Scholarly Works, Mathematics

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Research articles, presentations, and other scholarship

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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)
Zolotarev bounds for the singular values of Loewner matrices
Garcia Hilares, Nilton; Embree, Mark (2020-11-20)
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.
Relationship between Climate Variables and Dengue Incidence in Argentina
Lopez, Maria S.; Gomez, Andre A.; Müller, Gabriela V.; Walker, Elisabet; Robert, Michael A.; Estallo, Elizabet L. (US HHS, 2023-05-24)
BACKGROUND: Climate change is an important driver of the increased spread of dengue from tropical and subtropical regions to temperate areas around the world. Climate variables such as temperature and precipitation influence the dengue vector’s biology, physiology, abundance, and life cycle. Thus, an analysis is needed of changes in climate change and their possible relationships with dengue incidence and the growing occurrence of epidemics recorded in recent decades. OBJECTIVES: This study aimed to assess the increasing incidence of dengue driven by climate change at the southern limits of dengue virus transmission in South America. METHODS: We analyzed the evolution of climatological, epidemiological, and biological variables by comparing a period of time without the presence of dengue cases (1976–1997) to a more recent period of time in which dengue cases and important outbreaks occurred (1998–2020). In our analysis, we consider climate variables associated with temperature and precipitation, epidemiological variables such as the number of reported dengue cases and incidence of dengue, and biological variables such as the optimal temperature ranges for transmission of dengue vector. RESULTS: The presence of dengue cases and epidemic outbreaks are observed to be consistent with positive trends in temperature and anomalies from long-term means. Dengue cases do not seem to be associated with precipitation trends and anomalies. The number of days with optimal temperatures for dengue transmission increased from the period without dengue cases to the period with occurrences of dengue cases. The number of months with optimal transmission temperatures also increased between periods but to a lesser extent. CONCLUSIONS: The higher incidence of dengue virus and its expansion to different regions of Argentina seem to be associated with temperature increases in the country during the past two decades. The active surveillance of both the vector and associated arboviruses, together with continued meteorological data collection, will facilitate the assessment and prediction of future epidemics that use trends in the accelerated changes in climate. Such surveillance should go hand in hand with efforts to improve the understanding of the mechanisms driving the geographic expansion of dengue and other arboviruses beyond the current limits. https://doi.org/10.1289/EHP11616.
Effects of Contact Tracing and Self-Reporting in a Network Disease Model
Gandhi, Punit; Robert, Michael A.; Palacios, John; Chan, David (2022-02-15)
We examine the effects of symptomatic individuals getting tested and the use of contact tracing in a network model of disease transmission on different epidemio-logical metrics. These metrics include the length of the epidemic, number of people infected, number of tests performed, and the likelihood of an epidemic occurring. We utilize a network model to resolve the influence of contact patterns between individuals as opposed to assuming mass action where all individuals are connected to each other. We find that the effects of self-reporting and contact tracing vary depending on the structure of the network. We also compare the results from the network model with an analogous ODE model that assumes mass action and demonstrate how the results can be dramatically and surprisingly different.
Fluctuation analysis for particle-based stochastic reaction-diffusion models
Heldman, Max; Spilopoulos, Konstantinos; Isaacson, Samuel; Ma, Jingwei (2024-01-17)
Recent works have derived and proven the large-population mean-field limit for several classes of particle-based stochastic reaction-diffusion (PBSRD) models. These limits correspond to systems of partial integral-differential equations (PIDEs) that generalize standard mass-action reaction-diffusion PDE models. In this work we derive and prove the next order fluctuation corrections to such limits, which we show satisfy systems of stochastic PIDEs with Gaussian noise. Numerical examples are presented to illustrate how including the fluctuation corrections can enable the accurate estimation of higher order statistics of the underlying PBSRD model.
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.
Topological nature of dislocation networks in two-dimensional moiré materials
Engelke, Rebecca; Yoo, Hyobin; Carr, Stephen; Xu, Kevin; Cazeaux, Paul; Allen, Richard; Valdivia, Andres Mier; Luskin, Mitchell; Kaxiras, Efthimios; Kim, Minhyong; Han, Jung Hoon; Kim, Philip (American Physical Society, 2023-03-16)
Moiré superlattice patterns at the interface of two-dimensional (2D) van der Waals (vdW) materials, arising from a small displacement between similar lattices, have been extensively studied over the past decade for their dramatic ability to tune material properties. However, previous work to understand the structure of these interfaces has largely focused on some special cases, particularly honeycomb lattices displaced by twist or isotropic scaling. In this work, we develop practical and analytical tools for understanding the moiré structure that can be generalized to other lattice distortions and lattice types. At large enough moiré lengths, all moiré systems relax into commensurated 2D domains separated by networks of dislocation lines. The nodes of the 2D dislocation line network can be considered as vortexlike topological defects. However, we find these topological defects to exist on a punctured torus, requiring different mathematical formalism than the topological defects in a superconductor or planar ferromagnet. In the case of twisted bilayer graphene, the defects are characterized by the free group on two generators. We find that antivortices occur in the presence of anisotropic heterostrain, such as a shear or anisotropic displacement, while arrays of vortices appear under a twist or isotropic displacement between vdW materials. Utilizing the dark field imaging capability of transmission electron microscopy (TEM), we experimentally demonstrate the existence of vortex and antivortex pair formation in a moiré system, caused by competition between different types of heterostrains in the vdW interfaces. We also present a methodology for mapping the underlying heterostrain of a moiré structure from experimental TEM data, which provides a quantitative relation between the various components of heterostrain and vortex-antivortex density in moiré systems.

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