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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.

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At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Prediction of Redox Potentials for the Late Actinides Cm to Lr Using Electronic Structure Methods

Our previously developed computational method for calculating the aqueous redox potentials of the early actinides has been extended to the later elements in the actinide series: Cm, Bk, Cf, Es, Fm, Md, No, and Lr in multiple oxidation states. These calculations were performed using density functional theory with small-core pseudopotentials and their associated basis sets. Solvation effects were considered via a supermolecule-continuum approach, with 30 water molecules representing two solvation shells. Both the COSMO and SMD implicit solvation models were utilized. The structural parameters and hydration numbers for Cm(III), Bk(III), Bk(IV), and Cf(III) are in reasonable agreement with the available experimental data. For redox processes involving atomic cations in solution, the B3LYP/COSMO approach predicted redox potentials to within ±0.2 V of experiment for most redox couples, consistent with our prior work. Inclusion of spin-orbit corrections in specific redox pairs, especially those with the later actinides in high oxidation states, yields improved results relative to calculations including only scalar-relativistic corrections. The An +m /An(0) redox potentials were calculated using a Born-Haber cycle incorporating sublimation, ionization, and hydration energies. Due to a lack of experimental data, three sets of ionization energies were used for the Born-Haber cycle. The calculated An(III/0) potentials showed better agreement with experimental data when using the COSMO solvation model and the test set comprising the NIST recommended ionization energies. Furthermore, the Md(II/0) potential was better described with the SMD model, whereas No(II/0) was not well described by all methods. Finally, the computational approach was able to predict redox potentials that for most cases agreed with the current available experimental or estimated data.

Actinides↗
DOE OSTI2021

Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers

We report accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are merely tiny perturbations on top of a close balance between gravity and the pressure gradient, such simulations place heavy demands on numerical hydrodynamics schemes. We demonstrate how discretization errors on grids of reasonable size can lead to spurious flows orders of magnitude faster than the physical flow. Well-balanced numerical schemes can deal with this problem. Three such schemes were applied in the implicit, finite-volume SEVEN-LEAGUE HYDRO code in combination with a low-Mach-number numerical flux function. We compare how the schemes perform in four numerical experiments addressing some of the challenges imposed by typical problems in stellar hydrodynamics. We find that the α-β and deviation well-balancing methods can accurately maintain hydrostatic solutions provided that gravitational potential energy is included in the total energy balance. They accurately conserve minuscule entropy fluctuations advected in an isentropic stratification, which enables the methods to reproduce the expected scaling of convective flow speed with the heating rate. The deviation method also substantially increases accuracy of maintaining stationary orbital motions in a Keplerian disk on long timescales. The Cargo–LeRoux method fares substantially worse in our tests, although its simplicity may still offer some merits in certain situations. Overall, we find the well-balanced treatment of gravity in combination with low Mach number flux functions essential to reproducing correct physical solutions to challenging stellar slow-flow problems on affordable collocated grids.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Learning Constrained Parametric Differentiable Predictive Control Policies With Guarantees

We present differentiable predictive control (DPC), a method for offline learning of constrained neural control policies for nonlinear dynamical systems with performance guarantees. We show that the sensitivities of the parametric optimal control problem can be used to obtain direct policy gradients. Specifically, we employ automatic differentiation (AD) to efficiently compute the sensitivities of the model predictive control (MPC) objective function and constraints penalties. To guarantee safety upon deployment, we derive probabilistic guarantees on closed-loop stability and constraint satisfaction based on indicator functions and Hoeffding’s inequality. We empirically demonstrate that the proposed method can learn neural control policies for various parametric optimal control tasks. In particular, we show that the proposed DPC method can stabilize systems with unstable dynamics, track time-varying references, and satisfy nonlinear state and input constraints. Our DPC method has practical time savings compared to alternative approaches for fast and memory-efficient controller design. Specifically, DPC does not depend on a supervisory controller as opposed to approximate MPC based on imitation learning. We demonstrate that, without losing performance, DPC is scalable with greatly reduced demands on memory and computation compared to implicit and explicit MPC while being more sample efficient than model-free reinforcement learning (RL) algorithms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

ESMs Latent Space Exploration for Uncertainty Quantification and Spatiotemporal Downscaling

This final report for DOE Award DE-SC0023044 presents advances in two key areas of climate modeling: (1) representative climate model selection and (2) Earth System Model (ESM) downscaling using hybrid AI methods. The first section introduces a reordered, three-stage workflow to select representative GCM runs that more effectively balance historical skill with ensemble spread, validated across Texas, Bihar, and New York. The second section introduces two novel super-resolution frameworks, ViSIR and ViFOR, that integrate Vision Transformers with sinusoidal and Fourier-based implicit neural representations. These models achieve state-of-the-art reconstruction accuracy for ESM variables including surface temperature and heat fluxes. The report includes detailed methodology, benchmarks, and results, demonstrating significant gains in uncertainty quantification, spatial fidelity, and scalability for climate-impact studies.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons

Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C 60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Reducing uncertainty of high-latitude ecosystem models through identification of key parameters

Abstract Climate change is having significant impacts on Earth’s ecosystems and carbon budgets, and in the Arctic may drive a shift from an historic carbon sink to a source. Large uncertainties in terrestrial biosphere models (TBMs) used to forecast Arctic changes demonstrate the challenges of determining the timing and extent of this possible switch. This spread in model predictions can limit the ability of TBMs to guide management and policy decisions. One of the most influential sources of model uncertainty is model parameterization. Parameter uncertainty results in part from a mismatch between available data in databases and model needs. We identify that mismatch for three TBMs, DVM-DOS-TEM, SIPNET and ED2, and four databases with information on Arctic and boreal above- and belowground traits that may be applied to model parametrization. However, focusing solely on such data gaps can introduce biases towards simple models and ignores structural model uncertainty, another main source for model uncertainty. Therefore, we develop a causal loop diagram (CLD) of the Arctic and boreal ecosystem that includes unquantified, and thus unmodeled, processes. We map model parameters to processes in the CLD and assess parameter vulnerability via the internal network structure. One important substructure, feed forward loops (FFLs), describe processes that are linked both directly and indirectly. When the model parameters are data-informed, these indirect processes might be implicitly included in the model, but if not, they have the potential to introduce significant model uncertainty. We find that the parameters describing the impact of local temperature on microbial activity are associated with a particularly high number of FFLs but are not constrained well by existing data. By employing ecological models of varying complexity, databases, and network methods, we identify the key parameters responsible for limited model accuracy. They should be prioritized for future data sampling to reduce model uncertainty.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Commercial, industrial, and institutional discount rate estimation for efficiency standards analysis: Sector-level data 1998–2021

Underlying each of the Department of Energy’s (DOE’s) federal appliance and equipment energy conservation standards are a set of complex analyses of the projected costs and benefits of regulation. Any new or amended standard must be designed to achieve significant additional energy conservation, provided that it is technologically feasible and economically justified (42 U.S.C. 6295(o)(2)(A)). DOE determines economic justification based on whether the benefits exceed the burdens, considering a variety of factors, including the economic impact of the standard on consumers of the product and the savings in lifetime operating cost compared to any increase in price or maintenance expenses (42 U.S.C. 6295(o)(2)(B)). As part of this determination, DOE conducts a Life-Cycle Cost (LCC) analysis, which models the combined impact of appliance first cost and operating cost changes on a representative commercial building sample in order to identify the fraction of customers achieving LCC savings or incurring net cost at the considered efficiency levels. Thus, the commercial discount rate value(s) used to calculate the present value of energy cost savings within the LCC model implicitly plays a role in estimating the economic impact of potential standard levels. This report provides an in-depth discussion of the commercial discount rate estimation process. It is an update to previous reports on estimating commercial discount rates from firm-level financial data (Fujita, 2016). Major topics covered in this report include: Discount rate estimation methods and rationale; -Data sources used and data limitations; -Discount rate distributions for use in standards analysis; -Discount rate estimation methods and distributions specific to the small business subgroup analysis. Going forward, this report will be updated as data allow and analyses necessitate.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2022

Validation and parameterization of a novel physics-constrained neural dynamics model applied to turbulent fluid flow

We report, in fluid physics, data-driven models to enhance or accelerate time to solution are becoming increasingly popular for many application domains, such as alternatives to turbulence closures, system surrogates, or for new physics discovery. In the context of reduced order models of high-dimensional time-dependent fluid systems, machine learning methods grant the benefit of automated learning from data, but the burden of a model lies on its reduced-order representation of both the fluid state and physical dynamics. In this work, we build a physics-constrained, data-driven reduced order model for Navier–Stokes equations to approximate spatiotemporal fluid dynamics in the canonical case of isotropic turbulence in a triply periodic box. The model design choices mimic numerical and physical constraints by, for example, implicitly enforcing the incompressibility constraint and utilizing continuous neural ordinary differential equations for tracking the evolution of the governing differential equation. We demonstrate this technique on a three-dimensional, moderate Reynolds number turbulent fluid flow. In assessing the statistical quality and characteristics of the machine-learned model through rigorous diagnostic tests, we find that our model is capable of reconstructing the dynamics of the flow over large integral timescales, favoring accuracy at the larger length scales. More significantly, comprehensive diagnostics suggest that physically interpretable model parameters, corresponding to the representations of the fluid state and dynamics, have attributable and quantifiable impact on the quality of the model predictions and computational complexity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

STSR-INR: Spatiotemporal super-resolution for multivariate time-varying volumetric data via implicit neural representation

Implicit neural representation (INR) has surfaced as a promising direction for solving different scientific visualization tasks due to its continuous representation and flexible input and output settings. We present STSR-INR, an INR solution for generating simultaneous spatiotemporal super-resolution for multivariate time-varying volumetric data. Inheriting the benefits of the INR-based approach, STSR-INR supports unsupervised learning and permits data upscaling with arbitrary spatial and temporal scale factors. Unlike existing GAN- or INR-based super-resolution methods, STSR-INR focuses on tackling variables or ensembles and enabling joint training across datasets of various spatiotemporal resolutions. Here we achieve this capability via a variable embedding scheme that learns latent vectors for different variables. In conjunction with a modulated structure in the network design, we employ a variational auto-decoder to optimize the learnable latent vectors to enable latent-space interpolation. To combat the slow training of INR, we leverage a multi-head strategy to improve training and inference speed with significant speedup. We demonstrate the effectiveness of STSR-INR with multiple scalar field datasets and compare it with conventional tricubic+linear interpolation and state-of-the-art deep-learning-based solutions (STNet and CoordNet).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

The Transportation Energy and Mobility Pathway Options (TEMPO) Model: Overview and Validation of V1.0 [Slides]

This presentation documents the approaches and methods used in the Transportation Energy & Mobility Pathway Options™ (TEMPO) model to evaluate passenger and freight demand for transportation and mobility services, project vehicle ownership and technology adoption decisions, and determine transport mode choices to derive scenarios of future energy use and emissions. TEMPO is an all-inclusive transportation demand model that covers the entire United States, with an implicit spatial resolution and an hourly temporal resolution that allows for generating time-resolved energy use profiles to assess multisectoral integration aspects. Key features of the TEMPO model include the ability to perform endogenous out-of-sample forecasting to extrapolate recent emerging trends and analyze impacts of disruptive technological breakthroughs and behavioral changes. TEMPO employs an innovative representation of passenger mobility demand stemming from household-level decisions that determine vehicle adoption, ownership, and use based on sociodemographics (e.g., income, household composition), technology attributes (e.g., travel cost, time), geography (e.g., urban, suburban, rural) and population-specific multiday mobility and travel requirements. This representation enables a more forward-looking perspective on the use of new mobility options and the adoption of alternative fuel vehicles, as well as a more accurate representation of their energy usage profiles than previous modeling approaches. A comparison with the U.S. Energy Information Administration’s Annual Energy Outlook showcases the ability of TEMPO to accurately replicate widely accepted projections by representing the key elements of the entire transportation sector at the appropriate level of resolution. TEMPO is intended to generate future scenarios of technology adoption, energy use, and emissions in the transportation sector to compare alternatives, inform decision makers, and assess integration with energy infrastructure and supply systems at an appropriate spatiotemporal resolution.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2022

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Clustering-Based Predictive Analytics to Improve Scientific Data Discovery

Given the sheer volume of scientific data archived within the data-intensive projects at the US Department of Energy's Oak Ridge National Laboratory, finding precisely what data we are looking for may not be a trivial task; conversely, we may also miss a more prominent data product. To address such issues, we propose improving the data discovery system and using data analytics methods to comprehend what specific users might be interested in based on their physiological state, search patterns, and past data usage history. This work's primary goal is to prune the complexity, increase the visibility of popular data products, and direct users toward the data that best meet their needs. The proposed algorithm constructs a user profile based on the user's explicit or implicit interactions with the system, such as items they are currently looking at on-site and the key metadata mappings related to the data set. The pattern is then used to build a training data set, which will help find relevant data to recommend to the user.

Devarakonda, Ranjeet↗
DOE OSTI2020

Calibration of parallel bond parameters in bonded particle models via physics-informed adaptive moment optimisation

This study proposes an automated calibration procedure for bond parameters in bonded discrete element modelling. By exploring the underlying physical correlations between microscopic parameters of bonds and macroscopic strength parameters of the continuum to be modelled, the microscopic shear strength and tensile strength are identified as independent variables for calibration purpose. Then a physics-informed iterative scheme is proposed to automatically approximate the bond parameters by viewing the micro-macro relation as an implicitly defined mathematical mapping function. As a result of highly non-convex features of this implicit mapping, the adaptive moment estimation (Adam), which is especially suitable for problems with noisy gradients, is adopted as the basic iterative scheme, in conjunction with other numerical techniques to approximately evaluate the partial derivatives involved. The whole procedure offers a simple and effective framework for bond parameter calibration. Finally, a numerical example of SiC ceramic is provided for validation. By compared with some existing calibration methods, the proposed method shows significant advantages in terms of calibration efficiency and accuracy.

42 ENGINEERING↗
DOE OSTI2023

Commercial, industrial, and institutional discount rate estimation for efficiency standards analysis: Sector-level data 1998–2022

Underlying each of the U.S. Department of Energy’s (DOE’s) federal appliance and equipment energy conservation standards are a set of complex analyses of the projected costs and benefits of regulation. Any new or amended standard must be designed to achieve significant additional energy conservation, provided that it is technologically feasible and economically justified (42 U.S.C. 6295(o)(2)(A)). DOE determines economic justification based on whether the benefits exceed the burdens, considering a variety of factors, including the economic impact of the standard on consumers of the product and the savings in lifetime operating cost compared to any increase in price or maintenance expenses (42 U.S.C. 6295(o)(2)(B)). As part of this determination, DOE conducts a life-cycle cost (LCC) analysis, which models the combined impact of appliance first cost and operating cost changes on a representative commercial building sample to identify the fraction of customers achieving LCC savings or incurring net cost at the considered efficiency levels. Thus, the commercial discount rate value(s) used to calculate the present value of energy cost savings within the LCC model implicitly plays a role in estimating the economic impact of potential standard levels. This report provides an in-depth discussion of the commercial discount rate estimation process. It is an update to previous reports on estimating commercial discount rates from firm-level and sector-level financial data (e.g., Fujita, 2021, 2016). Major topics covered in this report include the following: -Discount rate estimation methods and rationale -Data sources used and data limitations -Discount rate distributions for use in standards analysis -Discount rate estimation methods and distributions specific to the small business subgroup analysis A version of this analysis was most recently released in 2022. Going forward, this report will be updated as data allow and analyses necessitate.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2022

Time-dependent THMC properties and microstructural evolution of damaged rocks in excavation damage zone

Modeling coupled thermo-hydro-mechanical-chemical (THMC) processes in host rocks near high-level nuclear waste (HLW) repositories at various time scales is an extremely challenging task. The current study integrates experimental, theoretical, and numerical methods in assessing the evolution of excavation damage zone (EDZ) over time and its implication on the long-term migration of hazardous species. Argillite and rock salt and are the focus of this study. The first part of the report presents a novel time-dependent directional microcrack damage theory for generic brittle rocks. It features detailed statistical description of the microcracks within a damaged solid, permitting a direct upscaling of microscale processes such as crack growth kinetics, crack closure/opening, sliding friction to explain the macroscopic creep, nonlinear elasticity, shear dilation, and loading-unloading hysteresis. This provides a basic platform for describing the anisotropic mechanical and transport properties of damaged rocks during excavation and subsequent THMC loadings. The model is validated and numerically implemented to Finite Element (FE) package ABAQUS through the user-defined material (UMAT) interface and have demonstrated great potential in resolving the time-dependent and anisotropic evolution of damage in the EDZ. The second part of the report presents a multi-scale experimental effort in characterizing the thermal, hydraulic, and mechanical properties of Mancos shale and Avery Island salt. For the Mancos shale, triaxial compression tests are performed at different confining pressures and temperatures to probe its thermomechanical properties relevant to HLW repositories. The obtained stress-strain data are interpreted using the proposed directional damage theory. Post-test specimens are subjected to gas permeability tests to reveal the correlation between permeability and the degree of microcracking. At microscale, temperature-controlled nanoindentation tests are performed and found a linear correlation between fracture toughness and elastic modulus from 25°C to 300°C. For Avery Island salt, we have designed and manufactured a novel relative-humidity controlled uniaxial creep device. Long-term creep tests at low stresses (< 5 MPa) are performed at different levels of relative humidity (RH). Besides confirming the much higher creep rates as one would expect through extrapolating the high-stress creep data, the results reveal that the steady-state creep rate of rock salt is strongly dependent on the ambient RH, an aspect that is often neglected in the literature. Both behaviors can be attributed to the pressure-solution creep mechanism which dominates at low stress and high RH levels. The third part of the report explores a set of numerical strategies in modeling the THMC behavior of porous geomaterials. A fully implicit, monolithic FE solution that can flexibly interface with different material models and coupling mechanisms for THMC problems is developed and verified through the ABAQUS user-defined element (UEL) interface. The scheme is used to study the THM response of a hypothetical HLW storage site with reference to an existing in-situ heater test. Strategies for integrating the UEL and the microcrack UMAT are suggested. Another numerical endeavor of this study is to implement a higher-order asymptotic homogenization method to account for the heterogeneous porous structures. The same method is then extended to perform microstructure-informed thermo-mechanical modeling of generalized continua. The above outcomes of this project provide a strong thrust towards enhancing the fundamental understanding and modeling capability of the long-term evolution of host rocks in EDZ, thus helping achieve the design goal of 1-million-year isolation of high-level nuclear wastes.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2025

Distributed Stochastic Optimization of a Neural Representation Network for Time-Space Tomography Reconstruction

4D time-space reconstruction of dynamic events or deforming objects using X-ray computed tomography (CT) is an important inverse problem in non-destructive evaluation. Conventional back-projection based reconstruction methods assume that the object remains static for the duration of several tens or hundreds of X-ray projection measurement images (reconstruction of consecutive limited-angle CT scans). However, this is an unrealistic assumption for many in-situ experiments that causes spurious artifacts and inaccurate morphological reconstructions of the object. To solve this problem, we propose to perform a 4D time-space reconstruction using a distributed implicit neural representation (DINR) network that is trained using a novel distributed stochastic training algorithm. Our DINR network learns to reconstruct the object at its output by iterative optimization of its network parameters such that the measured projection images best match the output of the CT forward measurement model. Here, we use a forward measurement model that is a function of the DINR outputs at a sparsely sampled set of continuous valued 4D object coordinates. Unlike previous neural representation architectures that forward and back propagate through dense voxel grids that sample the object's entire time-space coordinates, we only propagate through the DINR at a small subset of object coordinates in each iteration resulting in an order-of-magnitude reduction in memory and compute for training. DINR leverages distributed computation across several compute nodes and GPUs to produce high-fidelity 4D time-space reconstructions. We use both simulated parallel-beam and experimental cone-beam X-ray CT datasets to demonstrate the superior performance of our approach.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. Here we use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.

97 MATHEMATICS AND COMPUTING↗