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DOE OSTI2023

Subsetted model output for tracer-related variables from a LASSO-CACTI WRF simulation

Inert-tracer-related model output from a Weather Research and Forecasting (WRF) simulation from the Large-Eddy Simulation (LES) Atmospheric Radiation Measurement (ARM) Symbiotic Simulation and Observation (LASSO) deep-convection scenario for the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign. The LASSO-CACTI simulations span grid spacings from 7.5 km to 100 m for convection near the Sierras de Córdoba mountain range, roughly centered on the ARM Mobile Facility. More information can be found at https://www.arm.gov/capabilities/modeling/lasso. This version of the output is a collection of variables subsetted from the raw WRF output to reduce the file overhead for users not needing the full raw dataset. The subset files are in netCDF format with the height-based variables on the raw, model eta levels.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRSDate not supplied

An Observation-Driven Framework for Modeling Post-Fire Hydrologic Response: Evaluation for Two Central California Case Studies

In a warming climate, wildfires are becoming increasingly common, especially in semi-arid environments. Wildfires can disrupt forest ecosystems and induce changes to the land surface. Collectively, these impacts can alter the hydrologic response of a catchment following a fire, resulting in increased potential for surface runoff, reduced evapotranspiration, and, ultimately, a higher risk for flash flooding and mass wasting. The timescale for post-fire recovery of hydrological processes to return to pre-fire conditions is not well established due to the lack of ground measurements. Accurate characterization of fire impacts on hydrologic response is also challenging to simulate, given the complex interplay of various processes. Here, we present a generalized framework to quantify wildfire impacts on runoff generation. We consider the vegetation and soil disturbances as the primary contributors to post-fire floods. Using an ensemble modeling structure to account for parameter uncertainty and consider model sensitivity, remotely sensed leaf area index (LAI) is assimilated into a land surface model (LSM) to simulate vegetation disturbance, and the maximum land surface saturation LSM parameter is decreased to parameterize the soil disturbance following observed fires. We evaluated the impacts of fire-induced changes to LAI and soil saturation on hydrologic states like runoff and evapotranspiration. Analysis of two events in central California shows that 1) the LAI assimilation has a greater impact on water balance; and 2) the soil scheme captures a range of outcomes, with substantial impacts on surface runoff for some ensemble members, which could affect flood potential.

Timothy M Lahmers↗

DOE OSTI2026

Storylines for the 1997 New Year’s Flood: The role of watershed antecedent conditions and future warming in shaping discharge in the Truckee River watershed

The 1997 New Year’s flood was among the most devastating floods in the Truckee River watershed located in western Nevada. This event resulted from complex interactions of flood drivers, such as extreme precipitation, wet antecedent watershed conditions, warm temperatures and rapid snowmelt. We leveraged simulated forcings from the regionally refined mesh capabilities of the Energy Exascale Earth System Model (RRM-E3SM) and a process-based hydrological model to recreate the 1997 New Year’s flood for the Truckee River watershed across four climate warming levels ranging from the current temperatures to + 4° C. For each scenario, we conducted ensemble simulations with the same forcing but with 100 different seasonal watershed antecedent conditions, which were randomly sampled from long-term hydrological simulations. The results show that the 1997 New Year’s flood can be reproduced or exceeded consistently only when the antecedent watershed conditions are wet, specifically when streamflows are above the 75th percentile of the climatological value. There is negligible change in ensemble mean peakflows for Truckee River near Reno; however, there are increases of 18% and 14% under the warming levels of + 3° C and + 4° C, respectively. The increases in peakflows under future climate warming are attributed to wetter antecedent watershed conditions and enhanced snowmelt. Furthermore, the largest increases in peakflows occur at small, high-elevation headwater basins along the Sierra Nevada crest. This study highlights that changes in extreme flood events will result from the complex interplay of multiple flood drivers. It also demonstrates the potential of storyline approaches to analyze future realizations of these extreme events under different climate scenarios.

Climate change impact study↗

DOE OSTI2023

Spatially-resolved dynamic sampling of different phasic magnetic resonances of nanoparticle ensembles in a magnetotactic bacterium Magnetospirillum magnetotacticum

Abstract Nanoscaled magnetic particle ensembles are promising building blocks for realizing magnon based binary logic. Element-specific real-space monitoring of magnetic resonance modes with sampling rates in the GHz regime is imperative for the experimental verification of future complex magnonic devices. Here we present the observation of different phasic magnetic resonance modes using the element-specific technique of time-resolved scanning transmission x-ray microscopy within a chain of dipolarly coupled Fe 3 O 4 nanoparticles (40–50 nm particle size) inside a single cell of a magnetotactic bacterium Magnetospirillum magnetotacticum . The particles are probed with 25 nm resolution at the Fe L 3 x-ray absorption edge in response to a microwave excitation of 4.07 GHz. A plethora of resonance modes is observed within multiple particle segments oscillating in- and out-of-phase, well resembled by micromagnetic simulations.

time-resolved scanning transmission x-ray microsco↗

NASA NTRS2002

A Bayesian Approach to Estimating Coupling Between Neural Components: Evaluation of the Multiple Component, Event-Related Potential (mcERP) Algorithm

Accurate measurement of single-trial responses is key to a definitive use of complex electromagnetic and hemodynamic measurements in the investigation of brain dynamics. We developed the multiple component, Event-Related Potential (mcERP) approach to single-trial response estimation. To improve our resolution of dynamic interactions between neuronal ensembles located in different layers within a cortical region and/or in different cortical regions. The mcERP model assets that multiple components defined as stereotypic waveforms comprise the stimulus-evoked response and that these components may vary in amplitude and latency from trial to trial. Maximum a posteriori (MAP) solutions for the model are obtained by iterating a set of equations derived from the posterior probability. Our first goal was to use the ANTWERP algorithm to analyze interactions (specifically latency and amplitude correlation) between responses in different layers within a cortical region. Thus, we evaluated the model by applying the algorithm to synthetic data containing two correlated local components and one independent far-field component. Three cases were considered: the local components were correlated by an interaction in their single-trial amplitudes, by an interaction in their single-trial latencies, or by an interaction in both amplitude and latency. We then analyzed the accuracy with which the algorithm estimated the component waveshapes and the single-trial parameters as a function of the linearity of each of these relationships. Extensions of these analyses to real data are discussed as well as ongoing work to incorporate more detailed prior information.

Shah, Ankoor S.↗

DOE OSTI2022

Emerging Electrochemical Techniques for Probing Site Behavior in Single-Atom Electrocatalysts

Single-atom catalysts (SACs) have aroused tremendous interest over the past decade, particularly in the community of energy and environment-related electrocatalysis. A rapidly growing number of recent publications have recognized it as a promising candidate with maximum atomic utilization, distinct activity, and selectivity in comparison to bulk catalysts and nanocatalysts. However, the complexity of localized coordination environments and the dispersion of isolated sites lead to significant difficulties when it comes to gaining insight into the intrinsic behavior of electrocatalytic reactions. Furthermore, the low metal loadings of most SACs make conventional ensemble measurements less likely to be accurate on the subnanoscale. Thus, it remains challenging to probe the activity and properties of individual atomic sites by available commercial instruments and analytical methods. In spite of this, continuing efforts have lately focused on the development of advanced measurement methodologies, which are very useful to the fundamental understanding of SACs. There have recently been a number of in situ/operando techniques applied to SACs, such as electron microscopy, spectroscopy, and other analysis methods, which support relevant functions to identify the active sites and reaction intermediates and to investigate the dynamic behavior of localized structures of the catalytic sites. This Account aims to present recent electrochemical probing techniques which can be used to identify single-atomic catalytic sites within solid supports. First, we describe the basic principles of molecular probe methods for the study and analysis of electrocatalytic site behavior. In particular, the in situ probing technique enabled by surface interrogation scanning electrochemical microscopy (SISECM) can measure the active site density and kinetic rate with high resolution. An alternative electrochemical probing technique is further demonstrated on the basis of single-entity electrochemistry, which allows the unique electrochemical imaging of the size and catalytic rate of single atoms, molecules, and clusters. Next, the merits and limitations of different electrochemical techniques are then discussed, along with perspectives for future prospects. Apart from this, we further showcase the powerful capability of emerging electrochemical probing techniques for determining significant effects and properties of SACs for various electrocatalytic reactions, including oxygen reduction and evolution, hydrogen evolution, and nitrate reduction. Overall, electrochemical techniques with atomic resolution have greatly increased opportunities for observing, measuring, and understanding the surface and interface chemistry during energy conversion. In the future, it is anticipated that the development of electrochemical probing techniques will be advanced with innovative perspectives on the behavior and features of SACs. We hope that this Account can contribute in several ways to promoting the fundamental knowledge and technical progress of emerging electrochemical measurements for studying SACs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Comparison of Equilibrium and Nonequilibrium Approaches for Relative Binding Free Energy Predictions

Alchemical relative binding free energy calculations have recently found important applications in drug optimization. A series of congeneric compounds are generated from a preidentified lead compound, and their relative binding affinities to a protein are assessed in order to optimize candidate drugs. While methods based on equilibrium thermodynamics have been extensively studied, an approach based on nonequilibrium methods has recently been reported together with claims of its superiority. However, these claims pay insufficient attention to the basis and reliability of both methods. Here we report a comparative study of the two approaches across a large data set, comprising more than 500 ligand transformations spanning in excess of 300 ligands binding to a set of 14 diverse protein targets. Ensemble methods are essential to quantify the uncertainty in these calculations, not only for the reasons already established in the equilibrium approach but also to ensure that the nonequilibrium calculations reside within their domain of validity. If and only if ensemble methods are applied, we find that the nonequilibrium method can achieve accuracy and precision comparable to those of the equilibrium approach. Compared to the equilibrium method, the nonequilibrium approach can reduce computational costs but introduces higher computational complexity and longer wall clock times. There are, however, cases where the standard length of a nonequilibrium transition is not sufficient, necessitating a complete rerun of the entire set of transitions. This significantly increases the computational cost and proves to be highly inconvenient during large-scale applications. Our findings provide a key set of recommendations that should be adopted for the reliable implementation of nonequilibrium approaches to relative binding free energy calculations in ligand-protein systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Probing supersymmetric black holes with surface defects

It has long been conjectured that the large N deconfinement phase transition of $\mathcal{N}$ = 4 SU(N) super-Yang-Mills corresponds via AdS/CFT to the Hawking-Page transition in which black holes dominate the thermal ensemble, and quantitative evidence of this has come through the recent matching of the superconformal index of $\frac{1}{16}$-BPS states to the supersymmetric black hole entropy. Here we introduce the half-BPS Gukov-Witten surface defect as a probe of the superconformal index, which also serves as an order parameter for the deconfinement transition. This can be studied directly in field theory as a modification of the usual unitary matrix model or in the dual description as a D3-brane probe in the background of a (complex) supersymmetric black hole. Using a saddle point approximation, we determine our defect index in the large N limit as a simple function of the chemical potentials and show independently that it is reproduced by the renormalized action of the brane in the black hole background. Along the way, we also comment on the Cardy limit and the thermodynamics of the D3-brane in the generalized ensemble. The defect index sharply distinguishes between the confining and the deconfining phases of the gauge theory and thus is a supersymmetric non-perturbative order parameter for these large N phase transitions which deserves further investigation. Finally, our work provides an example where the properties of a black hole coupled to an external system can be analyzed precisely.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2026

Statistics of base polytopes in F-theory

We propose a new statistical ensemble of toric bases for elliptic Calabi-Yaus used in F-theory models, by focusing on only the convex hull of the base, i.e., the base polytope. This physically motivated coarse-graining greatly simplifies the combinatorial complexity of the part of the 4d F-theory landscape with toric bases. We develop a Monte Carlo approach that randomly samples the base polytopes within fixed boxes, with proper statistical weights. We first apply the algorithm to the set of 2d base polytopes, generating an enlarged set of toric 2d bases that include certain types of codimension-two (4,6) points, and we validate our approach against exact numbers. We then explore the set of 3d base polytopes which fit in a set of “maximal” 3d boxes, and estimate the total number of inequivalent 3d base polytopes to be 10 85 –10 90 . We provide statistical data such as the distribution of non-Higgsable gauge groups on these bases. Amusingly, a similar method can also be applied to generate reflexive polytopes in various dimensions. In both the reflexive and base polytope cases, the number of relevant polytopes obeys a Gaussian distribution as a function of the number of vertices, which can be understood in terms of other results on random polytopes in the math literature.

Differential and algebraic geometry↗

NASA NTRSDate not supplied

A Novel Machine Learning Method for Surface PM2.5 Estimations from Geostationary Satellites

Particulate matter (PM) with a diameter of less or equal to 2.5 μm, known as PM , affects human health as it penetrates the respiratory system. The Environmental Protection Agency (EPA) measures the atmospheric concentration of PM using air quality monitors stationed throughout the Continental United States (CONUS). Such measurements are points on a spatial domain and therefore, might not be representative of the air quality at nearby areas considering that the composition of the atmosphere is highly variable from place to place. Satellite based AOD permits a spatially uniform means of estimating PM and new geostationary satellites provide high temporal and spatial resolution estimation of AOD. However, the concentration of PM is non-linearly dependent on other atmospheric parameters that include relative humidity, temperature, and height of the planetary boundary layer. This information may be estimated at similar spatial and temporal resolutions as AOD from numerical modeling such as from the National Oceanic and Atmospheric Administration’s (NOAA) High Resolution Rapid Refresh (HRRR) model which resolves near real-time atmospheric conditions over the CONUS. The estimation of PM concentration is a multi-parametric problem that considers the effect of temporal dependencies among the different parameters. Deep learning approaches are appropriate for such complex estimation problems as they intrinsically capture relations among multiple non-linear parameters. This study compares deep-learning methods to traditional regression analysis to demonstrate the capabilities of these methods in predicting PM2.5 concentrations. Additionally, a novel ensemble learning approach is employed to identify scientific processes that could further improve the estimation of PM concentration. Utilizing Long Short-Term Memory (LSTM) neural networks, which are suitable for multivariate time series estimation problems as they are capable of learning long-term dependencies, individual models are created for each EPA station and trained on the aforementioned dataset collocated over each station. Individual station models are merged if the model's performance is improved by reducing the root mean squared error (RMSE) metric. This ensemble training method ultimately reduces the RMSE value. Evaluation of these results provide insights into physical processes and related observable parameters that may contribute to PM concentrations. Identified parameters evaluated to be statistically different between the merged and unmerged models are expected to improve overall performance. These new parameters are then utilized for reevaluation of the deep learning methods with an extreme gradient boosting model with an RMSE of 5.5 providing the best results.

George Priftis↗

DOE OSTI2024

Active learning of ternary alloy structures and energies

Abstract Machine learning models with uncertainty quantification have recently emerged as attractive tools to accelerate the navigation of catalyst design spaces in a data-efficient manner. Here, we combine active learning with a dropout graph convolutional network (dGCN) as a surrogate model to explore the complex materials space of high-entropy alloys (HEAs). We train the dGCN on the formation energies of disordered binary alloy structures in the Pd-Pt-Sn ternary alloy system and improve predictions on ternary structures by performing reduced optimization of the formation free energy, the target property that determines HEA stability, over ensembles of ternary structures constructed based on two coordinate systems: (a) a physics-informed ternary composition space, and (b) data-driven coordinates discovered by the Diffusion Maps manifold learning scheme. Both reduced optimization techniques improve predictions of the formation free energy in the ternary alloy space with a significantly reduced number of DFT calculations compared to a high-fidelity model. The physics-based scheme converges to the target property in a manner akin to a depth-first strategy, whereas the data-driven scheme appears more akin to a breadth-first approach. Both sampling schemes, coupled with our acquisition function, successfully exploit a database of DFT-calculated binary alloy structures and energies, augmented with a relatively small number of ternary alloy calculations, to identify stable ternary HEA compositions and structures. This generalized framework can be extended to incorporate more complex bulk and surface structural motifs, and the results demonstrate that significant dimensionality reduction is possible in thermodynamic sampling problems when suitable active learning schemes are employed.

Chemistry↗

DOE OSTI2022

Dilute Alloys Based on Au, Ag, or Cu for Efficient Catalysis: From Synthesis to Active Sites

The development of new catalyst materials for energy-efficient chemical synthesis is critical as over 80% of industrial processes rely on catalysts, with many of the most energy-intensive processes specifically using heterogeneous catalysis. Catalytic performance is a complex interplay of phenomena involving temperature, pressure, gas composition, surface composition and structure over multiple length and time scales. In response to this complexity, the integrated approach to heterogeneous dilute-alloy catalysis reviewed here brings together materials synthesis, mechanistic surface chemistry, reaction kinetics, in-situ and operando characterization, and theoretical calculations in a coordinated effort to develop design principles to predict and improve catalytic selectivity. Dilute alloy catalysts—in which isolated atoms or small ensembles of the minority metal on the host metal lead to enhanced reactivity while retaining selectivity—are particularly promising as selective catalysts. Several dilute alloy materials using Au, Ag and Cu as the majority host element, including more recently introduced support-free nanoporous metals and oxide-supported nanoparticle "raspberry colloid templated (RCT)" materials, are reviewed for selective oxidation and hydrogenation reactions. Progress in understanding how such dilute alloy catalysts can be used to enhance selectivity of key synthetic reactions is reviewed, including quantitative scaling from model studies to catalytic conditions. The dynamic evolution of catalyst structure and composition studied in surface science and catalytic conditions and their relationship to catalytic function are also discussed, followed by advanced characterization and theoretical modeling that have been developed to determine the distribution of minority metal atoms at or near the surface. Furthermore, the integrated approach demonstrates the success of bridging the divide between fundamental knowledge and design of catalytic processes in complex catalytic systems, which can accelerate the development of new and efficient catalytic processes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

An overview of the Western United States Dynamically Downscaled Dataset (WUS-D3)

Abstract. Predicting future climate change over a region of complex terrain, such as the western United States (US), remains challenging due to the low resolution of global climate models (GCMs). Yet the climate extremes of recent years in this region, such as floods, wildfires, and drought, are likely to intensify further as climate warms, underscoring the need for high-quality and high-resolution predictions. Here, we present an ensemble of dynamically downscaled simulations over the western US from 1980–2100 at 9 km grid spacing, driven by 16 latest-generation GCMs. This dataset is titled the Western US Dynamically Downscaled Dataset (WUS-D3). We describe the challenges of producing WUS-D3, including GCM selection and technical issues, and we evaluate the simulations' realism by comparing historical results to temperature and precipitation observations. The future downscaled climate change signals are shaped in physically credible ways by the regional model's more realistic coastlines and topography. (1) The mean warming signals are heavily influenced by more realistic snowpack. (2) Mean precipitation changes are often consistent with wetting on the windward side of mountain complexes, as warmer, moister air masses are uplifted orographically during precipitation events. (3) There are large fractional precipitation increases on the lee side of mountain complexes, leading to potentially significant changes in water resources and ecology in these arid landscapes. (4) Increases in precipitation extremes are generally larger than in the GCMs, driven by locally intensified atmospheric updrafts tied to sharper, more realistic gradients in topography. (5) Changes in temperature extremes are different from what is expected by a shift in mean temperature and are shaped by local atmospheric dynamics and land surface feedbacks. Because of its high resolution, comprehensiveness, and representation of relevant physical processes, this dataset presents a unique opportunity to evaluate societally relevant future changes in western US climate.

Rahimi, Stefan (ORCID:0000000331884462)↗

NASA NTRS2018

Exoplanet Biosignatures: Future Directions

We introduce a Bayesian method for guiding future directions for detection of life on exoplanets. We describe empirical and theoretical work necessary to place constraints on the relevant likelihoods, including those emerging from better understanding stellar environment, planetary climate and geophysics, geochemical cycling, the universalities of physics and chemistry, the contingencies of evolutionary history, the properties of life as an emergent complex system, and the mechanisms driving the emergence of life. We provide examples for how the Bayesian formalism could guide future search strategies, including determining observations to prioritize or deciding between targeted searches or larger lower resolution surveys to generate ensemble statistics and address how a Bayesian methodology could constrain the prior probability of life with or without a positive detection.

Bayesian analysis↗

NASA NTRSDate not supplied

Improving the Representation of Land Surface Processes using the Data Assimilation Research Testbed (DART)

The land surface is a critical part of the earth system as processes related to water, carbon, energy and nitrogen cycling have important implications for climate forcing, air quality, water availability and seasonal atmospheric forecasting. Despite advances in land surface modeling, land surface model performance is often limited because of errors related to initial and boundary conditions, model structure, and parameters. Data assimilation (DA) techniques combined with an expanding network of earth system observations present an opportunity to reduce these errors and improve simulations. Here we apply an Ensemble Kalman Filter DA system as part of the Data Assimilation Research Testbed to a variety of land surface simulations. First, we describe the use of remotely sensed biomass observations to provide improved simulations of plant phenology, carbon and water cycling for regions highly sensitive to climate change (Western US, China, and Arctic). We discuss approaches to account for systemic biases between models and observations, including the use of spatially-varying adaptive ensemble inflation as an alternative approach to re-scaling soil moisture observations. Finally, we discuss a strategy to incorporate complementary observations (snow water equivalent, solar-induced fluorescence) to better constrain the representation of carbon and water cycling across complex terrain.

DART↗

DOE OSTI2021

Disentangling Rotational Dynamics and Ordering Transitions in a System of Self-Organizing Protein Nanorods via Rotationally Invariant Latent Representations

The dynamics of complex ordering systems with active rotational degrees of freedom exemplified by protein self-assembly is explored using a machine learning workflow that combines deep learning-based semantic segmentation and rotationally invariant variational autoencoder-based analysis of orientation and shape evolution. The latter allows for disentanglement of the particle orientation from other degrees of freedom and compensates for lateral shifts. The disentangled representations in the latent space encode the rich spectrum of local transitions that can now be visualized and explored via continuous variables. The time dependence of ensemble averages allows insight into the time dynamics of the system and, in particular, illustrates the presence of the potential ordering transition. Finally, analysis of the latent variables along the single-particle trajectory allows tracing these parameters on a single-particle level. The proposed approach is expected to be universally applicable for the description of the imaging data in optical, scanning probe, and electron microscopy seeking to understand the dynamics of complex systems where rotations are a significant part of the process.

representation learning↗

DOE OSTI2021

Heterogeneity and Hysteresis in the Polymer Collapse of Single Core–Shell Stimuli-Responsive Plasmonic Nanohybrids

Broad application of polymeric stimuli-responsive smart nanohybrids requires understanding the mechanisms governing active control. Ensemble techniques have identified inhomogeneous polymer collapse in microgels that potentially arise from heterogeneous interchain interactions and differences in core size. A single-particle examination would establish the influence of core size and internal polymer network heterogeneity on local interactions that contribute to the observed inhomogeneous polymer collapse dynamics of nanohybrids. Using single-particle dark-field spectroscopy, we investigated the complex polymer collapse profiles of core–shell plasmonic nanohybrids comprising thermoresponsive poly(N-isopropylacrylamide) (pNIPAM)-encapsulated gold nanorods (AuNRs). We report that the polymer collapse behavior was independent of the core size. For thinner polymer shells, we observed hysteresis in the collapse of AuNR@pNIPAMs, likely related to local pNIPAM aggregation due to interchain hydrogen bonding. For thicker polymer shells, we observed a broad polymer collapse distribution that we attributed to a two-step phase transition that arises from a polymer network density gradient. Furthermore, our single-particle approach relates the internal heterogeneity of the polymer network of nanohybrids to the mechanisms underlying heterogeneous phase transitions that traditional, ensemble-averaged approaches are unable to discern.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Machine learning visualization tool for exploring parameterized hydrodynamics

We are interested in the computational study of shock hydrodynamics, i.e. problems involving compressible solids, liquids, and gases that undergo large deformation. These problems are dynamic and nonlinear and can exhibit complex instabilities. Due to advances in high performance computing it is possible to parameterize a hydrodynamic problem and perform a computational study yielding $\mathscr{O} (TB)$ of simulation state data. We present an interactive machine learning tool that can be used to compress, browse, and interpolate these large simulation datasets. This tool allows computational scientists and researchers to quickly visualize 'what-if' situations, perform sensitivity analyses, and optimize complex hydrodynamic experiments.

97 MATHEMATICS AND COMPUTING↗