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(9,201 - 9,220 of 9,317)
Pages
- Title
- Large-Signal Transient Stability and Control of Inverter-Based Resources
- Creator
- Wang, Duo
- Date
- 2024
- Description
-
Renewable generation, including solar photovoltaic (PV) systems, type 3 and 4 wind turbine generation systems (WTG), battery energy storage...
Show moreRenewable generation, including solar photovoltaic (PV) systems, type 3 and 4 wind turbine generation systems (WTG), battery energy storage systems (BESS), as well as high voltage direct current (HVDC) and flexible alternating current (FACT) transmission system devices with increasing penetration level are being connected to the bulk power systems (BPS) via power electronic (PE) converters as the interface, referred to as the inverter-based resources (IBRs) on the transmission and sub-transmission levels or distributed energy resources (DERs) located on the distribution level. The IBR is almost entirely defined by the control algorithms and found to be more prone to experiencing large disturbances due to the lack of the conventional synchronous machine (SM) intrinsic synchronous characteristics and mechanical inertia, as well as being in smaller capacity sizes. Thus, these reasons motivate this dissertation to study the large-signal transient stability and control of IBRs for reliable grid integration and rapid grid transformation. For large-signal stability analysis methods, Lyapunov-based methods are the fundamental theory used to characterize the stability issues with analytical solutions, although other non-Lyapunov methods could also be very helpful. A main difficulty hindering the widespread adoption of the Lyapunov stability analysis method is the difficulty of finding the proper Lyapunov function candidate for a higher dimensional nonlinear system. The Port-Hamiltonian (PH) nonlinear control theory is explored in this dissertation as a promising theoretical framework solution addressing this challenging issue. A PH-based tracking and robust control method is proposed to facilitate the practical application of the PH framework in IBR controls. In addition, considering the typical grid-forming (GFM) IBR control with a first-order low pass filter (LPF) block is usually involved with control saturation function for protection purposes under abnormal operating conditions with anti-windup issue in practical implementation, a PH-based bounded LPF (PH-BLPF) control is proposed to incorporate this in the large-signal PH interconnection modeling framework while preserving the robust tracking Lyapunov stability with improved transient dynamic performance and stability margin.Moreover, specific real-world transient synchronization stability issues, such as the grid voltage large fault disturbance case, are studied. In addition, to meet the recent emerging IBR grid code requirements, such as the current magnitude limitation, grid support function, and fault recovery capability of GFM-VSCs, a virtual impedance-based current-limiting GFM control with enhanced transient stability and grid support is proposed.
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- Title
- Retrospective Quantitative T1 Imaging to Examine Characteristics of Multiple Sclerosis Lesions
- Creator
- Young, Griffin James
- Date
- 2024
- Description
-
Quantitative MRI plays an essential role in assessing tissue abnormality and diseaseprogression in multiple sclerosis (MS). Specifically, T1...
Show moreQuantitative MRI plays an essential role in assessing tissue abnormality and diseaseprogression in multiple sclerosis (MS). Specifically, T1 relaxometry is gaining popularity as elevated T1 values have been shown to correlate with increased inflammation, demyelination, and gliosis. The predominant issue is that relaxometry requires parametric mapping through advanced imaging techniques not commonly included in standard clinical protocols. This leaves an information gap in large clinical datasets from which quantitative mapping could have been performed. We introduce T1-REQUIRE, a retrospective T1 mapping method that approximates T1 values from a single T1-weighted MR image. This method has already been shown to be accurate within 10% of a clinically available reference standard in healthy controls but will be further validated in MS cohorts. We also further aim to determine T1-REQUIRE’s statistical significance as a unique biomarker for the assessment of MS lesions as they relate to clinical disability and disease burden. A 14-subject comparison between T1-REQUIRE maps derived from 3D T1 weighted turbo field echoes (3D T1w TFE) and an inversion-recovery fast field echo (IRFFE) revealed a whole-brain voxel-wise Pearson’s correlation of r = 0.89 (p < 0.001) and mean bias of 3.99%. In MS white matter lesions, r = 0.81, R2 = 0.65 (p < 0.001, N = 159), bias = 10.07%, and in normal appearing white matter (NAWM), r = 0.82, R 2 = 0.67 (p < 0.001), bias = 9.48%. Mean lesional T1-REQUIRE and MTR correlated significantly (r = -0.68, p < 0.001, N = 587) similar to previously published literature. Median lesional MTR correlated significantly with EDSS (rho = -0.34, p = 0.037), and lesional T1-REQUIRE exhibited xiii significant correlations with global brain tissue atrophy as measured by brain parenchymal fraction (BPF) (r = -0.41, p = 0.010, N = 38). Multivariate linear regressions between T1- REQUIRE NAWM provided meaningful statistical relationships with EDSS (β = 0.03, p = 0.027, N = 38), as well as did mean MTR values in the Thalamus (β = -0.27, p = 0.037, N = 38). A new spoiled gradient echo variation of T1-REQUIRE was assessed as a proof of concept in a small 5-subject MS cohort compared with IR-FFE T1 maps, with a whole brain voxel-wise correlation of r = 0.88, R2 = 0.77 (p < 0.001), and Bias = 0.19%. Lesional T1 comparisons reached a correlation of r = 0.75, R2 = 0.56 (p < 0.001, N = 42), and Bias = 10.81%. The significance of these findings means that there is the potential to provide supplementary quantitative information in clinical datasets where quantitative protocols were not implemented. Large MS data repositories previously only containing structural T1 weighted images now may be used in big data relaxometric studies with the potential to lead to new findings in newly uncovered datasets. Furthermore, T1-REQUIRE has the potential for immediate use in clinics where standard T1 mapping sequences aren’t able to be readily implemented.
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- Title
- Two Essays on Mergers and Acquisitions
- Creator
- Xu, Yang
- Date
- 2024
- Description
-
This dissertation is composed of two self-contained chapters that both relate to mergers and acquisitions (M&A). In the first essay, we...
Show moreThis dissertation is composed of two self-contained chapters that both relate to mergers and acquisitions (M&A). In the first essay, we examine the Delaware (DE) reincorporation effect on firms’ post-IPO behaviors on mergers and acquisitions. We find that firms’ DE reincorporation decisions enhance the likelihood of engaging in M&A as targets. However, as a tradeoff, DE reincorporated firms get lower takeover valuations compared to stay-at-home-state firms, and the acquisition of reincorporated firms is less likely to be successful. Our second essay aims to explore the role of the options market in price discovery for M&A. We find that the predictive power of the changes in implied volatility of the target firm stock for the takeover outcome is statistically and economically significant. The risk arbitrage portfolios incorporating filters derived from the options on stocks of the target firms generate annualized risk-adjusted abnormal returns between 2.6% and 5%, depending on the portfolio weighting method, the threshold of filters for the implied volatility change, and the asset pricing models applied for abnormal returns. The results are robust to different empirical setups and are not explained by traditional factors.
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- Title
- Heterogeneous Workloads Study towards Large-scale Interconnect Network Simulation
- Creator
- Wang, Xin
- Date
- 2023
- Description
-
High-bandwidth, low-latency interconnect networks play a key role in the design of modern high- performance computing (HPC) systems. The ever...
Show moreHigh-bandwidth, low-latency interconnect networks play a key role in the design of modern high- performance computing (HPC) systems. The ever-increasing need for higher bandwidth and higher message rate has driven the design of low-diameter interconnect topologies like variants of dragonfly. As these hierarchical networks become increasingly dominant, interference caused by resource sharing can lead to significant network congestion and performance variability. Meanwhile, with the rapid growth of the machine learning applications, the workloads of future HPC systems are anticipated to be a mix of scientific simulation, big data analytics, and machine learning applications. However, little work has been conducted to understand performance implications of co-running heterogeneous workloads on large-scale dragonfly systems. There is a greater need to study how different interconnect technologies affect workload performance, and how conventional scientific applications interact with emerging big data applications at the underlying interconnect level. In this work, we firstly present a comparative analysis exploring the communication interference for traditional HPC applications by analyzing the trade-off between localizing communication and balancing network traffic. We conduct trace-based simulations for applications with different communication patterns, using multiple job placement policies and routing mechanisms. Then we develop a scalable workload manager that provides an automatic framework to facilitate hybrid workload simulation. We investigate various hybrid workloads and navigate various application-system configurations for a deeper understanding of performance implications of a diverse mix of workloads on current and future supercomputers. Finally, we propose a scalable framework, Union+, that enables simulation of communication and I/O simultaneously. By combining different levels of abstraction, Union+ is able to efficiently co-model the communication and I/O traffic on HPC systems that equipped with flash-based storage. We conduct experiments with different system configurations, showing how Union+ can help system designers to assess the usefulness of future technologies in next-generation HPC machines.
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- Title
- Empowering Visually Impaired Individuals With Holistic Assistance Using Real-Time Spatial Awareness System
- Creator
- Yu, Xinrui
- Date
- 2024
- Description
-
The integration of artificial intelligence (AI) into daily life opens unprecedented avenues for enhancing the experiences of visually impaired...
Show moreThe integration of artificial intelligence (AI) into daily life opens unprecedented avenues for enhancing the experiences of visually impaired individuals, offering them greater autonomy and quality of life. This thesis introduces a Visually Impaired Spatial Awareness (VISA) system designed to assist visually impaired individuals holistically through a structured approach. At the foundational level, the VISA system incorporates several key technologies to interpret the surroundings and assist in basic navigation tasks. It utilizes Augmented Reality (AR) markers to facilitate recognition of places and aid in navigation, employs neural network models for advanced object detection and tracking, and leverages depth information for accurate object localization. Progressing to the intermediate level, the VISA system integrates the data obtained from object detection and depth sensing to assist in more complex navigational tasks such as obstacle avoidance and pathfinding toward a desired destination. At the advanced level, the VISA system synthesizes the capabilities developed at the foundational and intermediate levels to enhance the spatial awareness of visually impaired users, allowing them to undertake complex tasks, such as navigating complex environments and locating specific items. The VISA system also emphasizes efficient human-machine interaction, incorporating text-to-speech and speech-to-text technologies to facilitate natural and intuitive communication between the user and the system. The VISA system's performance was evaluated in different environments simulating real-world scenarios. The experimental results show that the user can interact with our system intuitively with minimal effort, and affirm that the VISA system can effectively assist the visually impaired user in locating and reaching for objects, navigating indoors, identifying merchandise, and recognizing both handwritten and printed texts.
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- Title
- Utilizing Image Processing in Evaluation of Fibroblast Stimulation for Collagen Remodeling
- Creator
- Yoon, Shin Hae
- Date
- 2023
- Description
-
This research delves into the realm of image processing as a pivotal component in the evaluation of fibroblast stimulation for collagen...
Show moreThis research delves into the realm of image processing as a pivotal component in the evaluation of fibroblast stimulation for collagen remodeling. The study focuses on unraveling the intricate synergy between electrospun silk fibroin-carbon nanotube (SF-CNT) fibers and electrical stimulation, working in harmony to enhance tissue regeneration. Building upon our previous work, we successfully engineered SF-CNT fibers through the electrospinning process, yielding highly aligned structures reminiscent of natural extracellular matrix proteins. These fibers were fortified with water stability through post-treatment with ethanol vapor, while subtle additions of carbon nanotubes (CNTs) significantly improved fiber alignment, strength, and conductivity without compromising biocompatibility. This innovative platform served as a cell culture matrix for fibroblasts harvested from pelvic organ prolapse (POP) patients, facilitating electrical stimulation that triggered a substantial increase in collagen production. In this study, we harnessed the power of various image-processing software tools, including ImageJ and Python, to analyze immunostained images of fibroblasts obtained from POP patients. Under carefully tailored electrical stimulation conditions, the stimulated cells exhibited an astonishing up to 11.97-fold increase in alpha-smooth muscle actin (α-SMA) expression, unequivocally signifying the successful activation of myofibroblasts. Additionally, in an animal model employing LOX-knockout mice to mimic collagen disorders associated with POP, the application of optimized electrical stimulation conditions for patient 003 led to a remarkable surge in collagen production and structural enhancement, underlining the potential of electrical stimulation in expediting tissue remodeling. Intriguingly, fibroblasts from patient 005 and patient 006 exhibited a distinct response, shedding light on the influence of POP severity on cellular behavior. This study firmly reinforces the imperative of personalized therapeutic approaches, emphasizing the need to customize treatment strategies to align with individual patient characteristics through innovative biological image analysis techniques.
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- Title
- Effect of organic acid treatment in reducing Salmonella on six types of sprout seeds
- Creator
- Yang, Dachuan
- Date
- 2023
- Description
-
Fresh sprouts present a special food safety concern as their growing conditions also favor the growth of pathogens such as Salmonella....
Show moreFresh sprouts present a special food safety concern as their growing conditions also favor the growth of pathogens such as Salmonella. Contamination in sprouts often originates from the seeds used for sprouting. The Produce Safety Rule requires that seeds used to grow sprouts be treated to reduce pathogens. The treatments may be applied by sprout growers or by seed suppliers. Although 20,000 ppm calcium hypochlorite is the most used seed treatment method, the high chlorine level can be hazardous to workers and the environment. Alternative seed treatment methods that are safe and environmentally friendly are needed. In addition, a post-treatment drying step is needed when seed suppliers are using chemical seed treatment methods. This study evaluated the efficacy of an organic acid solution for reducing Salmonella on six types of seeds (alfalfa, clover, radish, mung bean, onion, and broccoli). The impact of treatment on seed germination and sprout yield was also examined. Ten grams of seeds inoculated with a five-serotype cocktail of Salmonella were pre-rinsed with 40 ml of water twice and treated with 75.7 ml of the organic acid solution for 1 hour. The treated seeds were either not rinsed or rinsed with 40 ml of water twice before being dried in the biological safety cabinet for 24 hours. The Salmonella level, germination percentage, and sprout yield of seeds treated with water, seeds treated with the organic acid solution, seeds treated with organic acid, dried, and rinsed, and seeds treated with organic acids, dried, and not rinsed were compared. Salmonella reductions that could be achieved with this organic acid solution treatment were less than 0.5 log CFU/g without drying, 0.6-2.0 log CFU/g with drying and rinse, or 1.6-2.9 log CFU/g with drying and no rinse. Drying significantly enhanced the treatment efficacy (p < 0.05 ) on alfalfa, radish, mung bean, and onion seeds. If seeds were not rinsed after treatment, the log reductions achieved on mung bean and onion seeds were significantly higher (p < 0.05). If seeds were treated and rinsed, the germination rates of six types of seeds were not affected (p > 0.05) regardless of whether the seeds were dried or not. All treatments significantly decreased the sprout yield of clover seeds by 13% (p < 0.05 ). If seeds were not rinsed after treatment, the germination rates of clover and broccoli seeds were reduced by 7 and 9%, respectively, and the sprout yield of alfalfa seeds was reduced by 35%. Overall, the organic acid solution was ineffective when compared with 20,000 ppm calcium hypochlorite in reducing Salmonella on sprout seeds, although the drying step after treatment could improve the treatment efficacy.
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- Title
- Grand View Blvd...Los Angeles Jan 91
- Creator
- Gordon, Robert Philip
- Date
- 1991-01
- Description
-
Watercolor painting of the intersection of an unknown street with Grand View Boulevard in Los Angeles, California.
- Collection
- Robert Philip Gordon papers, 1963-2010
- Title
- New Object
- Creator
- Zhang, He
- Title
- Ink and Pastel Drawing, undated
- Creator
- Henry, Mary Dill, 1913-2009
- Description
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Untitled ink and pastel drawing by Mary Henry of a hand, possibly her own. Drawing is undated.
- Collection
- Mary Dill Henry Papers, 1913-2021
- Title
- Bronzeville, Chicago, Illinois, 1961
- Date
- 1961
- Description
-
Color photograph depicting the Bronzeville neighborhood in 1961. Coverage includes the 3400 south blocks of Michigan Avenue, Indiana Avenue,...
Show moreColor photograph depicting the Bronzeville neighborhood in 1961. Coverage includes the 3400 south blocks of Michigan Avenue, Indiana Avenue, Prairie Avenue, and Calumet Avenue in Chicago, Illinois. One newer white commercial building at the left is the former Chicago Defender building. Photographer unknown.
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- Bronzeville photo, 1961
- Title
- Design on a Hill
- Creator
- Henry, Mary Dill, 1913-2009, Ziegler, William
- Description
-
Photograph of Design on a Hill (1940), a stone lithograph by Mary Dill Henry, completed as a part of her studies with Ray Bertrand....
Show morePhotograph of Design on a Hill (1940), a stone lithograph by Mary Dill Henry, completed as a part of her studies with Ray Bertrand. Handwritten inscription on verso: "Lithograph #2, 1940 Federal Art Project, litho pencil on stone. Ray Bertrand, San Francisco Art Project, was my teacher." Typewritten inscription on verso: "Mary Henry: Design on a Hill, 1940. Stone lithograph, 12" x 16". Courtesy of the artist, Freeland, WA. Photo: Wm. Ziegler." Photograph by William Ziegler. Date of photograph unknown.
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- Mary Dill Henry Papers, 1913-2021
- Title
- Plate 5
- Creator
- Henry, Mary Dill, 1913-2009
- Date
- 1936-1939
- Description
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Untitled ink drawing identified on recto as "Plate 5." Date of drawing is unknown. Date range listed is approximate.
- Collection
- Mary Dill Henry Papers, 1913-2021
- Title
- Ink Drawings, 1981
- Creator
- Henry, Mary Dill, 1913-2009
- Date
- 1981-02-17
- Description
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Untitled ink drawings by Mary Henry. Inscription on verso reads "8' x 6' Feb 17 81" and also contains what appear to be mathematical...
Show moreUntitled ink drawings by Mary Henry. Inscription on verso reads "8' x 6' Feb 17 81" and also contains what appear to be mathematical calculations.
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- Mary Dill Henry Papers, 1913-2021
- Title
- LOCAL VISCOELASTIC PROPERTIES OF SOFT ANISOTROPIC FIBROUS TISSUE
- Creator
- Gallo, Nicolas Remy
- Date
- 2020
- Description
-
The current aging population, with more than 80 million "baby boomers", will present a steep medical challenge for our society in a...
Show moreThe current aging population, with more than 80 million "baby boomers", will present a steep medical challenge for our society in a foreseeable future. Half of the adults over 85 years old are predicted to be diagnosed with Alzheimer's disease by 2050. With healthcare cost reaching over 700 billion dollars in the United States, early detection of Alzheimer's disease (AD) and other co-existing neurodegenerative diseases is crucial to improve the recovery odds in patients and to decrease individual care cost. This work seeks to tackle this problem by proposing a novel computational framework toward improving the measurement of shear visco-elastic properties of brain white matter (WM), which vary with age. These measurements practically represent the effective (average) response of many cells and are typically obtained by using rheology or elastography. Although the former is direct, the latter requires the solution of an inverse problem based on a priori mechanical tissue model. The mechanical anisotropy of WM has previously not been fully explored although many inconsistencies have been reported in brain MRE experiments. To account for these inconsistencies a transversely isotropic constitutive model for the brain WM is proposed to interpret prior experiments involving 7 young and 4 older healthy men. By employing a novel inversion scheme, we report the local variation of the effective transverse and axial shear moduli in two well aligned WM structures (corpus callosum: CC; and cortical spinal tract: CST) for both the young and old cohort of healthy subjects part of the study. This work reports statistically significant changes in local regional variation of the transverse modulus across the CC for the young cohort. In the older cohort, the trend was similar yet not statistically significant. A novel candidate biomarker, the shear anisotropy metric, defined as the ratio of the transverse and axial shear moduli, found statistically significant local regional variation across the CC but not in the CST. Healthy aging was observed to decrease both transverse and axial in both CC and CST, although the variation was significant only for the CC. Finally, in an effort to understand the cause of effective transverse mechanical properties variation in WM with aging, the connection between effective and intrinsic contribution of WM cellular constituents is established. The intrinsic mechanical contributions of axons and glial matrix are separated by fitting the estimates of the effective shear moduli to a microscopic composite fiber model of myelinated axons embedded in the glial matrix. This work provides a method to establish a baseline for healthy brain mechanical properties thus promising to increase the specificity of MRE toward early diagnosis of neurodegenerative diseases. Additional oscillating disc rheology experiments with decellularized porcine myocardium, and the fabrication of a stable heterogeneous phantom matching the mechanical, diffusional and electrical properties of the WM provide foundational knowledge for due development and validation of MRE methodologies employed in other tissues.
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- Title
- Aqua Hawk
- Creator
- Keller, Keith
- Date
- 1977
- Description
-
Aqua Hawk documents the creation of a concrete canoe by Civil Engineering students at Illinois Institute of Technology for a race sponsored by...
Show moreAqua Hawk documents the creation of a concrete canoe by Civil Engineering students at Illinois Institute of Technology for a race sponsored by the American Concrete Institute. The film shows the different stages of the construction of the canoe, named the Aqua Hawk, and its eventual competition at a regional race held near Champaign, Illinois on April 24, 1976. Directed by Illinois Tech student Keith Keller, the film was narrated by Institute of Design instructor Charles Dee Sharp.
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- Office of Marketing and Communications films, 1958-1988
- Title
- My IIT
- Creator
- Namburi, Kartheeka
- Date
- 2024
- Description
-
Master's Project focused on developing a user-friendly mobile application for Illinois Institute of Technology (IIT).
- Title
- Laser Powder Bed Fusion Of Cost-Effective Non-Spherical Ti-6Al-4V Powder
- Creator
- Asherloo, Mohammadreza
- Date
- 2023
- Description
-
This comprehensive research delves into the intricate dynamics of Laser Powder Bed Fusion (L-PBF) of Ti-6Al-4V powders, emphasizing the...
Show moreThis comprehensive research delves into the intricate dynamics of Laser Powder Bed Fusion (L-PBF) of Ti-6Al-4V powders, emphasizing the potential of non-spherical, hydride-dehydride (HDH) powders as a cost-efficient alternative to traditional spherical powders. The study systematically explores the interplay between powder morphology, granulometry, and various post-processing treatments in shaping the resultant microstructure, porosity, and mechanical properties of L-PBF fabricated Ti-6Al-4V components.Initial investigations focused on the flowability, packing density, and resultant density of L-PBF parts using HDH powders with varying size distributions. Through meticulous optimization of laser parameters, parts with a relative density exceeding 99.5% were achieved, even at production rates 1.5–2 times higher than conventional LPBF processes. Dynamic synchrotron X-ray imaging provided insights into laser-powder interactions, revealing key mechanisms of porosity formation associated with HDH powders. Further microstructural examinations highlighted the formation of columnar β grains with acicular α/α′ phases in the as-built condition. Mechanical tests, including fatigue assessments under fully-reversed tension-compression conditions, revealed the critical role of surface roughness in fatigue performance. Notably, mechanical grinding significantly improved fatigue strength, especially in the high cycle fatigue region, by eliminating surface micro-notches. X-ray diffraction analyses further elucidated the stress and micro-strain profiles, offering insights into the material's deformation mechanisms. A pivotal discovery was the presence of α/α′ on prior β/β grain boundaries, challenging the prevailing notion that high cooling rates in L-PBF preclude β/β grain boundary variant selection. Electron backscatter diffraction and synchrotron X-ray imaging illuminated the role of powder characteristics in locally modulating cooling rates, leading to β/β grain boundary α′ lath growth. Lastly, the research underscored the multifaceted interdependencies among contouring, powder granulometry, Hot Isostatic Pressing (HIP), and mechanical surface treatments. A pronounced increase in sub-surface porosities was identified when contouring was combined with fine powder granulometry. However, post-HIP treatments induced a phase transformation from martensitic α′ to a basket-weave α+β microstructure, enhancing the material's fatigue resistance to levels comparable to wrought Ti-6Al-4V. In summation, this doctoral research offers a holistic understanding of the L-PBF process for Ti-6Al-4V, emphasizing the viability of non-spherical HDH powders and providing a roadmap for parameter optimization, defect minimization, and mechanical property enhancement in L-PBF-fabricated Ti-6Al-4V structures.
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- Title
- Case Study: A Comparison of Pedagogical Content Knowledge Between Coaches and Coaches/Mentees
- Creator
- Barone, Ana MargaritaSalinas
- Date
- 2024
- Description
-
This multiple case study dissertation aimed to examine one of the domains of pedagogical content knowledge, knowledge of content and students,...
Show moreThis multiple case study dissertation aimed to examine one of the domains of pedagogical content knowledge, knowledge of content and students, between different types of elementary coaches and between coach and their respective collaborating teachers. It also investigated the impact a coaches’ background experiences have on the dynamic between coaches and teachers and the perceptions' teacher have on the effectiveness of coaching. The theoretical framework used in this qualitative study was Ball, Thames, and Phelps’ (2008) definition of PCK. Data was collected from six coaches–four instructional coaches and two math coaches–and eleven k-5th grade teachers. Data collection involved a survey, LMT assessment, and semi-structured interviews, and a thematic analysis method was conducted. The findings from the cross-case analysis resulted in ten themes, with the majority having multiple categories. One finding to one of the research questions was that there were no differences in knowledge of content and students between mathematics coaches and general instructional coaches, but other areas to further investigate emerged. Another finding was that coaches were either within the same capacity as their respective teachers or had extra knowledge of content and students. Although the majority of the coaches’ knowledge of content and students was at a higher level according to their LMT score, it does not necessarily mean that coaches are working with teachers in improving knowledge of content and students. In addition, more research is recommended in creating a pedagogical content knowledge instrument that is specific for coaches.
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