex in nucleic acid simulation. Here we introduced the reparameterization of the recently developed GB-Neck model to improve its accuracy. Compared to other pairwise GB models (e.g. GB-OBC and GB-Neck) the new GB models …

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Evidence For Genetic Contributions To Neurobehavioral Traits: Trait Fear/Fearlessness, Behavioral Inhibition, And The Late Positive Potential, Anna Ef Weinberg Dec 2014

Evidence For Genetic Contributions To Neurobehavioral Traits: Trait Fear/Fearlessness, Behavioral Inhibition, And The Late Positive Potential, Anna Ef Weinberg

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Psychiatric disorders are the most common and costly forms of disease and injury worldwide. Despite this, our ability to accurately identify and treat these disorders is limited, due in part to deficient understanding of the underlying mechanisms of dysfunction. The current study aimed to identify genetically-influenced neurobehavioral traits--individual differences evident in both self-report and neural response--that might help improve classification and treatment of mental disorders. Results from event-related potential (ERP) components sensitive to emotional content demonstrated that a) neural processing of emotional content is heritable, b) the heritable traits of heightened trait fear and behavioral disinhibition are associated with unique …

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The Macroeconomics Of Bank Competition Over The Business Cycle, Yan Liu Dec 2014

The Macroeconomics Of Bank Competition Over The Business Cycle, Yan Liu

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Two empirical regularities of bank lending practices stand out: interest rate spreads on loans and lending standards both are lower during booms than in recessions. I provide a unified explanation of these two facts, stressing procyclical competition of the banking sector as the driving force. I first develop a game of bank lending with screening. Borrowers have private information about the creditworthiness of their projects. Banks rely on a screening technology to distinguish good projects from bad ones by choosing the screening intensity, which I identify with the lending standards. Because screening is costly, in the optimum the screening intensities …

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Image Reconstruction Theory And Implementation For Low-Dose X-Ray Computed Tomography, Yan Liu Dec 2014

Image Reconstruction Theory And Implementation For Low-Dose X-Ray Computed Tomography, Yan Liu

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

The excessive X-ray radiation exposure during clinical examinations has been reported to be linked to increase lifetime risk of cancers in patients. Directly lower computed tomography (CT) dose without improving reconstruction technique will degrade the image quality and is not acceptable. The objective of this dissertation is investigating novel reconstruction methods to improve image quality in low-dose cases. In practice, it is usually more convenient to improve the conventional analytical methods by refining projection model and designing new filters due to the fast computing time and low computational complexity. However, the reconstructions from analytical methods are still sensitive to artifacts …

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Distributed Estimation In The Presence Of Correlation, Zhiyuan Weng Dec 2014

Distributed Estimation In The Presence Of Correlation, Zhiyuan Weng

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

We study the problem of distributed estimation, where a group of nodes are required to cooperate with each other to estimate some parameter of interest from noisy measurements without a fusion center. Distributed estimation algorithms are useful in several areas, including wireless sensor networks, where robustness, scalability, flexibility, and low power consumption are desirable. In this dissertation, we mainly focus on the cases where the node measurements are correlated. First, we consider the problem of fusing multiple estimates from different nodes. Cases of both known and unknown correlation are investigated. A Bayesian approach and a convex optimization approach are proposed. …

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Polymodal Grain-Size Modes In Long Island Sands, Silts, And Weathered Bedrock, Courtney Anne Melrose Dec 2014

Polymodal Grain-Size Modes In Long Island Sands, Silts, And Weathered Bedrock, Courtney Anne Melrose

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

The purpose of this study is to determine if grain size modes already present in weathered bedrock sediments can be retained and inherited in derived, or processed sediments. To test this theory, weathered bedrock sediment, as well as Cretaceous aged sands and silts, Glacial sands and silts, Loess-like silt sediment, and glacially transported granitic gneisses were collected and analyzed for grain size. This study is based on the implicit assumption that grain size modes are indicative only of process, and are not reflective of modes present in the parent material of the sediment. A distinct pattern of grain size modes …

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High-Throughput Single-Cell Copy Number Profiling For Cancer Heterogeneity Analysis, Taimour Baslan Dec 2014

High-Throughput Single-Cell Copy Number Profiling For Cancer Heterogeneity Analysis, Taimour Baslan

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Intra-tumoral genetic heterogeneity has long been recognized, yet remains poorly understood. This has primarily been due to the lack of sensitive technologies to measure it. Genome wide analysis at the level of single cells has recently emerged as a powerful tool to dissect cancer genome heterogeneity. However, to be truly transformative, single cell approaches must accommodate the analysis of large numbers of single cells. Here, using integrative informatics and molecular biology approaches this study presents a robust, low-cost, and high-throughput method to retrieve the genome-wide copy number landscape of hundreds of single cancer cells. Application of the method to human …

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Studies Of Hydrodynamic Processes In Alternative Magneto-Inertial Fusion Devices, Lina Zhang Dec 2014

Studies Of Hydrodynamic Processes In Alternative Magneto-Inertial Fusion Devices, Lina Zhang

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

The main goal of the research is evaluation of the plasma jet driven magneto-inertial fusion (PJMIF) concept via simulations. To achieve this goal, the development of mathematical models and numerical algorithms for PJMIF has been performed, and large-scale simulation studies have been conducted. In the PJMIF concept, a plasma liner, formed by the merger of a large number of radial, highly supersonic plasma jets, implodes on a magnetized plasma target and compresses it to conditions of the fusion ignition. 1- (spherically symmetric), 2- and 3-dimensional simulations of the implosion of plasma liners and compression of plasma targets have been performed …

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Propaedeutic To Philosophy: Dialogue And Truth In Philosophical Practice, Lauren Nelson Hesse Dec 2014

Propaedeutic To Philosophy: Dialogue And Truth In Philosophical Practice, Lauren Nelson Hesse

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

This dissertation advocates a model of dialogue as an approach to education and to philosophical discourse itself that is capable of acknowledging and accommodating the manifold possibilities for human thought. Contemporary philosophy is permeated by a loss of certainty and the increasingly problematic nature of the concept of truth, and the complication of these concepts exposes significant limitations in the ways we conceptualize thought itself. Working in this vein, Gilles Deleuze identifies what he calls an " image of thought," which characterizes the forms we have traditionally allowed thinking to take. His critique of this image, along with its subsequent …

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Questionable Form: An Inquiry Into The Relationship Between Philosophy And Literature, Amir Rauf Alduha Jaima Dec 2014

Questionable Form: An Inquiry Into The Relationship Between Philosophy And Literature, Amir Rauf Alduha Jaima

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

Through a comparative reading of philosophical and literary texts, I examine the distinction between the projects and practices of philosophy and literature. I argue that, like philosophy, novels make arguments and explicitly engage the range of philosophical questions; and like literature, essential elements of philosophy include aesthetic considerations. Moreover, the presumed distinction between philosophy and literature does a disservice to both. If we overlook the literary qualities of philosophy, we risk relegating to the margins important contributions to traditional philosophical problems; and if we overlook the philosophical qualities of literature, we risk passing over some of the most profound elements …

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The Placiality Of Thought, Brian A. Irwin Dec 2014

The Placiality Of Thought, Brian A. Irwin

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

All experience is embodied and, concomitantly, all experience is emplaced. It follows that abstract thought, as a mode of experience, is always emplaced. In what sense is this the case? Following the insights of Maurice Merleau-Ponty, this dissertation seeks to answer this question. A first key insight comes from Phenomenology of Perception: namely, the suggestion that every thought is a bodily experience, and that abstract thought expresses bodily capacities that, through imagination, enter into realms of the virtual. A second key insight comes from Merleau-Ponty's later philosophy, especially The Visible and the Invisible: this is his account of language as …

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Analysis Of Short Range Entangled Topological Phases Protected By Time-Reversal Symmetry, Sebastian Moritz Dick Dec 2014

Analysis Of Short Range Entangled Topological Phases Protected By Time-Reversal Symmetry, Sebastian Moritz Dick

Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions)

We discuss a short-range entangled topological phase in 3+1 dimensions that is pro- tected by time-reversal symmetry. Two models are compared that realize this phase: The first is a construction developed by Chen, Gu, Liu and Wen, which encodes the system’s topological properties in the representation of the symmetry group. The sec- ond theory uses a non-linear sigma model in which the distinct topological phases differ by the way the symmetry acts on the order parameter. Both theories have in common that the modeled phases are in one to one correspondence with the elements of the co- homology group H …

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