Scholarly Works, Mathematics

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  • Energy-Conserving Hermite Methods for Maxwell's Equations
    Appelö, Daniel; Hagstrom, Thomas; Law-Kam-Cio, Yann-Meing (Springer, 2024-01-22)
    Energy-conserving Hermite methods for solving Maxwell's equations in dielectric and dispersive media are described and analyzed. In three space dimensions methods of order $2m$ to $2m+2$ require $(m+1)^3$ degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of $m$. We prove stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special seminorm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of electromagnetic wave propagation over thousands of wavelengths.
  • Block-Circulant Approximation of the Precision Matrix for Assimilating SWOT Altimetry Data
    Yaremchuk, Max; Beattie, Christopher; Panteleev, Gleb; D’Addezio, Joseph (MDPI, 2024-05-29)
    The recently deployed Surface Water and Ocean Topography (SWOT) mission for the first time has observed the ocean surface at a spatial resolution of 1 km, thus giving an opportunity to directly monitor submesoscale sea surface height (SSH) variations that have a typical magnitude of a few centimeters. This progress comes at the expense of the necessity to take into account numerous uncertainties in calibration of the quality-controlled altimeter data. Of particular importance is the proper filtering of spatially correlated errors caused by the uncertainties in geometry and orientation of the on-board interferometer. These “systematic” errors dominate the SWOT error budget and are likely to have a notable signature in the SSH products available to the oceanographic community. In this study, we explore the utility of the block-circulant (BC) approximation of the SWOT precision matrix developed by the Jet Propulsion Laboratory for assessment of a mission’s accuracy, including the possible impact of the systematic errors on the assimilation of the wide-swath altimeter data into numerical models. It is found that BC approximation of the precision matrix has sufficient (90–99%) accuracy for a wide range of significant wave heights of the ocean surface, and, therefore, could potentially serve as an efficient preconditioner for data assimilation problems involving altimetry observations by space-borne interferometers. An extensive set of variational data assimilation (DA) experiments demonstrates that BC approximation provides more accurate SSH retrievals compared to approximations, assuming a spatially uncorrelated observation error field as is currently adopted in operational DA systems.
  • Incorporating Intracellular Processes in Virus Dynamics Models
    Ciupe, Stanca M.; Conway, Jessica M. (MDPI, 2024-04-30)
    In-host models have been essential for understanding the dynamics of virus infection inside an infected individual. When used together with biological data, they provide insight into viral life cycle, intracellular and cellular virus–host interactions, and the role, efficacy, and mode of action of therapeutics. In this review, we present the standard model of virus dynamics and highlight situations where added model complexity accounting for intracellular processes is needed. We present several examples from acute and chronic viral infections where such inclusion in explicit and implicit manner has led to improvement in parameter estimates, unification of conclusions, guidance for targeted therapeutics, and crossover among model systems. We also discuss trade-offs between model realism and predictive power and highlight the need of increased data collection at finer scale of resolution to better validate complex models.
  • An Integrated Methodological Approach for Documenting Individual and Collective Mathematical Progress: Reinventing the Euler Method Algorithmic Tool
    Rasmussen, Chris; Wawro, Megan; Zandieh, Michelle (MDPI, 2024-03-21)
    In this paper we advance a methodological approach for documenting the mathematical progress of learners as an integrated analysis of individual and collective activity. Our approach is grounded in and expands the emergent perspective by integrating four analytic constructs: individual meanings, individual participation, collective mathematical practices, and collective disciplinary practices. Using video data of one small group of four students in an inquiry-oriented differential equations classroom, we analyze a 10 min segment in which one small group reinvent Euler’s method, an algorithmic tool for approximating solutions to differential equations. A central intellectual contribution of this work is elaborating and coordinating the four methodological constructs with greater integration, cohesiveness, and coherence.
  • General boundary value problems of a class of fifth order KdV equations on a bounded interval
    Sriskandasingam, Mayuran; Sun, Shu Ming; Zhang, Bing-yu Y. (2024)
  • Electronic Health Record: Comparative analysis of HL7 and open EHR approaches
    Nestor, Mamani Macedo; Garcia Hilares, Nilton Alan; Julio, Pariona Quispe; R, Alarcon Matutti (IEEE, 2010-06-01)
    This paper presents a comparative analysis of the main proposals to automatize a Patient’s Health Record in any Medical Center: HL7 and OpenEHR. The methodology includes analyzing each approach, defining some criteria of evaluation, doing a comparative chart, and showing the main conclusions.
  • Resilient s-ACD for Asynchronous Collaborative Solutions of Systems of Linear Equations
    Erlandson, Lucas; Atkins, Zachary; Fox, Alyson; Vogl, Christopher; Miedlar, Agnieszka; Ponce, Colin (IEEE, 2023-09-26)
    Solving systems of linear equations is a critical component of nearly all scientific computing methods. Traditional algorithms that rely on synchronization become prohibitively expensive in computing paradigms where communication is costly, such as heterogeneous hardware, edge computing, and unreliable environments. In this paper, we introduce an s-step Approximate Conjugate Directions (s-ACD) method and develop resiliency measures that can address a variety of different data error scenarios. This method leverages a Conjugate Gradient (CG) approach locally while using Conjugate Directions (CD) globally to achieve asynchronicity. We demonstrate with numerical experiments that s-ACD admits scaling with respect to the condition number that is comparable with CG on the tested 2D Poisson problem. Furthermore, through the addition of resiliency measures, our method is able to cope with data errors, allowing it to be used effectively in unreliable environments.
  • Detection of passageways in natural foliage using biomimetic sonar
    Wang, Ruihao; Liu, Yimeng; Müller, Rolf (IOP, 2022-08-10)
    The ability of certain bat species to navigate in dense vegetation based on trains of short biosonar echoes could provide for an alternative parsimonious approach to obtaining the sensory information that is needed to achieve autonomy in complex natural environments. Although bat biosonar has much lower data rates and spatial (angular) resolution than commonly used human-made sensing systems such as LiDAR or stereo cameras, bat species that live in dense habitats have the ability to reliably detect narrow passageways in foliage. To study the sensory information that the animals may have available to accomplish this, we have used a biomimetic sonar system that was combined with a camera to record echoes and synchronized images from 10 different field sites that featured narrow passageways in foliage. The synchronized camera and sonar data allowed us to create a large data set (130 000 samples) of labeled echoes using a teacher-student approach that used class labels derived from the images to provide training data for echo-based classifiers. The performance achieved in detecting passageways based on the field data closely matched previous results obtained for gaps in an artificial foliage setup in the laboratory. With a deep feature extraction neural network (VGG16) a foliage-versus-passageway classification accuracy of 96.64% was obtained. A transparent artificial intelligence approach (class-activation mapping) indicated that the classifier network relied heavily on the initial rising flank of the echoes. This finding could be exploited with a neuromorphic echo representation that consisted of times where the echo envelope crossed a certain amplitude threshold in a given frequency channel. Whereas a single amplitude threshold was sufficient for this in the previous laboratory study, multiple thresholds were needed to achieve an accuracy of 92.23%. These findings indicate that despite many sources of variability that shape clutter echoes from natural environments, these signals contain sufficient sensory information to enable the detection of passageways in foliage.
  • Characteristics of departments with high-use of active learning in introductory STEM courses: implications for departmental transformation
    Lau, Alexandra C.; Henderson, Charles; Stains, Marilyne; Dancy, Melissa; Merino, Christian; Apkarian, Naneh; Raker, Jeffrey R.; Johnson, Estrella (2024-02-12)
    Background: It is well established in the literature that active learning instruction in introductory STEM courses results in many desired student outcomes. Yet, regular use of high-quality active learning is not the norm in many STEM departments. Using results of a national survey, we identified 16 departments where multiple instructors reported using high levels of active learning in their introductory chemistry, mathematics, or physics courses. We conducted interviews with 27 instructors in these 16 departments to better understand the characteristics of such departments. Results: Using grounded theory methodology, we developed a model that highlights relevant characteristics of departments with high use of active learning instruction in their introductory courses. According to this model, there are four main, interconnected characteristics of such departments: motivated people, knowledge about active learning, opportunities, and cultures and structures that support active learning. These departments have one or more people who are motivated to promote the use of active learning. These motivated people have knowledge about active learning as well as access to opportunities to promote the use of active learning. Finally, these departments have cultures and structures that support the use of active learning. In these departments, there is a positive feedback loop that works iteratively over time, where motivated people shape cultures/structures and these cultures/structures in turn increase the number and level of commitment of the motivated people. A second positive feedback loop was found between the positive outcome of using active learning instruction and the strengthening of cultures/structures supportive of active learning. Conclusions: According to the model, there are two main take-away messages for those interested in promoting the use of active learning. The first is that all four components of the model are important. A weak or missing component may limit the desired outcome. The second is that desired outcomes are obtained and strengthened over time through two positive feedback loops. Thus, there is a temporal aspect to change. In all of the departments that were part of our study, the changes took at minimum several years to enact. While our model was developed using only high-use of active learning departments and future work is needed to develop the model into a full change theory, our results do suggest that change efforts may be made more effective by increasing the robustness of the four components and the connections between them.
  • Quantum K theory of partial flag manifolds
    Mihalcea, Constantin; Sharpe, Eric; Gu, Wei; Zhang, Hao; Xu, Weihong; Zou, Hao (Elsevier, 2024-04)
    In this paper we use three-dimensional gauged linear sigma models to make physical predictions for Whitney-type presentations of equivariant quantum K theory rings of partial flag manifolds, as quantum products of universal subbundles and various ratios, extending previous work for Grassmannians. Physically, these arise as OPEs of Wilson lines for certain Chern-Simons levels. We also include a simplified method for computing Chern-Simons levels pertinent to standard quantum K theory.
  • The isomorphism problem for Grassmannian Schubert varieties
    Tarigradschi, Mihail; Xu, Weihong (Academic Press – Elsevier, 2023-11-01)
    We prove that Schubert varieties in potentially different Grassmannians are isomorphic as varieties if and only if their corresponding Young diagrams are identical up to a transposition. We also discuss a generalization of this result to Grassmannian Richardson varieties. In particular, we prove that Richardson varieties in potentially different Grassmannians are isomorphic as varieties if their corresponding skew diagrams are semi-isomorphic as posets, and we conjecture the converse. Here, two posets are said to be semi-isomorphic if there is a bijection between their sets of connected components such that the corresponding components are either isomorphic or opposite.
  • Quantum K Whitney relations for partial flag varieties
    Gu, Wei; Mihalcea, Leonardo C.; Sharpe, Eric; Xu, Weihong; Zhang, Hao; Zou, Hao (2023-10-05)
    In a recent paper, we stated conjectural presentations for the equivariant quantum K ring of partial flag varieties, motivated by physics considerations. In this companion paper, we analyze these presentations mathematically. We prove that if the conjectured relations hold, then they must form a complete set of relations. Our main result is a proof of the conjectured presentation in the case of the incidence varieties. We also show that if a quantum K divisor axiom holds (as conjectured by Buch and Mihalcea), then the conjectured presentation also holds for the complete flag variety.
  • On an Equivalence of Divisors on (M)over-bar(0,n) from Gromov-Witten Theory and Conformal Blocks
    Chen, L.; Gibney, A.; Heller, L.; Kalashnikov, E.; Larson, H.; Xu, W. (Springer, 2022-08-16)
    We consider a conjecture that identifies two types of base point free divisors on M ¯ ,n. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on M ¯ ,n to the same statement for n = 4. A reinterpretation leads to a proof of the conjecture on M ¯ ,n for a large class, and we give sufficient conditions for the non-vanishing of these divisors.
  • Numerical simulation for the bone remodeling cycle
    Garcia Hilares, Nilton (2015-04-25)
  • Image segmentation by graph partitioning
    Garcia Hilares, Nilton (2021-10-07)
  • A Parallel Aggregation Algorithm in Algebraic Multigrid
    Garcia Hilares, Nilton (2019-10-09)
  • Zolotarev bounds for the singular values of Loewner matrices
    Garcia Hilares, Nilton; Embree, Mark (2020-11-20)
  • Dynamical Mode Decomposition (DMD) for Power Networks
    Garcia Hilares, Nilton (2022-12-02)
  • An Adaptive Zolotarev Upper-Bound for the Singular Values of Loewner Matrices
    Garcia Hilares, Nilton; Embree, Mark (2021-11-12)