Search NASA⌕ Search

SEARCH · Search NASA

Results for “spatio-temporal data”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Efficient data-driven regression for reduced-order modeling of spatial pattern formation

We present an efficient data-driven regression approach for constructing reduced-order models (ROMs) of reaction-diffusion systems exhibiting pattern formation. The ROMs are learned non-intrusively from available training data of physically accurate numerical simulations. The method can be applied to general nonlinear systems through the use of polynomial model form, while not requiring knowledge of the underlying physical model, governing equations, or numerical solvers. The process of learning ROMs is posed as a low-cost least-squares problem in a reduced-order subspace identified via Proper Orthogonal Decomposition (POD). Numerical experiments on classical pattern-forming systems–including the Schnakenberg and Mimura–Tsujikawa models–demonstrate that higher-order surrogate models significantly improve prediction accuracy while maintaining low computational cost. The proposed method provides a flexible, non-intrusive model reduction framework, well suited for the analysis of complex spatio-temporal pattern formation phenomena.

Data-driven modeling↗
DOE OSTI2024

Analysis of Superconducting Magnet Quench Antenna Data

Quenching poses a serious problem for superconducting magnets operating at high currents. It occurs when the material transitions from the superconducting to the normal state, which leads to heating and potential damage to the magnet. To understand and mitigate quenching, the Magnet Department at Fermilab is developing and testing superconducting magnet quench antenna arrays. This study delves into the anomalous events preceding the quench during magnet training by analyzing the collected data. With the moving average and Fast Fourier Transform techniques, we investigate the trends and frequency patterns of the data. Moreover, we introduce an unsupervised anomaly detection algorithm based on Principal Component Analysis and DBSCAN clustering. It can autonomously identify events within background noise, without relying on any predefined event features. Our analysis reveals that the spatio-temporal distribution of these anomalous events has little connection to the quench location, indicating that a majority of them bear no relation to the quenching process.

43 PARTICLE ACCELERATORS↗
DOE OSTI2025

Mapping wall-to-wall fractional cover of Arctic tundra plant functional types in Alaska using 20-m spatial resolution satellite imagery and harmonized plot observations

Estimates of fractional cover (fCover) across given land surfaces are used to assess, and often model, vegetation composition and diversity, which are crucial for understanding the health and functioning of terrestrial ecosystems. Remote sensing provides a useful means for scaling local, plot-measured fCover estimates to regional scales. Leveraging a recently synthesized and harmonized plot database, this study generated wall-to-wall maps of fCover for six Alaskan-Arctic plant functional types (PFT), including non-vascular plants, forbs, graminoids, and deciduous and evergreen shrubs, using 20-m satellite data (Sentinel-1, Sentinel-2, ArcticDEM) using a machine learning regression approach, specifically the random forest (RF) algorithm, which is well-suited for handling nonlinear relationships and high-dimensional satellite datasets. This study additionally addressed the spatio-temporal inconsistencies e.g., sampling scale, plot size, and collection year in plot measured fCover by adopting a multivariate outlier detection approach—Cook’s distance—to identify high-quality plots for model training and validation. Our approach achieves high accuracy (R 2 = 0.59–0.93, root mean squared errors = 0.02–0.10 for all PFTs) between plot-observed and satellite-derived fCover when using high-quality plot samples. The mapped fCover characterizes the spatial patterns of different PFTs across the tundra biome at a 20-m resolution, providing key information needed for improved representation of Arctic tundra vegetation in terrestrial biosphere models to better understand climate-vegetation feedback across the Arctic tundra.

Arctic tundra↗
DOE OSTI2026

Implications of Sea Breeze Circulations on boundary layer aerosols in the southern coastal Texas region

The Sea Breeze Circulation (SBC) influences atmospheric processes at multiple scales in coastal regions. Understanding how SBCs impact the aerosol number budget and aerosol–cloud interaction processes is essential. This study investigates sea breeze–aerosol interactions (SAIs) during 46 summertime SBC events using data from the TRacking Aerosol Convection Interactions Experiment (TRACER) field campaign across urban (main) and rural (supplemental) coastal sites in southern Texas. Weather Research and Forecasting model coupled with Chemistry (WRF-Chem) simulations complement observations to explore spatio-temporal meteorological controls on boundary layer aerosols. During the summertime, Sea Breeze Fronts (SBF) penetrating inland transported cool, moist air over the land, introducing air masses with distinct properties compared to the preexisting continental air. These SAIs cause variability in number concentrations of up to a factor of two, with events typically lasting ∼ 5 h before returning to background conditions. SAI impact on aerosols varies with site proximity to water and the preceding sea breeze (SB) history, primarily affecting the marine-influenced accumulation mode. The main site, influenced by both Galveston Bay and the Gulf of Mexico, reflects a stronger marine influence. In contrast, a supplemental site, at a similar shoreline distance but exposed only to the Gulf of Mexico and typically upstream of the urban core, samples SB air that has traversed land and partially regained continental characteristics. Simulations show that the regional SAIs extend ∼ 50 km inland and reach up to the boundary layer height. SAIs further decrease cloud condensation nuclei relevant aerosol number concentrations in ∼ 20 % of events during SBF passage.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

STFM: Accurate Spatio-Temporal Fusion Model for Weather Forecasting

Meteorological prediction is crucial for various sectors, including agriculture, navigation, daily life, disaster prevention, and scientific research. However, traditional numerical weather prediction (NWP) models are constrained by their high computational resource requirements, while the accuracy of deep learning models remains suboptimal. In response to these challenges, we propose a novel deep learning-based model, the Spatiotemporal Fusion Model (STFM), designed to enhance the accuracy of meteorological predictions. Our model leverages Fifth-Generation ECMWF Reanalysis (ERA5) data and introduces two key components: a spatiotemporal encoder module and a spatiotemporal fusion module. The spatiotemporal encoder integrates the strengths of convolutional neural networks (CNNs) and recurrent neural networks (RNNs), effectively capturing both spatial and temporal dependencies. Meanwhile, the spatiotemporal fusion module employs a dual attention mechanism, decomposing spatial attention into global static attention and channel dynamic attention. This approach ensures comprehensive extraction of spatial features from meteorological data. The combination of these modules significantly improves prediction performance. Experimental results demonstrate that STFM excels in extracting spatiotemporal features from reanalysis data, yielding predictions that closely align with observed values. In comparative studies, STFM outperformed other models, achieving a 7% improvement in ground and high-altitude temperature predictions, a 5% enhancement in the prediction of the u/v components of 10 m wind speed, and an increase in the accuracy of potential height and relative humidity predictions by 3% and 1%, respectively. This enhanced performance highlights STFM’s potential to advance the accuracy and reliability of meteorological forecasting.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Link Scheduling in Satellite Networks via Machine Learning Over Riemannian Manifolds

Low Earth Orbit (LEO) satellites play a crucial role in enhancing global connectivity, serving a complementary solution to existing terrestrial systems. In wireless networks, scheduling is a vital process that allocates time-frequency resources to users for interference management. However, LEO satellite networks face significant challenges in scheduling their links towards ground users due to the satellites’ mobility and overlapping coverage. This paper addresses the dynamic link scheduling problem in LEO satellite networks by considering spatio-temporal correlations introduced by the satellites’ movements. The first step in the proposed solution involves modeling the network over Riemannian manifolds, thanks to their representation as symmetric positive definite matrices. We introduce two machine learning (ML)-based link scheduling techniques that model the dynamic evolution of satellite positions and link conditions over time and space. To accurately predict satellite link states, we present a recurrent neural network (RNN) over Riemannian manifolds, which captures spatio-temporal characteristics over time. Furthermore, we introduce a separate model, the convolutional neural network (CNN) over Riemannian manifolds, which captures geometric relationships between satellites and users by extracting spatial features from the network topology across all links. Simulation results demonstrate that both RNN and CNN over Riemannian manifolds deliver comparable performance to the fractional programming-based link scheduling (FPLinQ) benchmark. Remarkably, unlike other ML-based models that require extensive training data, both models only need 30 training samples to achieve over 99% of the sum rate while maintaining similar computational complexity relative to the benchmark.

42 ENGINEERING↗
DOE OSTI2025

Data-Driven Clustering and Classification of Outage Patterns with Insights into their Links to Extreme Events

At a global level extreme events have increased in both scale and impact. These events have the potential to affect the electrical grid infrastructure and cause a wide range of outages, which can lead to a disruption in daily patterns, cost millions of dollars and also the loss of life. Currently, to track these outage events there have been various approaches developed ranging from regional to national level quantifications for what defines an outage. However, this variation in methods can potentially lead to subjective decision-making and a lack of proper management in relation to the event. While previous work has made strides in determining spatio-temporal patterns, minimal attention has been given to the type and number of outages an area may be exposed to. The differences in incurred cost and the overall severity of an event between a transformer box malfunction and a hurricane are drastic, and by finding historical signals, we can allow for more efficient management, potentially saving lives and millions of dollars. Here, we leverage unsupervised machine learning techniques to delineate outage patterns among 22 counties within the United States and find that there are clear, segregated clusters (0.93 silhouette) of data which are related by event behavior and underlying cause. This finding will allow for energy stakeholders, policy makers, and researchers to gain a deeper understanding of the extent and severity of historic events and to better prepare for electrical grid infrastructure planning and management.

Koob, Benjamin [ORNL]↗
DOE OSTI2024

PSInet: a new global water potential network

Abstract Given the pressing challenges posed by climate change, it is crucial to develop a deeper understanding of the impacts of escalating drought and heat stress on terrestrial ecosystems and the vital services they offer. Soil and plant water potential play a pivotal role in governing the dynamics of water within ecosystems and exert direct control over plant function and mortality risk during periods of ecological stress. However, existing observations of water potential suffer from significant limitations, including their sporadic and discontinuous nature, inconsistent representation of relevant spatio-temporal scales and numerous methodological challenges. These limitations hinder the comprehensive and synthetic research needed to enhance our conceptual understanding and predictive models of plant function and survival under limited moisture availability. In this article, we present PSInet (PSI—for the Greek letter Ψ used to denote water potential), a novel collaborative network of researchers and data, designed to bridge the current critical information gap in water potential data. The primary objectives of PSInet are as follows. (i) Establishing the first openly accessible global database for time series of plant and soil water potential measurements, while providing important linkages with other relevant observation networks. (ii) Fostering an inclusive and diverse collaborative environment for all scientists studying water potential in various stages of their careers. (iii) Standardizing methodologies, processing and interpretation of water potential data through the engagement of a global community of scientists, facilitated by the dissemination of standardized protocols, best practices and early career training opportunities. (iv) Facilitating the use of the PSInet database for synthesizing knowledge and addressing prominent gaps in our understanding of plants’ physiological responses to various environmental stressors. The PSInet initiative is integral to meeting the fundamental research challenge of discerning which plant species will thrive and which will be vulnerable in a world undergoing rapid warming and increasing aridification.

Forestry↗
DOE OSTI2025

Deep learning-based spatio-temporal fusion for high-fidelity ultra-high-speed X-ray radiography

Full-field ultra-high-speed (UHS) X-ray imaging experiments have been well established to characterize various processes and phenomena. However, the potential of UHS experiments through the joint acquisition of X-ray videos with distinct configurations has not been fully exploited. In this paper, we investigate the use of a deep learning-based spatio-temporal fusion (STF) framework to fuse two complementary sequences of X-ray images and reconstruct the target image sequence with high spatial resolution, high frame rate and high fidelity. We applied a transfer learning strategy to train the model and compared the peak signal-to-noise ratio (PSNR), average absolute difference (AAD) and structural similarity (SSIM) of the proposed framework on two independent X-ray data sets with those obtained from a baseline deep learning model, a Bayesian fusion framework and the bicubic interpolation method. The proposed framework outperformed the other methods with various configurations of the input frame separations and image noise levels. With three subsequent images from the low-resolution (LR) sequence of a four times lower spatial resolution and another two images from the high-resolution (HR) sequence of a 20 times lower frame rate, the proposed approach achieved average PSNRs of 37.57 dB and 35.15 dB, respectively. When coupled with the appropriate combination of high-speed cameras, the proposed approach will enhance the performance and therefore the scientific value of UHS X-ray imaging experiments.

deep learning↗
DOE OSTI2025

PV backsheets survey protocol: A framework for geo-spatial field surveys for bulk material characterization and reliability analysis applied across 41 PV systems

As widespread adoption of photovoltaic (PV) technologies continues, understanding the lifetime of modules is paramount to the viability of the industry as an environmentally conscious alternative to traditional energy generation. Although power degradation can affect the total energy production of a module over its lifetime, module safety failures necessitate the removal of a module leading to a loss of not only the particular asset, but the earning potential of the device. Therefore, it is critical to ensure that the components that provide essential safety functions for PV module operate for their entire rated lifetime. PV backsheets provide necessary electrical insulation to the completed device and failure of this component is cause for a immediate removal of the module. Degradation of the PV module backsheet has led to module safety failures in large-scale installations, costing millions of dollars in damages and lost potential revenue. The spatio-temporal degradation of fielded PV modules is important to study in order to identify which modules within installations are experiencing the greatest exposure conditions and in turn have the highest chance of failure. This paper describes a comprehensive field survey protocol developed for monitoring PV module backsheet performance using solely non-destructive methods in commercial PV fields. The protocol establishes a field naming convention, sampling method, data handling requirements, and measurement procedures. By ensuring consistent data collection practices, the field survey protocol enables research groups to obtain data of uniform quality on backsheet performance over multiple years and locations. In this study, the developed protocol was implemented at forty-one PV sites. Eight different types of airside layer backsheet materials including poly(vinylidene fluoride) (PVDF), acrylic PVDF, poly(tetrafluoroethylene-co-hexafluoropropylene-co-vinylidene fluoride) (THV), poly(vinyl fluoride) (PVF), poly(ethylene terephthalate) (PET), fluoroethylene vinyl ether (FEVE), polyethylene naphthalate (PEN), and glass were identified using attenuated total reflection Fourier transform infrared (ATR-FTIR) spectroscopy. The field survey results show that the spatial distribution of degradation indicators are non-uniform within a particular module, individual site, and across site locations. The degradation of PV modules increased in severity for modules mounted at the edge of rows (across a field) and near the junction box (within a module). This study demonstrates the sensitivity of material performance to exposure length across different materials and climates.

14 SOLAR ENERGY↗
DOE OSTI2025

Assessing the impact of climate change on rainfall-triggered landslides: a case study in California

Landslides are widespread natural hazards which take a heavy toll on lives, property, and infrastructure each year. In January 2023 and 2024, widespread landslides occurred in California as a result of extreme winter precipitation. Estimating the regional evolution of landslide hazard in a changing climate is essential for adaptation planning and risk mitigation efforts. The complex effects of climate change on landslide hazard, however, are poorly understood. Here, we use climate and landslide observations to develop a novel data-driven approach for landslide susceptibility assessment under historical and future climate conditions. We apply this framework to investigate the evolution of landslide susceptibility in California and its impacts on the roadway system based on downscaled climate projections under moderate and high greenhouse gas emission scenarios. On average, the results indicate an increase in landslide susceptibility through 2100 in regions where historical susceptibility is moderate to very high. Although the spatio-temporal variations in landslide susceptibility are sensitive to climate model uncertainties in predicting extreme rainfall, the direst impacts are largely circumvented in lower emission scenarios.

Climate Change↗
DOE OSTI2024

A flexible class of priors for orthonormal matrices with basis function-specific structure

Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system’s internal variability.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Causality-respecting adaptive refinement for PINNs: enabling precise interface evolution in phase field modeling

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving physical systems described by partial differential equations (PDEs). However, their accuracy in dynamical systems, particularly those involving sharp moving boundaries with complex initial morphologies, remains a challenge. Here, this study introduces an approach combining residual-based adaptive refinement (RBAR) with causality-informed training to enhance the performance of PINNs in solving spatio-temporal PDEs. Our method employs a three-step iterative process: initial causality-based training, RBAR-guided domain refinement, and subsequent causality training on the refined mesh. Applied to the Allen-Cahn equation, a widely-used model in phase field simulations, our approach demonstrates significant improvements in solution accuracy and computational efficiency over traditional PINNs. Notably, we observe an ‘overshoot and relocate’ phenomenon in dynamic cases with complex morphologies, showcasing the method’s adaptive error correction capabilities. This synergistic interaction between RBAR and causality training enables accurate capture of interface evolution, even in challenging scenarios where traditional PINNs fail. Our framework not only resolves the limitations of uniform refinement strategies but also provides a generalizable methodology for solving a broad range of spatio-temporal PDEs. The enhanced performance of the RBAR–causality combined framework demonstrates its strong potential for advancing PINN-based modeling of physical systems characterized by complex, evolving interfaces.

Allen-Cahn equations↗
DOE OSTI2025

Distributed Fiber Optic Sensing for in-well hydraulic fracture monitoring

This study presents the results from in-well hydraulic fracture monitoring within a horizontal well in an unconventional reservoir utilizing Distributed Fiber Optic Sensing (DFOS). An in-house-developed Brillouin-based Distributed Strain Sensing (DSS) interrogator was deployed to obtain strain measurements, complemented by a commercial Raman-based Distributed Temperature Sensing (DTS) interrogator for temperature measurements and a commercial Rayleigh-based Low-Frequency Distributed Acoustic Sensing (LF-DAS) interrogator for strain-rate measurements. Examined over a ten-day period, the spatio-temporal distribution of temperature-compensated strain obtained from DSS and DTS revealed distinct signatures of the multi-stage hydraulic fracturing process. These signatures were analyzed with respect to fracture width growth and closure, residual strain effects, and fracture conductivity near the wellbore. Fracture widths within the fracture zone were estimated for individual stages. The findings were assessed with LF-DAS measurements for further evaluation. This work integrates DFOS-measured strain, temperature, and strain-rate data for monitoring in-well hydraulic fracturing, with the goal of supporting future studies in interpreting DFOS measurements for improved understanding of hydraulic fracturing in unconventional reservoirs.

58 GEOSCIENCES↗
DOE OSTI2024

Quantitative phenotyping of crop roots with spectral electrical impedance tomography: a rhizotron study with optimized measurement design

Background: Root systems are key contributors to plant health, resilience, and, ultimately, yield of agricultural crops. To optimize plant performance, phenotyping trials are conducted to breed plants with diverse root traits. However, traditional analysis methods are often labour-intensive and invasive to the root system, therefore limiting high-throughput phenotyping. Spectral electrical impedance tomography (sEIT) could help as a non-invasive and cost-efficient alternative to optical root analysis, potentially providing 2D or 3D spatio-temporal information on root development and activity. Although impedance measurements have been shown to be sensitive to root biomass, nutrient status, and diurnal activity, only few attempts have been made to employ tomographic algorithms to recover spatially resolved information on root systems. In this study, we aim to establish relationships between tomographic electrical polarization signatures and root traits of different fine root systems (maize, pinto bean, black bean, and soy bean) under hydroponic conditions. Results: Our results show that, with the use of an optimized data acquisition scheme, sEIT is capable of providing spatially resolved information on root biomass and root surface area for all investigated root systems. We found strong correlations between the total polarization strength and the root biomass (R 2 = 0.82) and root surface area (R 2 = 0.8). Our findings suggest that the captured polarization signature is dominated by cell-scale polarization processes. Additionally, we demonstrate that the resolution characteristics of the measurement scheme can have a significant impact on the tomographic reconstruction of root traits. Conclusion: Our findings showcase that sEIT is a promising tool for the tomographic reconstruction of root traits in high-throughput root phenotyping trials and should be evaluated as a substitute for traditional, often time-consuming, root characterization methods.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2024

Visualizing an Exascale Data Center Digital Twin: Considerations, Challenges and Opportunities

Digital twins are an excellent tool to model, visualize, and simulate complex systems, to understand and optimize their operation. In this work, we present the technical challenges of real-time visualization of a digital twin of the Frontier supercomputer.We show the initial prototype and current state of the twin and highlight technical design challenges of visualizing such a large High Performance Computing (HPC) system. The goal is to understand the use of augmented reality as a primary way to extract information and collaborate on digital twins of complex systems. This leverages the spatio-temporal aspect of a 3D representation of a digital twin, with the ability to view historical and real-time telemetry, triggering simulations of a system state and viewing the results, which can be augmented via dashboards for details. Finally, we discuss considerations and opportunities for augmented reality of digital twins of large-scale, parallel computers.

Maiterth, Matthias↗
DOE OSTI2025

Vortical interactions in turbulent thermoacoustic systems

This study examines the dynamics of vortical interactions and their implications for mitigating thermoacoustic instability in a turbulent combustor. The regions of intense vortical interactions are identified as vortical communities in the network space of weighted directed vortical networks constructed from two-dimensional experimental velocity data. One can expect vortical interactions in the combustor to be strongest near the moment of vortex shedding, as the shed vortices gradually weaken due to dissipation while convecting downstream. However, we show that, during the state of thermoacoustic instability, there is a non-trivial consistent phase lag of approximately 52° between the shedding of the coherent structures from the backward-facing step and the time instant when the vortical interactions attain their local maximum value. We explain this phase lag by investigating the correlation between acoustic pressure fluctuations, spatio-temporal dynamics of coherent structures and vortical interactions in the reaction field of the combustor. We also show the aperiodic variation of vortical interactions during the states of combustion noise and aperiodic epochs of intermittency. Furthermore, the spatio-temporal evolution of pairs of vortical communities with the maximum inter-community interactions provides insight into explaining the critical regions detected in the reaction field during the states of intermittency and thermoacoustic instability, also identified in previous studies. As a result, we further show that the most efficient suppression of thermoacoustic instability via air microjet injection is achieved when steady air jets are introduced to disrupt the maximum inter-community interactions present during the state of thermoacoustic instability.

Sahay, Ankit [Indian Institute of Technology Madra↗
DOE OSTI2025

In situ multi-tier auto-ignition detection applied to dual-fuel combustion simulations

Here we use an anomaly detection methodology that is centered on analyzing fourth-order joint moments (co-kurtosis), particularly focusing on its application in auto-ignition of combustion problems with large numbers of species. Unsupervised anomaly detection is challenging to generalize across problem types and domains. A recent technique, centered on analyzing information in the fourth-order joint moment co-kurtosis, has shown promise, especially for high-dimensional scientific data. In this work we present developments to the co-kurtosis based anomaly detection method needed to make it effective and scalable for large-scale distributed scientific data, such as those generated by massively parallel simulations. An in situ co-kurtosis algorithm is employed as the anomaly detection method for identifying ignition kernels in simulations of turbulent combustion. Here, we extend an existing methodology which identifies regions of the domain where anomalies are present, and add another tier of anomaly detection where the individual samples contributing to the anomaly are identified. We apply this algorithm on-the-fly to a variety of turbulent reacting flow problems and compare it to the widely used (but significantly more expensive) chemical explosive mode analysis (CEMA). We demonstrate the ability of the method to detect and identify the onset of low and high temperature ignition which can be used for computational steering, as chemical and combustion anomalies occur intermittently at spatio-temporal locations unknown a priori. Finally, we apply our lightweight in situ algorithm to an exascale high-fidelity simulation with a total of 2.4 Trillion degrees of freedom, performed using an adaptive mesh refinement solver. Furthermore, through a scalability analysis, we show that the relative computational cost of this in-situ anomaly detection algorithm compared to an iteration of the reacting flow solver is negligible.

97 MATHEMATICS AND COMPUTING↗