Results for “Bounds” — Page 41

DOE OSTI2021

Advancing Tassel Detection and Counting: Annotation and Algorithms

Tassel counts provide valuable information related to flowering and yield prediction in maize, but are expensive and time-consuming to acquire via traditional manual approaches. High-resolution RGB imagery acquired by unmanned aerial vehicles (UAVs), coupled with advanced machine learning approaches, including deep learning (DL), provides a new capability for monitoring flowering. In this article, three state-of-the-art DL techniques, CenterNet based on point annotation, task-aware spatial disentanglement (TSD), and detecting objects with recursive feature pyramids and switchable atrous convolution (DetectoRS) based on bounding box annotation, are modified to improve their performance for this application and evaluated for tassel detection relative to Tasselnetv2+. The dataset for the experiments is comprised of RGB images of maize tassels from plant breeding experiments, which vary in size, complexity, and overlap. Results show that the point annotations are more accurate and simpler to acquire than the bounding boxes, and bounding box-based approaches are more sensitive to the size of the bounding boxes and background than point-based approaches. Overall, CenterNet has high accuracy in comparison to the other techniques, but DetectoRS can better detect early-stage tassels. The results for these experiments were more robust than Tasselnetv2+, which is sensitive to the number of tassels in the image.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

On the Minimum Radius of Very Massive Neutron Stars

Prospects of establishing the radii of massive neutron stars in PSR J1614 – 2230 and PSR J0740 + 6620 from Neutron Star Interior Composition ExploreR and Chandra observatories hold the potential to constrain the equation of state (EoS) of matter to densities well beyond those encountered in canonical stars of mass $\sim 1.4\,{M}_{\odot }$. In this work, we investigate the relation between the radii of very massive neutron stars up to the maximum mass, ${M}_{\max }$, supported by dense matter EoSs. Results from models with hadronic matter are contrasted with those that include a first-order hadron-to-quark phase transition. We find that a lower bound on ${M}_{\max }$ with an upper bound on the radius of massive pulsars serves to rule out quark matter that is too soft, and an upper bound on ${M}_{\max }$ with a lower bound on the radius of massive pulsars strongly disfavors a transition into quark matter that is too stiff appearing at low densities. In conclusion, the complementary role played by radius inferences from future gravitational-wave events of inspiraling binary neutron stars is also briefly discussed.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Data accompanying the publication "Sensitivity of grass fires burning in marginal conditions to atmospheric turbulence" in the Journal of Geophysical Research-Atmospheres

Data accompanying Jonko, Yedinak, Conley, and Linn (2021): "Sensitivity of grass fires burning in marginal conditions to atmospheric turbulence", JGR-Atmospheres. LA-UR-21-24832. The following files were generated from simulations performed with Los Alamos National Laboratory coupled atmosphere-fire behavior model FIRETEC v1.3: -area_burned.csv contains the area burned per second in m^2 for 45 CORE simulations, 20 INIT simulations and 20 BOUND simulations (85 columns), and 600 seconds of simulation (600 rows). -fire_perimeter.csv containing the fire perimeters in m for 45 CORE simulations, 20 INIT simulations and 20 BOUND simulations (85 columns), and 600 seconds of simulation (600 rows). -consumption_per_second.csv containing consumption in kg/s for 45 CORE simulations, 20 INIT simulations and 20 BOUND simulations (85 columns), and 600 seconds of simulation (600 rows). -rate_of_spread.csv containing the distance of the fire front from the ignition line in m for 45 CORE simulations, 20 INIT simulations and 20 BOUND simulations (85 columns), and 600 seconds of simulation (600 rows). -CORE1_u_10m_300_sec.hdf, INIT3_u_10m_300_sec.hdf, and BOUND3_u_10m_300_sec.hdf contain u wind speeds at 10 m height for 300 seconds (3D: time, x, y) of simulation for simulations CORE1, INIT3, and BOUND3, respectively. -CORE.final.fuel.density.hdf contains final fuel densities in the surface layer of the model (3D: ensemble member, x, y) for all 45 CORE simulations. -CORE{1-45}_3D_u_120_sec.hdf: 45 files containing vertically and horizontally resolved u winds for 120 seconds of simulation (4D: time, x, y, z) for the 45 CORE ensemble members. For additional information about this data, please contact the corresponding author at ajonko@lanl.gov.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

A Hybrid Gradient Method to Designing Bayesian Experiments for Implicit Models

Bayesian experimental design (BED) aims at designing an experiment to maximize the information gathering from the collected data. The optimal design is usually achieved by maximizing the mutual information (MI) between the data and the model parameters. When the analytical expression of the MI is unavailable, e.g.,having implicit models with intractable data distributions, a neural network-based lower bound of the MI was recently proposed and a gradient ascent method was used to maximize the lower bound [1]. However, the approach in [1] requires a pathwise sampling path to compute the gradient of the MI lower bound with respect to the design variables, and such a pathwise sampling path is usually inaccessible for implicit models. In this work, we propose a hybrid gradient approach that leverages recent advances in variational MI estimator and evolution strategies (ES)combined with black-box stochastic gradient ascent (SGA) to maximize the MI lower bound. This allows the design process to be achieved through a unified scalable procedure for implicit models without sampling path gradients. Several experiments demonstrate that our approach significantly improves the scalability of BED for implicit models in high-dimensional design space.

Zhang, Jiaxin↗

DOE OSTI2022

Centimeter human skeleton pose estimation

A method, apparatus and system for human skeleton pose estimation includes synchronously capturing images of a human moving through an area from a plurality of different points of view, for each of the plurality of captured images, determining a bounding box that bounds the human in the captured image and identifying pixel locations of the bounding box in the image, for each of the plurality of captured images, determining 2D and single-view 3D skeletons from the pixel locations of the bounding box, determining a first, multi-view 3D skeleton using a combination of the 2D and single-view 3D skeletons, and optimizing the first, multi-view 3D skeleton to determine a final 3D skeleton pose for the human. The method, apparatus and system can further include illuminating the area with structured light during the capturing of the images of the human moving through the area.

Brookshire, Jonathan D.↗

DOE OSTI2023

Privacy Amplification for Episodic Training Methods

It has been shown that differential privacy bounds improve when subsampling within a randomized mechanism. Episodic training, utilized in many standard machine learning techniques, uses a multistage subsampling procedure which has not been previously analyzed for privacy bound amplification. In this paper, we focus on improving the calculation of privacy bounds in episodic training by thoroughly analyzing privacy amplification due to subsampling with a multi-stage subsampling procedure. The newly developed bound can be incorporated into existing privacy accounting methods.

Tombs, Vandy↗

DOE OSTI2026

Excited-state uncertainties in lattice-QCD calculations of hadron masses and scattering phase shifts

Lattice QCD has historically produced energy results interpretable as either estimates relying on implicit assumptions about asymptotic behavior or one-sided upper bounds. New Lanczos methods providing two-sided bounds with less-restrictive assumptions are introduced and quantified in a high-statistics calculation with unphysical quark masses. Two-sided bounds without spectral assumptions provide sub-percent constraints on the nucleon mass. Other bounds, which assume all states in a given energy window are resolved, provide meaningful two-sided constraints on nucleon-nucleon scattering phase shifts.

Detmold, William [MIT, Cambridge, CTP]↗

DOE OSTI2025

Time correlations from steady-state expectation values

Recovering properties of correlation functions is typically challenging. On one hand, experimentally, it requires measurements with a temporal resolution finer than the system's dynamics. On the other hand, analytical or numerical analysis requires solving the system evolution. Here, we use recent results of quantum metrology with continuous measurements to derive general lower bounds on the relaxation and second-order correlation times that are both easy to calculate and measure. These bounds are based solely on steady-state expectation values and their derivatives with respect to a control parameter, and can be readily extended to the autocorrelation of arbitrary observables. We validate our method on two examples of critical quantum systems: a critical driven-dissipative resonator, where the bound matches analytical results for the dynamics, and the infinite-range Ising model, where only the steady state is solvable and thus the bound provides information beyond the reach of existing analytical approaches. Our results can be applied to experimentally characterize ultrafast systems, and to theoretically analyze many-body models with dynamics that are analytically or numerically hard.

Górecki, Wojciech [INFN, Pavia]↗

DOE OSTI2025

Time correlations from steady-state expectation values

Recovering properties of correlation functions is typically challenging. On one hand, experimentally, it requires measurements with a temporal resolution finer than the system's dynamics. On the other hand, analytical or numerical analysis requires solving the system evolution. Here, we use recent results of quantum metrology with continuous measurements to derive general lower bounds on the relaxation and second-order correlation times that are both easy to calculate and measure. These bounds are based solely on steady-state expectation values and their derivatives with respect to a control parameter, and can be readily extended to the autocorrelation of arbitrary observables. We validate our method on two examples of critical quantum systems: a critical driven-dissipative resonator, where the bound matches analytical results for the dynamics, and the infinite-range Ising model, where only the steady state is solvable and thus the bound provides information beyond the reach of existing analytical approaches. Our results can be applied to experimentally characterize ultrafast systems, and to theoretically analyze many-body models with dynamics that are analytically or numerically hard.

Górecki, Wojciech [INFN, Pavia]↗

DOE OSTI2026

Entanglement Requirements for Coherent Enhancement in Detectors

Coherent enhancement is a powerful mechanism for improving the sensitivity of a wide range of detectors, but its practical use is often limited by the difficulty of preparing the required quantum states. We show that this difficulty has a fundamental origin: coherent enhancement of a signal interacting with a detector is quantitatively constrained by entanglement. We prove general bounds on how the strength of coherent effects can scale with system size, as a function of the single-mode entanglement entropy of the detector. These bounds smoothly interpolate between the incoherent and fully coherent regimes, and apply both to parameter-estimation problems and to scattering processes. We discuss these results from two complementary perspectives: First, they appear as bounds on the quantum Fisher information of many-body states, which translate directly into limits on parameter sensitivity via the quantum Cramér-Rao bound. Second, they can be interpreted as limits on a class of scattering cross sections, leading to predictions for how minimum detectable interaction strengths scale with target size. Together, these results provide a unified view of coherent enhancement in metrology and scattering experiments, and motivate the development of new techniques for generating entangled detector states.

Bogorad, Zachary [Fermilab] (ORCID:000000019913647↗

DOE OSTI2026

Hydrogeological assessment of CO2 containment assurance and wellbore integrity at a Gulf Coast storage site

Abstract A large-scale carbon capture and storage (CCS) initiative on the Texas Gulf Coast serves as a premier demonstration of the U.S. Department of Energy’s CarbonSAFE program. Targeting deep saline formations, specifically Oligo-Miocene deltaic sequences, the project aims to establish technical and commercial viability for geologic CO2 storage within a major industrial corridor. This study provides a rigorous hydrogeological assessment to support Class VI permitting by quantifying the high degree of containment security. Utilizing a compositional reservoir simulator, we developed a suite of 27 distinct simulation cases to evaluate vertical plume dynamics near both planned injection wells and proximal legacy infrastructure. To ensure numerical accuracy near wellbores, we implemented a refined mesh strategy, determining that a 5.6 ft × 5.6 ft grid refinement offered the optimal balance between computational efficiency and descriptive precision. The modeling framework utilized a systematic sensitivity-based approach to evaluate the mechanical redundancy of the subsurface system by performing a bounding analysis of wellbore interfaces against hypothetical high-permeability microannuli. By systematically isolating competing physical drivers, including permeability, porosity, gas hysteresis, thermal gradients, salinity, and solubility trapping (quantified via Henry’s law with dynamically adjusted coefficients), this work moves beyond binary assessments to establish a nuanced hierarchy of containment factors. The results confirm that primary trapping mechanisms (e.g., gas hysteresis and solubility), combined with the site's unique geomechanical stratigraphy, significantly restrict vertical mobility and reinforce the robust containment security of the reservoir. Baseline results demonstrate substantial vertical separation between the CO2 plume and the upper confining system, ensuring robust containment. Sensitivity analysis reveals that even under highly conservative bounding scenarios—assuming theoretical 10-Darcy pathways at specific wellbore locations—the 2,900-ft thick multi-layered confining zone remains a reliable barrier. In these hypothetical upper-bound cases, peak upward fluxes of CO2 and saltwater after 15 years of injection remain localized and dissipate rapidly within the lower sections of the confining interval, leaving the integrity of the seal uncompromised. Furthermore, the study identifies that while localized wellbore pathways define theoretical upper bounds of vertical migration, the Area of Review (AoR) is primarily sensitive to regional thermal gradients and hysteresis, which can influence the AoR by over 3,000 acres in pessimistic configurations. Also, primary trapping mechanisms, specifically gas hysteresis and solubility, work in tandem with the Gulf Coast’s unique geomechanical stratigraphy to significantly restrict vertical mobility. Ductile, smectite-rich mudstones facilitate natural borehole convergence and the self-healing of potential conduits, creating a natural geomechanical bridge that effectively mitigates migration potential at both current injection points and legacy-well locations. This comprehensive modeling effort demonstrates that the integration of high-resolution wellbore simulations and regional geomechanical observations confirms the long-term storage security of the studied site, providing a physics-based foundation for industrial-scale CCS deployments. This modeling framework establishes a baseline for future research into coupled geomechanical effects, such as time-dependent borehole convergence, to further refine long-term containment projections. Acknowledgements We thank the Gulf Coast Carbon Center (GCCC) at the Bureau of Economic Geology for foundational research support. We appreciate Alex Bump for technical guidance and David Hoffman for model mesh generation. This work used TACC’s Frontera cluster for simulations and CMG Ltd. software licenses provided to UT-Austin. This material is based upon work supported by the Department of Energy under Award Number DE-FE0032338. Disclaimer This material is based upon work supported by the U.S. Department of Energy’s Fossil Energy and Carbon Management Office under the CarbonSAFE program, award Number DE-FE0032338. The views expressed herein do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

58 GEOSCIENCES↗

DOE OSTI2026

Hydrogeological assessment of CO2 containment assurance and wellbore integrity at a Gulf Coast storage site

Abstract A large-scale carbon capture and storage (CCS) initiative on the Texas Gulf Coast serves as a premier demonstration of the U.S. Department of Energy’s CarbonSAFE program. Targeting deep saline formations, specifically Oligo-Miocene deltaic sequences, the project aims to establish technical and commercial viability for geologic CO2 storage within a major industrial corridor. This study provides a rigorous hydrogeological assessment to support Class VI permitting by quantifying the high degree of containment security. Utilizing a compositional reservoir simulator, we developed a suite of 27 distinct simulation cases to evaluate vertical plume dynamics near both planned injection wells and proximal legacy infrastructure. To ensure numerical accuracy near wellbores, we implemented a refined mesh strategy, determining that a 5.6 ft × 5.6 ft grid refinement offered the optimal balance between computational efficiency and descriptive precision. The modeling framework utilized a systematic sensitivity-based approach to evaluate the mechanical redundancy of the subsurface system by performing a bounding analysis of wellbore interfaces against hypothetical high-permeability microannuli. By systematically isolating competing physical drivers, including permeability, porosity, gas hysteresis, thermal gradients, salinity, and solubility trapping (quantified via Henry’s law with dynamically adjusted coefficients), this work moves beyond binary assessments to establish a nuanced hierarchy of containment factors. The results confirm that primary trapping mechanisms (e.g., gas hysteresis and solubility), combined with the site's unique geomechanical stratigraphy, significantly restrict vertical mobility and reinforce the robust containment security of the reservoir. Baseline results demonstrate substantial vertical separation between the CO2 plume and the upper confining system, ensuring robust containment. Sensitivity analysis reveals that even under highly conservative bounding scenarios—assuming theoretical 10-Darcy pathways at specific wellbore locations—the 2,900-ft thick multi-layered confining zone remains a reliable barrier. In these hypothetical upper-bound cases, peak upward fluxes of CO2 and saltwater after 15 years of injection remain localized and dissipate rapidly within the lower sections of the confining interval, leaving the integrity of the seal uncompromised. Furthermore, the study identifies that while localized wellbore pathways define theoretical upper bounds of vertical migration, the Area of Review (AoR) is primarily sensitive to regional thermal gradients and hysteresis, which can influence the AoR by over 3,000 acres in pessimistic configurations. Also, primary trapping mechanisms, specifically gas hysteresis and solubility, work in tandem with the Gulf Coast’s unique geomechanical stratigraphy to significantly restrict vertical mobility. Ductile, smectite-rich mudstones facilitate natural borehole convergence and the self-healing of potential conduits, creating a natural geomechanical bridge that effectively mitigates migration potential at both current injection points and legacy-well locations. This comprehensive modeling effort demonstrates that the integration of high-resolution wellbore simulations and regional geomechanical observations confirms the long-term storage security of the studied site, providing a physics-based foundation for industrial-scale CCS deployments. This modeling framework establishes a baseline for future research into coupled geomechanical effects, such as time-dependent borehole convergence, to further refine long-term containment projections. Acknowledgements We thank the Gulf Coast Carbon Center (GCCC) at the Bureau of Economic Geology for foundational research support. We appreciate Alex Bump for technical guidance and David Hoffman for model mesh generation. This work used TACC’s Frontera cluster for simulations and CMG Ltd. software licenses provided to UT-Austin. This material is based upon work supported by the Department of Energy under Award Number DE-FE0032338. Disclaimer This material is based upon work supported by the U.S. Department of Energy’s Fossil Energy and Carbon Management Office under the CarbonSAFE program, award Number DE-FE0032338. The views expressed herein do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

58 GEOSCIENCES↗

DOE OSTI2026

Strong-Drive Limits in Josephson Circuits: From Chaos to an Unbound-Resonance Threshold

Strong microwave drives enable fast measurement and parametric control in superconducting circuits but can induce transitions out of the intended low-energy manifold. We develop a unified description of strong-drive limits in flux- and charge-driven Josephson circuits across drive frequency and dc flux bias. Using classical phase-space analysis and Floquet--Markov simulations, we identify distinct low- and high-frequency mechanisms. At low frequency, we characterize bound-state resonances and separatrix chaos and find that the flux-drive chaos threshold depends strongly on dc flux bias. At high frequency, these mechanisms are suppressed, and the dissipative steady state transfers from the central bound-state sector to outer resonances formed by above-barrier running trajectories. The resulting unbound-resonance threshold is nearly independent of drive frequency and circuit parameters over the regime studied and is controlled primarily by dc flux bias. Coherent simulations show that parametric operation persists beyond this threshold, but at a reduced rate, setting an effective upper bound on the achievable operation speed. We derive analytical criteria for both thresholds, validate them numerically, and experimentally confirm the predicted dc-bias dependence of the low-frequency threshold in a flux-driven SQUID. We also determine the timescales for transfer into the unbound-resonance regime and relaxation back to the bound-state manifold after the drive is removed. Finally, we relate the stability limits to a complementary picture based on the junction critical current and extend the framework to multitone drives and inductively shunted circuits. Together, these results identify the mechanisms limiting strong driving and suggest routes to extend the stable operating range of Josephson circuits.

You, Xinyuan [Fermilab] (ORCID:0000000291789419)↗

DOE OSTI2021

Data-driven enhancement of coherent structure-based models for predicting instantaneous wall turbulence

Predictions of the spatial representation of instantaneous wall-bounded flows, via coherent structure-based models, are highly sensitive to the geometry of the representative structures employed by them. In this study, we propose a methodology to extract the three-dimensional (3-D) geometry of the statistically significant eddies from multi-point wall-turbulence datasets, for direct implementation into these models to improve their predictions. The methodology is employed here for reconstructing a 3-D statistical picture of the inertial wall coherent turbulence for all canonical wall-bounded flows, across a decade of friction Reynolds number (Re T ). These structures are responsible for the Re T -dependence of the skin-friction drag and also facilitate the inner-outer interactions, making them key targets of structure-based models. The empirical analysis brings out the geometric self-similarity of the large-scale wall-coherent motions and also suggests the hairpin packet as the representative flow structure for all wall-bounded flows, thereby aligning with the framework on which the attached eddy model (AEM) is based. The same framework is extended here to also model the very-large-scaled motions, with a consideration of their differences in internal versus external flows. Implementation of the empirically-obtained geometric scalings for these large structures into the AEM is shown to enhance the instantaneous flow predictions for all three velocity components. Finally, an active flow control system driven by the same geometric scalings is conceptualized, towards favourably altering the influence of the wall coherent motions on the skin-friction drag.

42 ENGINEERING↗

DOE OSTI2021

Multi-armed bandit with sub-exponential rewards

Not provided.

Operations Research & Management Science↗

DOE OSTI2022

Local conformations and heterogeneities in structures and dynamics of isotactic polypropylene adsorbed onto carbon fiber

Carbon fiber (CF) reinforced polymers (CFRPs) have experienced widespread use in various industries. One of the most important parameters that controls the macroscopic property of CFRPs is the interface between a polymer matrix and CF. There is growing evidence to suggest the formation of a bound polymer layer (BPL), i.e., polymer chains that physically adsorb on a filler surface. However, this interface is always in contact with the thicker part of a polymer matrix, rendering its understanding a difficult task. Therein, we use CF-reinforced isotactic polypropylene (iPP) as a rational CFRP. To characterize the BPL on the CF surface, we extracted it from the CFRP using solvent-rinsing with p-xylene. The physical and thermal properties of the BPL were characterized by differential scanning calorimetry and thermogravimetric analysis, while its microscopic structures and dynamics were probed by small-angle neutron scattering and quasi-elastic neutron scattering (QENS) techniques. Subsequently, we employed atomistic molecular dynamics (MD) simulations to complement the QENS results above the bulk melting temperature and reveal details that were experimentally inaccessible. In this work, we observed that the degree of crystallinity of the BPL was quite lower than the bulk, while the melting temperature of the BPL remained the same as the bulks. Within the given length and time scales probed by QENS and MD, we also observed that most of the bound chains were mobile, with the formation of a high-density region (less than 1 nm in thickness) near the CF surface. The segmental dynamics of the bound chains probed by both QENS and MD were also much faster than those of the free chains, possibly due to the presence of a free surface region at the topmost surface of the BPL. Furthermore, the MD results demonstrated that the backbone chains and side groups lie nearly flat on the CF surface, which is the driving force for the flattening process of the iPP BPL to overcome the conformational entropy loss in the total free energy.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Peer-to-Peer Communication Trade-Offs for Smart Grid Applications: Preprint

Peer-to-peer energy management systems for smart grids require developers to consider the trade-offs between the amount of communication traffic generated and the quality and speed of convergence of the control algorithms that are deployed. Employing a fully connected communication causes messages to scale exponentially with the number of nodes, while using a sparse connectivity causes less information dissemination leading to degradation of the algorithm performance. The best communication topology for a particular application lies somewhere in between and often requires empirical evaluation by application designers. Existing methods do not put focus on the needs for smart grid applications, which is information dissemination throughout the network and they do not provide a flexible solution for application developers to prototype and deploy different topologies without modifying the application code. This paper introduces a configurable virtual communication topology framework TopLinkMgr, allowing users to specify any chosen communication topology and deploy peer-to-peer applications using it. It also introduces a self-adaptive, fault-tolerant topology management algorithm, Bounded Path Dissemination that can ensure the dissemination of information to all peers within a specified threshold for a sparsely connected topology. Experiments show that the algorithm improves on convergence speed and accuracy over state-of-the-art methods and is also robust against node failures. The results indicate the possibility of achieving a close-to optimal convergence without overloading the network allowing the realization of peer-to-peer control platforms covering larger and more complex power systems.

Bounded Path Dissemination↗

DOE OSTI2022

OptZConfig: Efficient Parallel Optimization of Lossy Compression Configuration

Lossless compressors have very low compression ratios that do not meet the needs of today's large-scale scientific applications that produce vast volumes of data. Error-bounded lossy compression (EBLC) is considered a critical technique for the success of scientific research. Although EBLC allows users to set an error bound for the compression, users have been unable to specify the requirements on the compression quality, limiting practical use. Our contributions are: (1) We formulate the problem of configuring EBLC to preserve a user-defined metric as an optimization problem. This allows many classes of new metrics to be preserved, which improves over current practices. (2) We present a framework, OptZConfig, that can adapt to improvements in the search algorithm, compressor, and metrics with minimal changes, enabling future advancements in this area. (3) We demonstrate the advantages of our approach against the leading methods to configure compressors to preserve specific metrics. Here, our approach improves compression ratios against a specialized compressor by up to 3 x, has a 56x speedup over FRaZ, 1000x speedup over MGARD-QOI post tuning, and 110x speedup over systematic approaches which had not been bounded by compressors before.

97 MATHEMATICS AND COMPUTING↗