Scholarly Works, Mathematics

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  • 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.
  • 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.
  • 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.
  • Image segmentation by graph partitioning
    Garcia Hilares, Nilton (2021-10-07)
  • A Parallel Aggregation Algorithm in Algebraic Multigrid
    Garcia Hilares, Nilton (2019-10-09)
  • An Adaptive Zolotarev Upper-Bound for the Singular Values of Loewner Matrices
    Garcia Hilares, Nilton; Embree, Mark (2021-11-12)
  • Ingested histamine and serotonin interact to alter Anopheles stephensi feeding and flight behavior and infection with Plasmodium parasites
    Coles, Taylor A.; Briggs, Anna M.; Hambly, Malayna G.; Cespedes, Nora; Fellows, Abigail M.; Kaylor, Hannah L.; Adams, Alexandria D.; Van Susteren, Grace; Bentil, Ronald E.; Robert, Michael A.; Riffell, Jeffrey A.; Lewis, Edwin E.; Luckhart, Shirley (Frontiers, 2023-07-24)
    Blood levels of histamine and serotonin (5-HT) are altered in human malaria, and, at these levels, we have shown they have broad, independent effects on Anopheles stephensi following ingestion by this invasive mosquito. Given that histamine and 5-HT are ingested together under natural conditions and that histaminergic and serotonergic signaling are networked in other organisms, we examined effects of combinations of these biogenic amines provisioned to A. stephensi at healthy human levels (high 5-HT, low histamine) or levels associated with severe malaria (low 5-HT, high histamine). Treatments were delivered in water (priming) before feeding A. stephensi on Plasmodium yoelii-infected mice or via artificial blood meal. Relative to effects of histamine and 5-HT alone, effects of biogenic amine combinations were complex. Biogenic amine treatments had the greatest impact on the first oviposition cycle, with high histamine moderating low 5-HT effects in combination. In contrast, clutch sizes were similar across combination and individual treatments. While high histamine alone increased uninfected A. stephensi weekly lifetime blood feeding, neither combination altered this tendency relative to controls. The tendency to re-feed 2 weeks after the first blood meal was altered by combination treatments, but this depended on mode of delivery. For blood delivery, malaria-associated treatments yielded higher percentages of fed females relative to healthy-associated treatments, but the converse was true for priming. Female mosquitoes treated with the malaria-associated combination exhibited enhanced flight behavior and object inspection relative to controls and healthy combination treatment. Mosquitoes primed with the malaria-associated combination exhibited higher mean oocysts and sporozoite infection prevalence relative to the healthy combination, with high histamine having a dominant effect on these patterns. Compared with uninfected A. stephensi, the tendency of infected mosquitoes to take a second blood meal revealed an interaction of biogenic amines with infection. We used a mathematical model to project the impacts of different levels of biogenic amines and associated changes on outbreaks in human populations. While not all outbreak parameters were impacted the same, the sum of effects suggests that histamine and 5-HT alter the likelihood of transmission by mosquitoes that feed on hosts with symptomatic malaria versus a healthy host.
  • Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions
    Bezemek, Z. W.; Heldman, Max (Springer, 2023-11-22)
    We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton–Jacobi–Bellman (HJB) equation on Wasserstein space which arises in the theory of mean-field (McKean–Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB equation on Wasserstein space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB equation, our scheme can have vanishingly small relative error in the many particle limit.
  • Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability
    Chekroun, Mickaël D.; Liu, Honghu (Elsevier, 2024-04-01)
    Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics. To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle–node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems. These dynamical insights enable us in turn to design a stochastic model whose solutions – as the delay parameter drifts slowly through its critical values – produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO's interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.
  • Randomized Algorithms for Rounding in the Tensor-Train Format
    Al Daas, Hussam; Ballard, Grey; Cazeaux, Paul; Hallman, Eric; Miedlar, Agnieszka; Pasha, Mirjeta; Reid, Tim W.; Saibaba, Arvind K. (Siam Publications, 2023-01-27)
    The tensor-train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20\times speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.
  • Topology Optimization with Adaptive Mesh Refinement
    De Sturler, Eric; Paulino, Glaucio H.; Wang, Shun (2008)
  • Domain truncation, absorbing boundary conditions, Schur complements, and Padé approximation
    Gander, Martin J.; Jakabčin, Lukáš; Outrata, Michal (Osterreichische Akademie der Wissenschaften, Verlag, 2024)
    We show for a model problem that the truncation of an unbounded domain by an artificial Dirichlet boundary condition placed far away from the domain of interest is equivalent to a specific absorbing boundary condition placed closer to the domain of interest. This specific absorbing boundary condition can be implemented as a truncation layer terminated by a Dirichlet condition. We prove that the absorbing boundary condition thus obtained is a spectral Padé approximation about infinity of the transparent boundary condition. We also study numerically two improvements for this boundary condition, the truncation with an artificial Robin condition placed at the end of the truncation layer and a Padé approximation about a different point than infinity. Both of these give new and substantially better results compared to using the artificial Dirichlet boundary condition at the end of the truncation layer. We prove our results in the context of linear algebra, using spectral analysis of finite and infinite Schur complements, which we relate to continued fractions. We illustrate our results with numerical experiments.
  • Spectral analysis of implicit 2 stage block Runge-Kutta preconditioners
    Gander, Martin J.; Outrata, Michal (Elsevier, 2023-01-01)
    We analyze the recently introduced family of preconditioners in [15] for the stage equations of implicit Runge-Kutta methods for two stage methods. We give explicit formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general method and use these to give explicit convergence estimates of preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis also allows us to qualitatively predict and explain the main observed features of the GMRES convergence behavior, not only bound it. We illustrate our analysis with numerical experiments. We also consider the direction of numerical optimization for improving the preconditioners performance, as suggested in [15]. We consider two different ways – both distinct to the one introduced in [15] – and numerically optimize these, using the explicit bounds obtained beforehand.
  • Why Similar Policies Resulted In Different COVID-19 Outcomes: How Responsiveness And Culture Influenced Mortality Rates
    Lim, Tse Yang; Xu, Ran; Ruktanonchai, Nick; Saucedo, Omar; Childs, Lauren M.; Jalali, Mohammad S.; Rahmandad, Hazhir; Ghaffarzadegan, Navid (Health Affairs, 2023-12)
    In the first two years of the COVID-19 pandemic, per capita mortality varied by more than a hundredfold across countries, despite most implementing similar nonpharmaceutical interventions. Factors such as policy stringency, gross domestic product, and age distribution explain only a small fraction of mortality variation. To address this puzzle, we built on a previously validated pandemic model in which perceived risk altered societal responses affecting SARS-CoV-2 transmission. Using data from more than 100 countries, we found that a key factor explaining heterogeneous death rates was not the policy responses themselves but rather variation in responsiveness. Responsiveness measures how sensitive communities are to evolving mortality risks and how readily they adopt nonpharmaceutical interventions in response, to curb transmission.We further found that responsiveness correlated with two cultural constructs across countries: uncertainty avoidance and power distance. Our findings show that more responsive adoption of similar policies saves many lives, with important implications for the design and implementation of responses to future outbreaks.
  • Extraordinary parasite multiplication rates in human malaria infections
    Greischar, Megan A.; Childs, Lauren M. (Cell Press, 2023-08)
    For pathogenic organisms, faster rates of multiplication promote transmission success, the potential to harm hosts, and the evolution of drug resistance. Parasite multiplication rates (PMRs) are often quantified in malaria infections, given the relative ease of sampling. Using modern and historical human infection data, we show that established methods return extraordinarily – and implausibly – large PMRs. We illustrate how inflated PMRs arise from two facets of malaria biology that are far from unique: (i) some developmental ages are easier to sample than others; (ii) the distribution of developmental ages changes over the course of infection. The difficulty of accurately quantifying PMRs demonstrates a need for robust methods and a subsequent re-evaluation of what is known even in the well-studied system of malaria.