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2021

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Calabi-Yau Products: Graded Quivers For General Toric Calabi-Yaus, Sebastián Franco, Azeem Hasan Jan 2021

Calabi-Yau Products: Graded Quivers For General Toric Calabi-Yaus, Sebastián Franco, Azeem Hasan

Publications and Research

The open string sector of the topological B-model on CY (m + 2)-folds is described by m-graded quivers with superpotentials. This correspondence generalizes the connection between CY (m + 2)-folds and gauge theories on the worldvolume of D(5 − 2m)-branes for m = 0, . . . , 3 to arbitrary m. In this paper we introduce the Calabi-Yau product, a new algorithm that starting from the known quiver theories for a pair of toric CYm+2 and CYn+2 produces the quiver theory for a related CYm+n+3 …


Dimers, Orientifolds And Anomalies, Riccardo Argurio, Matteo Bertolini, Sebastían Franco, Eduardo Garcia-Valdecasas, Shani Meynet, Antoine Pasternak, Valdo Tatitscheff Jan 2021

Dimers, Orientifolds And Anomalies, Riccardo Argurio, Matteo Bertolini, Sebastían Franco, Eduardo Garcia-Valdecasas, Shani Meynet, Antoine Pasternak, Valdo Tatitscheff

Publications and Research

We study 4d N = 1 gauge theories engineered via D-branes at orientifolds of toric singularities, where gauge anomalies are cancelled without the introduction of non-compact flavor branes. Using dimer model techniques, we derive geometric criteria for establishing whether a given singularity can admit anomaly-free D-brane configurations purely based on its toric data and the type of orientifold projection. Our results therefore extend the dictionary between geometric properties of singularities and physical properties of the corresponding gauge theories.


Jazz Colors: Pigment Identification In The Gouaches Used By Henri Matisse, Ana Martins, Anne Catherine Prud’Hom, Maroussia Duranton, Abed Haddad, Celine Daher, Anne Genachte-Le Bail, Tiffany Tang Jan 2021

Jazz Colors: Pigment Identification In The Gouaches Used By Henri Matisse, Ana Martins, Anne Catherine Prud’Hom, Maroussia Duranton, Abed Haddad, Celine Daher, Anne Genachte-Le Bail, Tiffany Tang

Publications and Research

Jazz, the illustrated book by Henri Matisse, is a testament to the vitality of the artist in the last decade of his career. The book consists of twenty illustrations reproduced in 370 copies using a stencil-based printing technique and the same Linel gouaches the artist had used for the original maquettes. This study reports on the comprehensive analysis carried out to identify the pigments in the gouaches used in Jazz by transmitted and reflectance infrared, Raman, SERS, and X-ray fluorescence spectroscopies, and describes the lightfastness of these gouaches as evaluated by microfaedometry. This study also highlights the necessity of …


The Interface And Instrumentality Of Eurorack Modular Synthesis, Eliot Bates Jan 2021

The Interface And Instrumentality Of Eurorack Modular Synthesis, Eliot Bates

Publications and Research

In this essay, I will examine a couple of aspects of hardware modular synthesizers which, despite not being a “new” morphological form (most commercially sold synthesizers from the 1960s through the early 1970s were modular), experienced an extremely unlikely expansion of interest and activity in the mid-1990s. With a contemporary modular synth, users buy a powered rack that holds modules specific to one of several competing formats. For the two most widely adopted formats, Eurorack (sometimes called 3U in reference to the height of modules, or Doepfer A-100 after the first branded system to adopt this panel height) and MOTM/MU …


Quantifying The Impact Of Load-Following On Gas-Fired Power Plants, Yifan Wang Jan 2021

Quantifying The Impact Of Load-Following On Gas-Fired Power Plants, Yifan Wang

Graduate Theses, Dissertations, and Problem Reports (ETD)

Due to rapid penetration of renewables into the grid, natural gas combined cycle (NGCC) power plants are being forced to cycle their loads more frequently and rapidly than for which they were designed. However, the impact of load-following operation on plant efficiency and equipment health are currently poorly understood. The objective of this work is to quantify the impact of load-following on the gas-fired plants by developing high-fidelity multi-scale dynamic models. There are four main tasks in this project. First, dynamic model of an NGCC power plant has been developed. The main components of the NGCC plants are the gas …


A Systematic Method For Measuring Gentrification Using Building Permits Data: A Washington D.C Case Study, Andey Fomil Jan 2021

A Systematic Method For Measuring Gentrification Using Building Permits Data: A Washington D.C Case Study, Andey Fomil

Graduate Theses, Dissertations, and Problem Reports (ETD)

Gentrification can significantly alter the socioeconomic, demographic, and commercial aspects of a city. It is a complex process that transforms the characteristics of entire neighborhoods, modifying not only the observable physical aspects, but also the community structure. Traditional quantitative gentrification measurement approaches assess the process through analysis of Census demographic indicators coupled with field visit analysis of the physical built environment. This study proposes a new gentrification measuring approach that combines traditional Census indicators with a new indicator in the form of City Building Permits. Two GIS spatial analysis techniques are utilized to evaluate the effectiveness and accuracy of the …


Enabling Technologies For Medicinal Chemistry And Synthesis: I. Cannabinoids; Ii. Illudalic Acid; Iii. Microwave Chemistry, Harvey Fulo Jan 2021

Enabling Technologies For Medicinal Chemistry And Synthesis: I. Cannabinoids; Ii. Illudalic Acid; Iii. Microwave Chemistry, Harvey Fulo

Graduate Theses, Dissertations, and Problem Reports (ETD)

Innovations in synthetic organic chemistry such as methodology improvements, new chemical transformations and technology development significantly impact medicinal chemistry and continue to drive the drug discovery process. Here, new cyclization strategies to the pharmaceutically underexplored 3,3-dimethylcyclopentane-fused aromatic rings are described. One approach highlighted oxidative cycloisomerization, intermolecular [2+2+2] cyclotrimerization and 5-exo-dig cyclization providing indanoylindole cannabinoids that possess high affinity for human CB2 receptors at nanomolar concentration and good selectivity index. Another featured a [4+2]-type benzannulation reaction which enabled the 5-step synthesis of illudalic acid — compared to the current available sequences (≥16 steps) — and its equipotent and LAR-selective analogs. …


Single And Multi-Photon Laser Induced Fluorescence For Electric Thruster And Fusion Applications, Thomas E. Steinberger Jan 2021

Single And Multi-Photon Laser Induced Fluorescence For Electric Thruster And Fusion Applications, Thomas E. Steinberger

Graduate Theses, Dissertations, and Problem Reports (ETD)

Single and Multi-photon Laser Induced Fluorescence for Electric Thruster and Fusion Applications

Thomas Edward Steinberger

Laser-based diagnostics are increasingly sought after to investigate a variety of plasmas due to their non-perturbative capabilities. Specifically, laser induced fluorescence (LIF) provides a highly localized and precise spectroscopic technique to measure absolute density, temperature, and bulk flow. In this work, LIF and two-photon absorption laser induced fluorescence (TALIF) are used to investigate electric propulsion and fusion-relevant plasmas, respectively. Ion velocity distribution functions (IVDF) of singly ionized atomic iodine (I II) are measured for the first time and lineshape characteristics are presented for the diagnosis …


Dynamic Modelling And Control Of Grid-Level Energy Storage Systems, Sai Pushpitha Vudata Jan 2021

Dynamic Modelling And Control Of Grid-Level Energy Storage Systems, Sai Pushpitha Vudata

Graduate Theses, Dissertations, and Problem Reports (ETD)

The focus of this work is on two energy storage technologies, namely pumped storage hydroelectricity (PHS) and secondary batteries. Under secondary battery technologies, two potential technologies for grid-scale storage, namely high-temperature sodium-sulfur (NaS) battery and vanadium redox flow battery (VRFB), are investigated. PHS is a largescale (>100 MW) technology that stores and generates energy by transporting water between two reservoirs at different elevations. The goal is to develop a detailed dynamic model of PHS and then design the controllers to follow the desired load trajectory accurately with high efficiency. The NaS battery and VRFB are advanced secondary batteries which …


Analysis Of Cell Signaling Perturbations In Response To Chronic Localized Infections, Amber Nicole Prince Jan 2021

Analysis Of Cell Signaling Perturbations In Response To Chronic Localized Infections, Amber Nicole Prince

Graduate Theses, Dissertations, and Problem Reports (ETD)

The tissue-level response to pathogens involves an intricate series of signal transduction events, influenced by immune and healing mediators that alert the host to danger and eliminate the infection. Disruptions to normal signaling events can compromise the host’s ability to respond and lead to the development of chronic infections that cannot be resolved without clinical intervention. Prolonged inflammation due to chronic infection can damage tissues and compromise healing processes, thus, the interactions of immune and healing mediators in signaling cascades are intimately linked to tissue health outcomes. Studying signaling networks relevant to these responses provided a more thorough understanding of …


Experimental Investigations Of Contact Friction And Transport Properties Of Monolayer And Bilayer Graphene, Prakash Gajurel Jan 2021

Experimental Investigations Of Contact Friction And Transport Properties Of Monolayer And Bilayer Graphene, Prakash Gajurel

Graduate Theses, Dissertations, and Problem Reports (ETD)

Results obtained from experimental investigations of contact friction in monolayer and bilayers graphene and the related effects on their transport properties are presented here along with their discussion and interpretation. For this purpose, chemical vapor deposited (CVD) graphene samples on SiO2/Si were prepared. The samples were characterized by atomic force microscopy (AFM), Raman and X-ray photoelectron spectroscopy (XPS). Summaries of the results are given below.

Defects-controlled friction in graphene is of technological importance but the underlying mechanism remains a subject of debate. The new results obtained from the analysis of lateral force microscopy images revealed that the contact …


Perceptions And Instructional Practices Used By West Virginian Secondary Agricultural Teachers In Meat Science Education, Kindra D. Carr Jan 2021

Perceptions And Instructional Practices Used By West Virginian Secondary Agricultural Teachers In Meat Science Education, Kindra D. Carr

Graduate Theses, Dissertations, and Problem Reports (ETD)

A focus on classroom integration of science, technology, engineering, and math (STEM) concepts has resulted in secondary teachers broadening the scope and delivery of STEM principles in their classes. Because agriculture classes allow students to convert abstract ideas into concrete ideas, they provide a useful setting to teach STEM concepts. Using secondary agricultural classes to teach science is effective and has contributed to higher standardized science test scores, as opposed to a science class alone. West Virginia is known for meat processing facilities and meat technology learning activities in its public schools. However, little is known about how the content …


Development Of Collision Resilient Drone For Flying In Cluttered Environment, Al Mahmud Jan 2021

Development Of Collision Resilient Drone For Flying In Cluttered Environment, Al Mahmud

Graduate Theses, Dissertations, and Problem Reports (ETD)

The use of multi-rotor aerial vehicles, also known as drones, has increased significantly in the last decade due to their ease of use and applicability to an extended range of functions. Therefore, the importance of making these vehicles safe, for both themselves and the people around them, has also increased. A clear solution to this problem is the development of algorithms that generate collision-free paths using different sensors. However, even with such onboard computation, avoiding a collision is not always possible. Thus, the use of a protective structure (cage) around the aerial vehicle is a solution to keep it safe …


Fundamentals And Applications Of Charge Transfer Dissociation Mass Spectrometry, Zachary Joseph Sasiene Jan 2021

Fundamentals And Applications Of Charge Transfer Dissociation Mass Spectrometry, Zachary Joseph Sasiene

Graduate Theses, Dissertations, and Problem Reports (ETD)

The recent surge in the analysis of biomolecules has driven the need for robust, versatile, sensitive, and accurate techniques such as tandem mass spectrometry for compound structural elucidation. Many fragmentation techniques have been developed over the years, but each has their own limitations, such as the loss of post-translational modifications (PTMs), inability to fragment singly charged precursor ions or cost prohibitiveness. To help fill this gap, a novel fragmentation technique called charge transfer dissociation (CTD) has been developed to fragment stored precursor ions through radical-driven fragmentation processes using kiloelectronvolt reagent cations. CTD has shown success in the analysis of proteins, …


Multimodel Operability Framework For Design Of Modular And Intensified Energy Systems, Vitor Gazzaneo Jan 2021

Multimodel Operability Framework For Design Of Modular And Intensified Energy Systems, Vitor Gazzaneo

Graduate Theses, Dissertations, and Problem Reports (ETD)

In this dissertation, a novel operability framework is introduced for the process design of modular and intensified energy systems that are challenged by complexity and highly constrained environments. Previously developed process operability approaches are reviewed and further developed in terms of theory, application, and software infrastructure. An optimization-based multilayer operability framework is introduced for process design of nonlinear energy systems. In the first layer of this framework, a mixed-integer linear programming (MILP)-based iterative algorithm considers the minimization of footprint and achievement of process intensification targets. Then, in the second layer, an operability analysis is performed to incorporate key features of …


Ecology Of Freshwater Turtles And Other Wetland Wildlife In A North-Central West Virginia Watershed, Darien N. Lozon Jan 2021

Ecology Of Freshwater Turtles And Other Wetland Wildlife In A North-Central West Virginia Watershed, Darien N. Lozon

Graduate Theses, Dissertations, and Problem Reports (ETD)

The goal of wetland mitigation in the United States is to achieve a no-net-loss of wetlands; however, mitigated wetlands must be monitored to ensure wetland function is comparable to natural wetlands. In this study, relationships between land use practices and freshwater turtle abundance, wetland connectivity, and heavy metal bioaccumulation were investigated. In Chapter 1, a summary of pre-restoration species abundance and diversity is provided for anurans, birds, benthic macroinvertebrates, fishes, small mammals, plants, and turtles in riparian wetlands along Ruby Run, a tributary of Deckers Creek in north-central West Virginia, USA. These data provide a baseline for comparison after mitigation …


Identifying The Technical And Teacher Skills Needed By In-Service Teachers To Effectively Teach An Agricultural Mechanics Course In West Virginia, Coleton Todd Payne Jan 2021

Identifying The Technical And Teacher Skills Needed By In-Service Teachers To Effectively Teach An Agricultural Mechanics Course In West Virginia, Coleton Todd Payne

Graduate Theses, Dissertations, and Problem Reports (ETD)

Agricultural mechanics remains one of the largest content sections in school-based agriculture education (SBAE). Therefore, in-service and pre-service teachers need to be effective in teaching agricultural mechanics to their students (Burris et al., 2005; Hainline & Wells, 2019). A number of studies (McKim & Saucier, 2011; Saucier et al., 2014) have identified the importance of safety in agricultural mechanics laboratories, but not the content skills specifically needed to teach the courses. The only areas of content that most new in-service teachers possess are those of basic woodworking and basic metal working. This study bases its theoretical framework off of the …


Turning Visual Noise Into Hardware Efficiency: Systems Of Viewer And Content Aware Power-Quality Scalable Embedded Memories With Ecc-Adaptation For Big Videos And Deep Learning, Ali Ahmad Haidous Jan 2021

Turning Visual Noise Into Hardware Efficiency: Systems Of Viewer And Content Aware Power-Quality Scalable Embedded Memories With Ecc-Adaptation For Big Videos And Deep Learning, Ali Ahmad Haidous

Electrical and Computer Engineering

Mobile devices, such as smart phones, are being increasingly utilized for watching videos. Video processing requires frequent memory access that consume a significant amount of power due to large data size and intensive computational requirements. This limits battery life and frustrates users. Memory designers are focused on hardware-level power-optimization techniques without consideration of how hardware performance influences viewers' actual experience. The human visual system is limited in its ability to detect subtle degradations in image quality. For example, under conditions of high ambient illumination – such as outdoors in direct sunlight – the veiling luminance (i.e., glare) on the screen …


Effects Of Acute And Chronic Nicotine Administration On Choice Of Probabilistic Outcomes, Katya A. Nolder Jan 2021

Effects Of Acute And Chronic Nicotine Administration On Choice Of Probabilistic Outcomes, Katya A. Nolder

Graduate Theses, Dissertations, and Problem Reports (ETD)

Risky choice can be operationally defined as the choice for a larger, uncertain reinforcer over a smaller, certain reinforcer. Research suggests smokers engage in more risky or maladaptive decisions when compared to nonsmokers. The relation between nicotine and risky choice could benefit from further investigation, since nicotine is the active substance of tobacco products that maintains tobacco addiction. Acute nicotine administration has shown to alter risky choice; however, since the everyday smoker uses nicotine repeatedly, more research on chronic administration is warranted and would allow for assessment of tolerance or sensitization of these effects. The present study investigated effects of …


Variations In Produced Water Chemistry And Relation To Regional Geology And Production In The Marcellus Shale, Northcentral West Virginia, Jonathan M. Brady Jan 2021

Variations In Produced Water Chemistry And Relation To Regional Geology And Production In The Marcellus Shale, Northcentral West Virginia, Jonathan M. Brady

Graduate Theses, Dissertations, and Problem Reports (ETD)

An investigation of 74 Marcellus Shale wells across northcentral West Virginia indicates changes in produced water chemistry and quantity can be related to geologic conditions based on well logs and core data. These changes are determined by reviewing multiple produced water analyses for individual wells for periods up to ten years. Results show variations among the areas in this study. From west to east across central Harrison County to central Taylor County, then north into Monongalia County, gamma-ray logs show increasing intensity, especially in the middle and lower Marcellus. XRD mineralogy from core data shows increasing clay content from west …


Equilibrium And Non-Equilibrium Ultrafast Carrier Transport And Dynamics In Chalcopyrite Semiconductors, Rishmali Thanuja Sooriyagoda Jan 2021

Equilibrium And Non-Equilibrium Ultrafast Carrier Transport And Dynamics In Chalcopyrite Semiconductors, Rishmali Thanuja Sooriyagoda

Graduate Theses, Dissertations, and Problem Reports (ETD)

Chalcopyrite crystals in the II-IV-V2 family have received significant interest due to their high nonlinearity, composition-tunable bandgaps, wide transparency windows, and high damage threshold. These semiconductors have been explored for electromagnetic (EM) screening, spintronic and photovoltaic applications, making them good optical and optoelectronic materials. This thesis uses terahertz spectroscopy to understand optical, electronic, and vibrational dynamical processes in CdGeP2, ZnGeP2 and CdSiP2 chalcopyrite semiconductors.

We have employed Terahertz time-domain spectroscopy to investigate temperature-dependent ground-state properties of bulk chalcopyrite crystals that can be related to electronic transport and electron-lattice interactions. The complex spectra provide refraction and …


Environmental Controls On The Terrestrial Water Cycle In Forested Mountain Ecosystems., Luis Andrés Guillén Jan 2021

Environmental Controls On The Terrestrial Water Cycle In Forested Mountain Ecosystems., Luis Andrés Guillén

Graduate Theses, Dissertations, and Problem Reports (ETD)

Water is a key resource to natural ecosystems and human societies alike, and the water cycle is fundamentally linked to the climate and the characteristics of catchments. However, the challenges posed by environmental change makes it imperative to understand how the water cycle is affected by biotic and abiotic factors, in particular, in areas that are crucial sources of water like forested headwater catchments. Therefore, this doctoral dissertation aims to advance the knowledge on the dynamics between climate, vegetation and landscape that determine the water balance of forested mountain ecosystems. This document presents five chapters, an introductory chapter, three standalone …


Dynamic Optimization Algorithms For Baseload Power Plant Cycling Under Variable Renewable Energy, Rebecca Hyen Hae Kim Jan 2021

Dynamic Optimization Algorithms For Baseload Power Plant Cycling Under Variable Renewable Energy, Rebecca Hyen Hae Kim

Graduate Theses, Dissertations, and Problem Reports (ETD)

The growing deployment of variable renewable energy (VRE) sources, such as wind and solar, is mainly due to the decline in the cost of renewable technologies and the increase of societal and cultural pressures. Solar and wind power generation are also known to have zero marginal costs and fuel emissions during dispatch. Thereby, the VRE from these sources should be prioritized when available. However, the rapid deployment of VRE has heightened concerns regarding the challenges in the integration between fossil-fueled and renewable energy systems. The high variability introduced by the VRE as well as the limited alignment between demand and …


Soil Response Of Helicopter Liming In The Monongahela National Forest, Jarrett Douglas Fowler Jan 2021

Soil Response Of Helicopter Liming In The Monongahela National Forest, Jarrett Douglas Fowler

Graduate Theses, Dissertations, and Problem Reports (ETD)

Soils in the Monongahela National Forest (MNF) are acidic due to sandstone parent material, acid deposition, uptake of base cations by vegetation, and release of organic acids by organic matter (OM) decomposition. Increases in soil acidity have caused declines in forest health and changed species composition and nutrient status. Liming can neutralize soil acidity, but no large-scale liming projects have been done on acid forest soils in the USA. In anticipation of acquiring funding for a proposed liming project in the MNF, in 2007 and 2009 10 sites were selected to sample and analyze soils before lime was applied. In …


Improving Characterization Of Small Molecules Using Enhanced Molecular Reaction And Ionization Techniques, Sandra N. Majuta Jan 2021

Improving Characterization Of Small Molecules Using Enhanced Molecular Reaction And Ionization Techniques, Sandra N. Majuta

Graduate Theses, Dissertations, and Problem Reports (ETD)

Metabolomics is an emerging “omics” field, which is comprised of the study of small molecule metabolites in biological systems. It can be argued that the abundances of metabolites within biological samples, such as tissue and plasma, are indicative of various physiological states. Thus, metabolomics analysis has the potential in the clinical scientific field to be an effective tool for early disease diagnosis as well as for proposing and monitoring new treatments for diseases. However, full characterization of the metabolome complement is often hampered by the wide diversity of metabolites, which also exhibit an extremely large concentration range in biological samples. …


Photochemical Motion Control Of Belousov-Zhabotinsky Droplets And Turbulence-Like Patterns In The Aluminum Hydroxide Precipitation System, Syed Jazli Syed Jamaluddin Jan 2021

Photochemical Motion Control Of Belousov-Zhabotinsky Droplets And Turbulence-Like Patterns In The Aluminum Hydroxide Precipitation System, Syed Jazli Syed Jamaluddin

Graduate Theses, Dissertations, and Problem Reports (ETD)

A Belousov-Zhabotinsky (BZ) droplet that is immersed in an oil-phase can self-propel due to a surface tension gradient that exists on the droplet surface. The surface tension gradient is created as a result of the reactions that occur within and on the surface of the BZ droplet. By affecting the rates of these reactions, the BZ droplet direction can be controlled. In our experiments, we tune the reactions that occur within and on the BZ droplet surface by imposing a light intensity gradient on the droplet. We analyzed the BZ droplet motion and demonstrated that we can control the droplet …


Majorization And Minimal Energy On Spheres, Oleg R. Musin Jan 2021

Majorization And Minimal Energy On Spheres, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

In the present paper, we consider the majorization theorem (also known as Karamata's inequality) and the respective minima of the majorization (the so-called M-sets) for f-energy potentials of m-point configurations on the unit sphere. In particular, we show the optimality of regular simplexes, describe some M-sets of small cardinality, define and discuss spherical f-designs.


Tri-Hamiltonian Duality System Of Merola-Ragnisco-Tu Equation, Mengxia Zhang, Bao-Feng Feng, Jiawen Liu Jan 2021

Tri-Hamiltonian Duality System Of Merola-Ragnisco-Tu Equation, Mengxia Zhang, Bao-Feng Feng, Jiawen Liu

School of Mathematical & Statistical Sciences Faculty Publications

The tri-Hamiltonian method is applied to the Merola-Ragnisco-Tu equation. This enables us to construct a new integrable system, whose continuum limit is related to the AKNS system. Moreover, the system is proved to have linear spectral problems (Lax pair), bi-Hamiltonian structure and Darboux-Bäcklund transformation. Through the Darboux-Bäcklund transformation, we construct some exact solutions for the system.


Academic Supply And Occupational Demand Report 2019-2020, Tennessee. Higher Education Commission. Jan 2021

Academic Supply And Occupational Demand Report 2019-2020, Tennessee. Higher Education Commission.

Academic Programs and Workforce Reports

No abstract provided.


Examination Of Sex- And Intensity-Specific Fatigue During Bilateral Leg Extension Exercise, Taylor Kayln Dinyer Jan 2021

Examination Of Sex- And Intensity-Specific Fatigue During Bilateral Leg Extension Exercise, Taylor Kayln Dinyer

Theses and Dissertations--Kinesiology and Health Promotion

The purposes of this study were to: 1) examine sex-related differences in the performance of repetitions to failure below (CL-10%, CL-20%) and above (50%, 60%, 70%, and 80% 1RM) the critical load (CL) to make inferences regarding the mechanisms responsible for variability in the performance of repetitions to failure at varying loads; 2) examine the time course of neuromuscular and muscle oxygen saturation responses during the performance of repetitions to failure below and above the CL; 3) to make inferences regarding motor unit activation strategies used to maintain force during the completion of repetitions below and …