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2014

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Replication And Knowledge Production In Empirical Software Engineering Research, Jonathan L. Krein Dec 2014

Replication And Knowledge Production In Empirical Software Engineering Research, Jonathan L. Krein

Theses and Dissertations

Although replication is considered an indispensable part of the scientific method in software engineering, few replication studies are published each year. The rate of replication, however, is not surprising given that replication theory in software engineering is immature. Not only are replication taxonomies varied and difficult to reconcile, but opinions on the role of replication contradict. In general, we have no clear sense of how to build knowledge via replication, particularly given the practical realities of our research field. Consequently, most replications in software engineering yield little useful information. In particular, the vast majority of external replications (i.e., replications performed …

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The Existence Of A Discontinuous Homomorphism Requires A Strong Axiom Of Choice, Michael Steven Andersen Dec 2014

The Existence Of A Discontinuous Homomorphism Requires A Strong Axiom Of Choice, Michael Steven Andersen

Theses and Dissertations

Conner and Spencer used ultrafilters to construct homomorphisms between fundamental groups that could not be induced by continuous functions between the underlying spaces. We use methods from Shelah and Pawlikowski to prove that Conner and Spencer could not have constructed these homomorphisms with a weak version of the Axiom of Choice. This led us to define and examine a class of pathological objects that cannot be constructed without a strong version of the Axiom of Choice, which we call the class of inscrutable objects. Objects that do not need a strong version of the Axiom of Choice are scrutable. We …

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Microchip Thermal Gradient Gas Chromatography, Anzi Wang Dec 2014

Microchip Thermal Gradient Gas Chromatography, Anzi Wang

Theses and Dissertations

Although the airbath oven is a reliable heating method for gas chromatography (GC), resistive heating is needed for higher analytical throughput and on-site chemical analysis because of size, heating rate and power requirements. In the last thirty years, a variety of resistive heating methods were developed and implemented for both benchtop and portable GC systems. Although fast heating rates and low power consumption have been achieved, losses in column efficiency and resolution, complex construction processes and difficulties experienced in recovering damaged columns have also become problematic for routine use of resistively heated columns. To solve these problems, a new resistively …

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Framework To Implement Authentication, Authorization And Secure Communications In A Multiuser Collaborative Cax Environment, Francis Mensah Dec 2014

Framework To Implement Authentication, Authorization And Secure Communications In A Multiuser Collaborative Cax Environment, Francis Mensah

Theses and Dissertations

Computer Aided Design (CAD) applications have historically been based on a single user per application architecture. Although this architecture is still popular to date, it does have several drawbacks. First of all the single user CAD architecture inhibits a concurrent engineering design process where several designers can work on the same model simultaneously. This limitation introduces time inefficiency especially when a project involves geographically dispersed designers. A solution to these drawbacks could be a transition from the traditional single user CAD architecture to a multiuser collaborative architecture. Advances in computer networking technologies, especially relating to the Internet, have provided the …

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Fuel Response To Mechanical Mastication Of Pinyon-Juniper Woodlands In Utah, Alan Wyatt Shakespear Dec 2014

Fuel Response To Mechanical Mastication Of Pinyon-Juniper Woodlands In Utah, Alan Wyatt Shakespear

Theses and Dissertations

Pinyon-juniper woodland encroachment threatens ecosystem function and diversity on sagebrush steppe. Decreased fire frequency likely favors proliferation of pinyon-juniper woodlands and subsequent decline in desirable understory species. Increased tree cover produces hazardous canopy fuel loads that contribute to severe crown fires and threaten life and property at the wildland-urban-interface. Mechanical mastication converts large canopy fuels into small woody debris, altering wildfire dynamics from a potential crown fire to a more controllable surface fire. We measured fuel loading and cover on untreated, masticated, and masticated + burned treatments on 30-m transects within 30 X 33-m subplots, representing 45 different sites throughout …

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A Covering System With Minimum Modulus 42, Tyler Owens Dec 2014

A Covering System With Minimum Modulus 42, Tyler Owens

Theses and Dissertations

We construct a covering system whose minimum modulus is 42. This improves the previous record of 40 by P. Nielsen.

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Modeling And Testing Of Bistable Waterbomb Base Configurations, Brandon Holbrook Hanna Dec 2014

Modeling And Testing Of Bistable Waterbomb Base Configurations, Brandon Holbrook Hanna

Theses and Dissertations

Origami is making an impact in engineering as solutions to problems are being found by applying origami principles (eg. flat-foldability) and using specific crease patterns as inspiration. This thesis presents an in-depth analysis of a particular origami fold -- the waterbomb base -- to facilitate its use in future engineering problems. The watebomb base is of interest due to its familiarity to the origami community, simple topology (can be made by folding a single sheet of paper four times), scalability, generalizability, and interesting kinetic behavior. It can behave as a nonlinear spring as well as a one- or two-way bistable …

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The Bioluminescence Heterozygous Genome Assembler, Jared Calvin Price Dec 2014

The Bioluminescence Heterozygous Genome Assembler, Jared Calvin Price

Theses and Dissertations

High-throughput DNA sequencing technologies are currently revolutionizing the fields of biology and medicine by elucidating the structure and function of the components of life. Modern DNA sequencing machines typically produce relatively short reads of DNA which are then assembled by software in an attempt to produce a representation of the entire genome. Due to the complex structure of all but the smallest genomes, especially the abundant presence of exact or almost exact repeats, all genome assemblers introduce errors into the final sequence and output a relatively large set of contigs instead of full-length chromosomes (a contig is a DNA sequence …

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A Hybrid Method For Sensitivity Optimization With Application To Radio-Frequency Product Design, Abraham Lee Dec 2014

A Hybrid Method For Sensitivity Optimization With Application To Radio-Frequency Product Design, Abraham Lee

Theses and Dissertations

A method for performing robust optimal design that combines the efficiency of experimental designs and the accuracy of nonlinear programming (NLP) has been developed, called Search-and-Zoom. Two case studies from the RF and communications industry, a high-frequency micro-strip band-pass filter (BPF) and a rectangular, directional patch antenna, were used to show that sensitivity optimization could be effectively performed in this industry and to compare the computational efficiency of traditional NLP methods (using fmincon solver in MATLAB R2013a) and they hybrid method Search-and-Zoom. The sensitivity of the BPF's S11 response was reduced from 0.06666 at the (non-robust) nominal optimum to 0.01862 …

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A Desk Study Of The Education Policy Implications Of Using Data From Multiple Sources: Example Of Primary School Teacher Supply And Demand In Malawi, Moses Khombe Dec 2014

A Desk Study Of The Education Policy Implications Of Using Data From Multiple Sources: Example Of Primary School Teacher Supply And Demand In Malawi, Moses Khombe

Theses and Dissertations

Malawi, as a country with very limited resources, needs to have educational policies in place to maximize effectiveness of the public education system. Policymakers depend on accurate data, but variations in data between sources leaves policymakers uncertain as they attempt to craft policies to address the growing educational crisis in Malawi. A desk study was performed to evaluate the policy implications of employing data from multiple sources using primary school teacher supply and demand in Malawi as an illustration. This study examined one national organization, Malawi's Ministry of Education, Science, and Technology (MoEST); three international aid and assistance organizations (IAAOs), …

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The Practice Of Neogeography In Community-Based Organizations, Patrick Oberle Dec 2014

The Practice Of Neogeography In Community-Based Organizations, Patrick Oberle

Theses - ALL

Neogeography and Volunteered Geographic Information (VGI) are two terms that have emerged recently to describe the practice of geography by those not formally trained in it as a discipline and spatial data provided by individuals through social media and other Web-based tools. Both neogeography and VGI can be directly linked to the growth of various online mapping websites and applications that allow for the creation of electronic maps that are interactive, adaptable, and easily shared via the Internet and Web. As recent phenomena, the practice of neogeography and VGI is not well understood, nor are the links these new fields …

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Differences In Mathematics And Literacy Achievement Between African American Males And Other 11th-Grade Students, Jeffery Flanigan Dec 2014

Differences In Mathematics And Literacy Achievement Between African American Males And Other 11th-Grade Students, Jeffery Flanigan

College of Education Dissertations

The gap in school performance between African American students and students of other subpopulations has shown up in a range of academic success measures including course grades, test scores, course selection, and college graduation rates (Johnston & Viadero 2000); college and high school grade point averages [GPAs] (Banks & Banks, 2004); and dropout rates (Gordon, 1999; Green, 2001; Irvine & Armento, 2001; Jencks & Phillips, 1998; Kober, 2001; Lee, 2002). These indicators all point to a disturbing pattern in the educational system and beg the question: What is going on with the education of African American males in the United …

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Front Tracking Method With High-Order Enhancement And Its Application In Two-Phase Micromixing Of Incompressible Viscous Fluids, Yijie Zhou Dec 2014

Front Tracking Method With High-Order Enhancement And Its Application In Two-Phase Micromixing Of Incompressible Viscous Fluids, Yijie Zhou

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

We develop a front tracking method based on the hydrodynamic library FronTier for the solution of the governing equations of motion for two-phase micromixing of incompressible, viscous, liquid-liquid solvent extraction processes. The method is used for accurate simulation of the turbulent micromixing dynamics of an aqueous and an organic phase exposed to intense centrifugal force and shearing stress. The onset of mixing is the result of the combination of the classical Rayleigh-Taylor and Kelvin-Helmholtz instabilities. We demonstrate verification and convergence results for one-phase and unmixed, two-phase flows. For mixed, two-phase flow a mixing environment that emulates a sector of the …

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Event-Based Modeling And Control Of An Advanced Manufacturing System, Yang Li Dec 2014

Event-Based Modeling And Control Of An Advanced Manufacturing System, Yang Li

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

Although the widely application of advanced IT technology has enabled production information to become increasingly transparent, detailed and real-time, the utilization of the information for system modeling and control remains largely unexplored. In fact, instantaneously transforming information gathered from a vast array of sources into useful knowledge for effective decision making has been identified as one of the six grand challenges in the vision for manufacturing 2020 and beyond. It is necessary to have a real-time integrated system modeling and control method, which utilizes the production information to quickly respond to unpredictable disturbance to manufacturing system and ensure smooth production …

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Protein Dimerization Mechanisms Study With Molecular Dynamics Simulation, Yuan Yao Dec 2014

Protein Dimerization Mechanisms Study With Molecular Dynamics Simulation, Yuan Yao

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

Protein dimerization is involved in many essential biological processes like gene expression, allosteric regulation, enzymatic activation and signal transduction. However, our understanding about dimerization mechanisms is limited despite of the progress made. In this work, we applied molecular dynamics simulation with all-atom structure-based model to the investigation often commonly studied regulatory dimers. Through the combinational analysis of intrinsic energy funnels, density of states, thermodynamic free energy landscapes, phi-values of transition states, and kinetic simulations, new and detailed mechanisms of dimerization and their coupling with monomer folding are revealed, in good agreements with existing experimental evidences and suggestions. We found five …

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Representations Of Gender In Young Adult (Ya) Literature, 1960-2010, Maria I. Cipriani Dec 2014

Representations Of Gender In Young Adult (Ya) Literature, 1960-2010, Maria I. Cipriani

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

This dissertation focuses the lens of queer studies and trauma studies on YA literature published in the United States between 1960 and 2010 to demonstrate that authors embed cultural messages in YA texts. Those messages for YA readers intend to model behavior considered appropriate and teach about normative sex and gender roles. Individuals between the ages of 10 and 20 comprise YA readership, and the means of teaching these readers cultural norms include imagery and cautionary tales. The content of the literature inculcates views of acceptable behaviors with respect to sexuality and gender roles and generally conflates sex and gender. …

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Achieving Regulatory Compliance In Data Management, Sumeet Vijay Bajaj Dec 2014

Achieving Regulatory Compliance In Data Management, Sumeet Vijay Bajaj

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

Regulations mandate consistent procedures for information access, processing, and storage. In the United States alone, over 10,000 data management regulations exist in the financial, life sciences, health care and government sectors. A recurrent theme in data management regulations is the need for regulatory compliant storage to ensure data confidentiality, data integrity, audit trails maintenance, data retention, and guaranteed deletion. This thesis describes the design and implementation of several regulatory compliant relational databases and file systems. The systems increase efficiency and lower costs of regulatory compliance through the use of novel cryptographic and system security constructs. The first system described in …

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Ricci Flow And Its Applications, Min Zhang Dec 2014

Ricci Flow And Its Applications, Min Zhang

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

This thesis introduces the Ricci Flow and Its applications. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. General Ricci flow is defined on arbitrary dimensional Riemannian manifolds. Surface (2-manifold) Ricci flow has unique characteristics, which are crucial for developing discrete theories and designing computational algorithms. The unified theoretic framework for discrete Surface Ricci Flow is innovated, including all the common schemes: Tangential Circle Packing, Thurston's Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, we also introduce novel schemes, Virtual Radius Circle …

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Computational Modeling And Analysis Of Cardiac Excitation, Abhishek Murthy Dec 2014

Computational Modeling And Analysis Of Cardiac Excitation, Abhishek Murthy

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

Modeling, analysis, and control of cardiac excitation, the biological process cardiac cells undergo on a periodic basis, is indispensable for understanding and countering life-threatening electrical disturbances of the heart, such as atrial and ventricular fibrillation. Modeling involves the derivation of a mathematical/computational representation of the dynamics of cardiac cells and their diffusivity. The key to effective modeling lies in striking a balance between i) precision, the ability of the model to replicate the underlying biological phenomena, and ii) performance, in terms of amenability to automated analysis techniques, such as simulation and formal verification. The techniques of abstraction and compositionality have …

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Dividend Taxes, Financial Frictions, And The U.S. Great Depression, Lunan Jiang Dec 2014

Dividend Taxes, Financial Frictions, And The U.S. Great Depression, Lunan Jiang

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

As shown by McGrattan (2012), an anticipated increase in dividend taxes plays an important role in explaining the dramatic investment decline during the U.S. Great Depression. The first chapter of this dissertation attempts to test whether this conclusion is still robust when household heterogeneity and precautionary saving motives are taken into consideration. I build an Aiyagari model with dynamic firms, dividend taxes, and labor productivity shocks that accounts for the U.S. earnings and wealth inequality using 1930s data. The conclusion is that the impact of anticipated increases of dividend taxes on the investment is very sensitive to the presence of …

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Parallel And Flexible Hardware Implementation Of Fletcher Checksum, Maria Isabel Mera Collantes Dec 2014

Parallel And Flexible Hardware Implementation Of Fletcher Checksum, Maria Isabel Mera Collantes

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

Checksums are utilized in many contexts such as communications, storage and reliable processing. The balance between checksum strength, implementation cost and obtained throughput often pose a challenge for present day system designers. In this research we propose two new methods for implementing the Fletcher Checksum (FC) in a parallelized context. We determined an extended parallel definition from the original FC and applied it to two different hardware implementation approaches. We then created a generator that would automatically output parameterized designs. We controlled the input word length, number of parallel inputs and architecture of the designs, and we then synthesized these …

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Refining Mid-Infrared Emission Spectroscopy As A Tool For Understanding Planetary Surface Mineralogy Through Laboratory Studies, Computational Models, And Lunar Remote Sensing Data, Jessica Anne Arnold Dec 2014

Refining Mid-Infrared Emission Spectroscopy As A Tool For Understanding Planetary Surface Mineralogy Through Laboratory Studies, Computational Models, And Lunar Remote Sensing Data, Jessica Anne Arnold

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

Mid-infrared emissivity spectra, while useful for quantifying mineral abundance, are dependent on a variety of other factors, including grain size. At present, scattering models are not fully able reproduce this dependence, especially for very fine-grained powders (d~<60μ m) typical of planetary regolith. Previous studies have modeled emissivity spectra of mineral powders by using Mie single scattering theory coupled with a radiative transfer model, such as Hapke [1993]. Chapter 2 of this work uses multiple scattering T-matrix code (from Mackowski and Mishchenko, 2011) to calculate the scattering coefficients for a cluster of spheres, rather than a single sphere as in Mie theory. I acquired emissivity spectra of highly uniform silica glass spheres, normally used for sieve calibration, for comparison to modeled spectra. Then, I calculated optical constants of the glass from micro-FTIR reflectance spectra of polished grains mounted on a petrographic slide. Modeled emissivity spectra computed with this T-matrix/radiative transfer hybrid method show a marked improvement over Mie-radiative transfer hybrid models, especially for smaller grain sizes. The complex index of refraction (ñ =n+ik) as a function of wavelength, where n and k are often referred to as optical constants, is an essential input into radiative transfer models. Hence, quantitative estimates of mineral abundance from remote sensing data require that optical constants are known for a comprehensive set of relevant minerals. While many geologically abundant materials are monoclinic or triclinic, due to both the additional complexity of the models needed and the increased effort required to orient these samples, very few sets of optical constants have been measured for lower-symmetry minerals. Chapter 3 describes how optical constants were calculated from polarized mid-infrared reflectance spectra of oriented single-crystals of five natural samples in the monoclinic crystal system, which include four pyroxene samples and one orthoclase sample. Lastly, in Chapter 4, remote sensing emissivity data from the Diviner Lunar Radiometer Experiment, along with lunar environment laboratory measurements of mineral mixtures, are used to establish constraints on the absolute abundance of olivine at sites where it has been detected by the visible/near-infrared instruments Moon Mineralogy Mapper and Kaguya Spectral Profiler. The two main results of this section are: 1) spectra do not mix linearly at these wavelengths for particles sizes typical of lunar regolith and 2) Diviner data are consistent with a wide range in olivine abundances at these sites. | 170 pages

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Models And Numerical Algorithms For Hydrodynamics With Atomic Processes And High Energy Density Applications, Hyoungkeun Kim Dec 2014

Models And Numerical Algorithms For Hydrodynamics With Atomic Processes And High Energy Density Applications, Hyoungkeun Kim

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

The main goal of the research is the development of numerical equation of state model enabling hydrodynamic simulations of problems involving multiple ionization and it's application in the high energy density physics (HEDP) such as the plasma jet driven magnetoinertial fusion (PJMIF) concept and the pellet injection in Tokamak via simulations studies. For this goal, the mathematical models and numerical algorithms for ionization process has been developed, and they are demonstrated in several simulations with productive result. A numerical model for an average ionization equation-of-state (EOS) for high-Z plasmas undergoing multiple ionization processes has been developed by improving the classical …

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Software Tools For Stochastic Simulations Of Turbulence, Ryan Kaufman Dec 2014

Software Tools For Stochastic Simulations Of Turbulence, Ryan Kaufman

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

We present two software tools useful for the analysis of mesh based physics application data, and specifically for turbulent mixing simulations. Each has a broader, but separate scope, as we describe. Both features play a key role as we push computational science to its limits and thus the present work contributes to the frontier of research. The first tool is Wstar, a weak* comparison tool, which addresses the stochastic nature of turbulent flow. The goal is to compare underresolved turbulent data in convergence, parameter dependence, or validation studies. This is achieved by separating space-time data from state data (e.g. density, …

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Mathematical Modeling Of G-Protein-Coupled Receptor Signaling Pathways, Tao Jiang Dec 2014

Mathematical Modeling Of G-Protein-Coupled Receptor Signaling Pathways, Tao Jiang

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

G-protein-coupled receptor (GPCR) signaling plays an important role in converting extracellular stimuli into cellular responses. Many biological functions are regulated by GPCR signaling, and nearly 40\% of current pharmaceuticals target GPCRs. In this work, mathematical modeling is used to investigate the initial steps of GPCR signaling with two major aims: 1) to understand the causes for non-canonical signaling behaviors; 2) to understand how signaling specificity is reacted to in the individual reactions of the pathway. The classic ternary complex model describing the interaction between a single ligand, receptor and G-protein served as our basic model for the initial investigation. Dose …

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Neural Correlates And Peripheral Mirnas Associated With Stress-Related Telomere Shortening, Stephanie Izzi Dec 2014

Neural Correlates And Peripheral Mirnas Associated With Stress-Related Telomere Shortening, Stephanie Izzi

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

Telomeres are repetitive nucleoprotein complexes at the ends of chromosomes that provide a protective cap for genomic DNA. Telomere shortening leads to cellular senescence or apoptosis. Accordingly, telomere length is considered both a mechanism and marker for aging. Increased exposure to chronic stress is associated with shorter telomere length. In addition, differences in stress reactivity have been associated with telomere length. Here I will describe several studies identifying neural circuits associated with such telomere shortening, an evaluation of saliva as an appropriate biomaterial to measure telomere length, and an exploration of the role of microRNAs as mediators of stress-related telomere …

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Modeling Intra-Day Markets With An Application Of Risk Management And Optimal Order Execution, Yikang Chai Dec 2014

Modeling Intra-Day Markets With An Application Of Risk Management And Optimal Order Execution, Yikang Chai

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

Financial time series data exhibits heavy tailed, volatility clustering and long range dependence style facts. Traditional Gaussian distribution assumption based model failed to explain these phenomena. A unified framework model proposed in this thesis, fractionally integrated autoregressive moving average (FARIMA) and fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) with multivariate generalized hyperbolic distribution (MGHD), trying to capture all these phenomena together. We also examined this model by using intra-day market dataset to backtest of various risk measure. With rise of high frequency trading and algorithm trading in recent years, trading volume hugely increased and markets became more volatile. Order execution …

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Structure Prediction Of Carbon Dioxide Hydrate And Xenon Hydrate, Hongfei Xu Dec 2014

Structure Prediction Of Carbon Dioxide Hydrate And Xenon Hydrate, Hongfei Xu

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

Gas hydrate technology is one of research focus in recent years, and can be applied to solve problems in energy, environment, and other areas. Previous views had been that most gas hydrates decompose into gas and water (or ice) under 1 GPa or so, thus, research on hydrate structures in past few decades were conducted under low pressure conditions. Recent studies have shown the possibility that new structures of gas hydrate may exist under higher pressure, and this thesis aims to explore the structures of hydrate in carbon dioxide-water system and xenon-water system with the application of high pressure. The …

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Damage And Alteration Of Mangroves Inhabited By A Marine Wood-Borer, Timothy Mathias Davidson, Catherine E. De Rivera, Hwey-Lian Hsieh Dec 2014

Damage And Alteration Of Mangroves Inhabited By A Marine Wood-Borer, Timothy Mathias Davidson, Catherine E. De Rivera, Hwey-Lian Hsieh

Environmental Science and Management Faculty Publications and Presentations

Animals can exert a strong influence on the structure and function of foundation species such as mangroves. Because mangroves live at the interface of land and sea, both terrestrial and marine species affect them, including numerous herbivores and boring species. These organisms can affect the fecundity, performance, and morphology of mangroves. In a mangrove stand in southwestern Taiwan, we discovered that mangroves were extensively damaged by woodboring isopods Sphaeroma terebrans. We examined the relationships between burrowing damage from S. terebrans and metrics of mangrove fecundity, performance, and morphology. Individuals of Rhizophora stylosa that were more burrowed by isopods had significantly …

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Bhaskara’S Approximation For The Sine, Karel Stroethoff Dec 2014

Bhaskara’S Approximation For The Sine, Karel Stroethoff

The Mathematics Enthusiast

The 7th century Indian mathematician Bhaskara (c.600 – c.680) obtained a remarkable approximation for the sine function. Many subsequent ancient authors have given versions of this rule, but none provided a proof or described how the result was obtained. Grover [1] provides a possible explanation, but I think the rule can be explained more clearly. Rather than give the rule first, we will derive it, and then discuss its accuracy, and explore some alternative approximations. Our derivation is simply an exercise in modeling.

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