Mathematical Sciences

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Mathematical research at MSU is focused primarily on related topics in pure and applied mathematics. Research programs complement each other and are often applied to problems in science and engineering. Research in statistics encompasses a broad range of theoretical and applied topics. Because the statisticians are actively engaged in interdisciplinary work, much of the statistical research is directed toward practical problems. Mathematics education faculty are active in both qualitative and quantitative experimental research areas. These include teacher preparation, coaching and mentoring for in-service teachers, online learning and curriculum development.

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    Reconstructing Full-Field Flapping Wing Dynamics from Sparse Measurements
    (IOP Publishing, 2020-11) Johns, William; Davis, Lisa; Jankauski, Mark
    Flapping insect wings deform during flight. This deformation benefits the insect’s aerodynamic force production as well as energetic efficiency. However, it is challenging to measure wing displacement field in flying insects. Many points must be tracked over the wing’s surface to resolve its instantaneous shape. To reduce the number of points one is required to track, we propose a physics-based reconstruction method called System Equivalent Reduction Expansion Processes (SEREP) to estimate wing deformation and strain from sparse measurements. Measurement locations are determined using a Weighted Normalized Modal Displacement (NMD) method. We experimentally validate the reconstruction technique by flapping a paper wing from 5-9 Hz with 45° and measuring strain at three locations. Two measurements are used for the reconstruction and the third for validation. Strain reconstructions had a maximal error of 30% in amplitude. We extend this methodology to a more realistic insect wing through numerical simulation. We show that wing displacement can be estimated from sparse displacement or strain measurements, and that additional sensors spatially average measurement noise to improve reconstruction accuracy. This research helps overcome some of the challenges of measuring full-field dynamics in flying insects and provides a framework for strain-based sensing in insect-inspired flapping robots.
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    Physical and chemical mechanisms that influence the electrical conductivity of lignin-derived biochar
    (2021-10) Kane, Seth; Ulrich, Rachel; Harrington, Abigail; Stadie, Nicholas P.; Ryan, Cecily A.
    Lignin-derived biochar is a promising, sustainable alternative to petroleum-based carbon powders (e.g., carbon black) for polymer composite and energy storage applications. Prior studies of these biochars demonstrate that high electrical conductivity and good capacitive behavior are achievable. However, these studies also show high variability in electrical conductivity between biochars (– S/cm). The underlying mechanisms that lead to desirable electrical properties in these lignin-derived biochars are poorly understood. In this work, we examine the causes of the variation in conductivity of lignin-derived biochar to optimize the electrical conductivity of lignin-derived biochars. To this end, we produced biochar from three different lignins, a whole biomass source (wheat stem), and cellulose at two pyrolysis temperatures (900 °C, 1100 °C). These biochars have a similar range of conductivities (0.002 to 18.51 S/cm) to what has been reported in the literature. Results from examining the relationship between chemical and physical biochar properties and electrical conductivity indicate that decreases in oxygen content and changes in particle size are associated with increases in electrical conductivity. Importantly, high variation in electrical conductivity is seen between biochars produced from lignins isolated with similar processes, demonstrating the importance of the lignin’s properties on biochar electrical conductivity. These findings indicate how lignin composition and processing may be further selected and optimized to target specific applications of lignin-derived biochars.
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    Characterization of synovial fluid metabolomic phenotypes of cartilage morphological changes associated with osteoarthritis
    (2019-08) Carlson, Alyssa K.; Rawle, Rachel A.; Wallace, Cameron W.; Brooks, Ellen G.; Adams, Erik; Greenwood, Mark C.; Olmer, Merissa; Lotz, Martin K.; Bothner, Brian; June, Ronald K.
    "Objective Osteoarthritis (OA) is a multifactorial disease with etiological heterogeneity. The objective of this study was to classify OA subgroups by generating metabolomic phenotypes from human synovial fluid. Design: Post mortem synovial fluids (n = 75) were analyzed by high performance-liquid chromatography mass spectrometry (LC-MS) to measure changes in the global metabolome. Comparisons of healthy (grade 0), early OA (grades I-II), and late OA (grades III-IV) donor populations were considered to reveal phenotypes throughout disease progression. Results: Global metabolomic profiles in synovial fluid were distinct between healthy, early OA, and late OA donors. Pathways differentially activated among these groups included structural deterioration, glycerophospholipid metabolism, inflammation, central energy metabolism, oxidative stress, and vitamin metabolism. Within disease states (early and late OA), subgroups of donors revealed distinct phenotypes. Synovial fluid metabolomic phenotypes exhibited increased inflammation (early and late OA), oxidative stress (late OA), or structural deterioration (early and late OA) in the synovial fluid. Conclusion: These results revealed distinct metabolic phenotypes in human synovial fluid, provide insight into pathogenesis, represent novel biomarkers, and can move toward developing personalized interventions for subgroups of OA patients.
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