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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 199 records · Page 11
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Fast methods for multisite charge transfer processes. I. Constrained, state averaged CASSCF(1,n) and CASSCF(2n − 1,n) simulations

We design a dynamically weighted state-averaged constrained complete active space self-consistent field (DW-SA-cCASSCF) algorithm to treat electrons or holes moving between n molecular fragments (where n can be larger than 2). Within such a so-called eDSCn/hDSCn approach, we consider configurations that are mutually single excitations of each other, and we apply a generalized set of constraints to tailor the method for studying charge transfer problems. The constrained optimization problem is efficiently solved using a DIIS-SQP algorithm, thus maintaining computational efficiency. We demonstrate the method for a finite Su–Schrieffer–Heeger chain, successfully reproducing the expected exponential decay of diabatic couplings with distance. When combined with a gradient, the current extension immediately enables efficient nonadiabatic dynamics simulations of complex multi-state charge transfer processes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Recovered supernova Ia rate from simulated LSST images

Aims.TheVera C. RubinObservatory’s Legacy Survey of Space and Time (LSST) will revolutionize time-domain astronomy by detecting millions of different transients. In particular, it is expected to increase the number of known type Ia supernovae (SN Ia) by a factor of 100 compared to existing samples up to redshift ∼1.2. Such a high number of events will dramatically reduce statistical uncertainties in the analysis of the properties and rates of these objects. However, the impact of all other sources of uncertainty on the measurement of the SN Ia rate must still be evaluated. The comprehension and reduction of such uncertainties will be fundamental both for cosmology and stellar evolution studies, as measuring the SN Ia rate can put constraints on the evolutionary scenarios of different SN Ia progenitors. Methods.We used simulated data from the Dark Energy Science Collaboration (DESC) Data Challenge 2 (DC2) and LSST Data Preview 0 to measure the SN Ia rate on a 15 deg 2 region of the “wide-fast-deep” area. We selected a sample of SN candidates detected in difference images, associated them to the host galaxy with a specially developed algorithm, and retrieved their photometric redshifts. We then tested different light-curve classification methods, with and without redshift priors (albeit ignoring contamination from other transients, as DC2 contains only SN Ia). We discuss how the distribution in redshift measured for the SN candidates changes according to the selected host galaxy and redshift estimate. Results.We measured the SN Ia rate, analyzing the impact of uncertainties due to photometric redshift, host-galaxy association and classification on the distribution in redshift of the starting sample. We find that we are missing 17% of the SN Ia, on average, with respect to the simulated sample. As 10% of the mismatch is due to the uncertainty on the photometric redshift alone (which also affects classification when used as a prior), we conclude that this parameter is the major source of uncertainty. We discuss possible reduction of the errors in the measurement of the SN Ia rate, including synergies with other surveys, which may help us to use the rate to discriminate different progenitor models.

Astronomy & Astrophysics↗
DOE OSTI2024

Characteristics of Precipitation and Mesoscale Convective Systems Over the Peruvian Central Andes in Multi 5‐Year Convection‐Permitting Simulations

Abstract Using the Weather Research and Forecasting model with two planetary boundary layer schemes, ACM2 and MYNN, convection‐permitting model (CPM) regional climate simulations were conducted for a 6‐year period, including a one‐year spin‐up period, at a 15‐km grid spacing covering entire South America and a nested convection‐permitting 3‐km grid spacing covering the Peruvian central Andes region. These two CPM simulations along with a 4‐km simulation covering South America produced by National Center for Atmospheric Research (NCAR), three gridded precipitation products, and rain gauge data in Peru and Brazil, are used to document the characteristics of precipitation and mesoscale convective systems (MCSs) in the Peruvian central Andes region. Results show that all km‐scale simulations generally capture the spatiotemporal patterns of precipitation and MCSs at both seasonal and diurnal scales, although biases exist in aspects such as precipitation intensity and MCS frequency, size, propagation speed, and associated precipitation intensity. The 3‐km simulation using MYNN scheme generally outperforms the other simulations in capturing seasonal and diurnal precipitation over the mountain, while both it and the 4‐km simulation demonstrate superior performance in the western Amazon Basin, based on the comparison to the gridded precipitation products and gauge data. Dynamic factors, primarily low‐level jet and terrain‐induced uplift, are the key drivers for precipitation and MCS genesis along the east slope of the Andes, while thermodynamic factors control the precipitation and MCS activity in the western Amazon Basin and over elevated mountainous regions. The study suggests model improvements and better model configurations for future regional climate projections.

Meteorology & Atmospheric Sciences↗
DOE OSTI2025

Benchmark microgravity experiments and computations for 3D dendritic-array stability in directional solidification

In this study, we present a comprehensive quantitative analysis of stability bands for dendritic arrays during directional solidification of a transparent succinonitrile-0.46 wt % camphor alloy, spanning a broad range of pulling velocities. Taking advantage of the microgravity environment aboard the International Space Station where most convection effects are suppressed, we obtain unique measurements that quantify the stable primary spacing range of spatially extended three-dimensional dendritic array structures under purely diffusive growth conditions. Through carefully designed velocity jump experiments and detailed examination of sub-grain boundary dynamics, we characterize key instabilities, including elimination and tertiary branching, shedding new light on the mechanisms governing dynamic dendritic spacing selection in extended 3D arrays. Phase field simulations are performed to characterize the stability limits of dendritic array structures for quantitative comparison with the flight experiments. Although the simulations capture general trends, significant deviations are noted at the upper stability boundary, indicating the influence of additional, unexplored factors. These findings contribute to a deeper understanding of dendritic growth dynamics and offer valuable benchmark data that could aid in refining predictive models and improving control of dendritic microstructures in metallurgical applications.

36 MATERIALS SCIENCE↗
DOE OSTI2025

A subgrid model for electron-scale turbulent transport in global ion-scale gyrokinetic simulations of tokamak plasmas

A subgrid electron–temperature–gradient (ETG) model is demonstrated here which averages local electron-scale turbulence over intermediate scales in space and time to include in global ion–temperature–gradient (ITG) simulations. This approach results in ion-scale equations, which incorporate the electron heat transport from ETG turbulence and the effects of electron-scale turbulence on the ion scale. Flux-tube ETG Cyclone Base Case simulations are performed at different radial locations and a kinetic form of the flux is added to global ion-scale (IS) simulations as a source term. Analytic radial profiles of ETG heat flux are constructed and compared to flux-tube results at multiple radial locations. Different ratios of ITG to ETG heat flux levels are considered and the results of capturing ETG heat transport in global ion-scale simulations are discussed. The potential coupling of the ETG streamer potential and intermediate-scale zonal flows to the ion scale is further addressed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Measurement and calibration of non-linear shear terms in galaxy cluster fields

Introduction Galaxy cluster lensing is a powerful tool for measuring the mass of galaxy clusters, but accurate shear measurement and calibration are critical to obtaining reliable results. This study focuses on the measurement and calibration of weak lensing shears to improve mass estimates in cluster lensing. To deal with the problem, we first developed an image simulation pipeline, jedisim, which utilizes galaxy images extracted from the Hubble Space Telescope (HST) Ultra Deep Field (UDF) and the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS). Methods The simulations represent realistic galaxy distributions and morphologies as input sources. The foreground halo with a Navarro–Frenk–White (NFW) profile is constructed such that the lensing signals of background galaxies can be measured by the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST) Science Pipelines. By comparing the measured reduced shear g meas and the true reduced shear g true , we observe non-linearity up to g ≲ 0.6 . We fit polynomials to the data with quadratic correction adequate to g ≲ 0.4 . Meanwhile, we conduct mass estimates using the pzmassfitter code on four different clusters. Results The mass estimate results are significantly improved after applying the shear calibration derived from the present work—from 4.954 ± 0.504 × 1 0 14 M ⊙ to 10.507 ± 0.498 × 1 0 14 M ⊙ after calibration for a simulated cluster with the mass of 10 × 1 0 14 M ⊙ . In multiple cases of validation, the estimated results are all consistent with true cluster mass. Discussion This study yields the first relationship between reality and shape measurement of the LSST Science Pipelines and serves as the first step toward the overall goal of mass calibration in cluster lensing. By addressing the challenges in shear measurement and calibration, we aim to enhance the accuracy and reliability of mass estimates in galaxy cluster lensing studies.

Astronomy & Astrophysics↗
DOE OSTI2024

Uncertainty Visualization Challenges in Decision Systems with Ensemble Data & Surrogate Models: Preprint

Uncertainty visualization is a key component in translating important insights from ensemble data into actionable decision-making by visually conveying various aspects of uncertainty within a system. With the recent advent of fast surrogate models for computationally expensive simulations, users can interact with more aspects of data spaces than ever before. However, the integration of ensemble data with surrogate models in a decision-making tool brings up new challenges for uncertainty visualization, namely how to reconcile and communicate the new and different types of uncertainties brought in by surrogates and how to utilize these new data estimates in actionable ways. In this work, we examine these issues as they relate to high-dimensional data visualization, the integration of discrete datasets and the continuous representations of those datasets, and the unique difficulties associated with systems that allow users to iterate between input and output spaces. We assess the role of uncertainty visualization in facilitating intuitive and actionable interaction with ensemble data and surrogate models, and highlight key challenges in this new frontier of computational simulation.

ensemble visualization↗
DOE OSTI2024

Super-X and conventional divertor configurations in MAST-U ohmic L-mode; a comparison facilitated by interpretative modelling

Measurements are presented, alongside corresponding interpretative SOLPS-ITER simulations, of the first MAST-U experiments comparing ohmically heated L-mode fuelling scans in Conventional divertor (CD) and Super-X divertor (SXD) configurations. In experiment, at comparable outer mid-plane separatrix electron density, $n_{e,\textrm{sep,OMP}}$, the maximum lower outer target heat load was found to be a factor 16 $\,\pm\,7$ lower in SXD compared to CD. In simulation, a factor 26.8 reduction was found (slightly higher than the experimental range), suggesting an additional reduction in SXD compared to the factor 9.3 expected from geometric considerations alone. According to the simulations, this additional reduction in the SXD is due to a net radial transport of the energy remaining downstream of the $T_e = 5$ eV location. This energy is carried out of the critical (highest heat load) flux tube by deuterium atoms, demonstrating the importance of a longer legged divertor which provides space for this to occur. Importantly, in both simulation and experiment, the SXD has minimal impact on the upstream n e and T e profiles. Spectral inferences of detachment front movement in SXD compare well between simulation and experiment. In regions of high magnetic field gradient, the parallel movement of the front towards the X-point becomes less sensitive to increasing $n_{e,\textrm{sep,OMP}}$, in qualitative agreement with simplified models and previous predictive simulations. Additional aspects, regarding the target ion flux rollover, upstream separatrix temperature and drift effects, are also presented and discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Efficient six-dimensional phase space reconstructions from experimental measurements using generative machine learning

Next-generation accelerator concepts, which hinge on the precise shaping of beam distributions, demand equally precise diagnostic methods capable of reconstructing beam distributions within six-dimensional position-momentum spaces. However, the characterization of intricate features within six-dimensional beam distributions using current diagnostic techniques necessitates a substantial number of measurements, using many hours of valuable beam time. Novel phase space reconstruction techniques are needed to reduce the number of measurements required to reconstruct detailed, high-dimensional beam features in order to resolve complex beam phenomena and as a feedback in precision beam shaping applications. In this study, we present a novel approach to reconstructing detailed six-dimensional phase space distributions from experimental measurements using generative machine learning and differentiable beam dynamics simulations. We demonstrate that this approach can be used to resolve six-dimensional phase space distributions from scratch, using basic beam manipulations and as few as 20 two-dimensional measurements of the beam profile. We also demonstrate an application of the reconstruction method in an experimental setting at the Argonne Wakefield Accelerator, where it is able to reconstruct the beam distribution and accurately predict previously unseen measurements 75× faster than previous methods.

43 PARTICLE ACCELERATORS↗
DOE OSTI2026

Quantum simulation of QED in Coulomb gauge

A recent work considered quantum simulation of Quantum Electrodynamics on a lattice in the Coulomb gauge with gauge degrees of freedom represented in the occupation basis in momentum space. In this work, we consider the more efficient representation of the gauge degrees of freedom in field basis in position space and develop a quantum algorithm for real-time simulation. We show that the continuum Coulomb gauge Hamiltonian is equivalent to the temporal gauge Hamiltonian when acting on physical states consisting of fermion and transverse gauge fields. The Coulomb gauge Hamiltonian is discretized by using the Green’s function of the discrete Laplacian operator under the Dirichlet boundary conditions. Both the continuum Coulomb gauge Hamiltonian and the discretized one proposed here guarantee that the unphysical longitudinal gauge fields are decoupled and commute with the corresponding Hamiltonian. Thus there is no need to impose any constraint. The local gauge field basis and the canonically conjugate variable basis are swapped efficiently using the quantum Fourier transform. We prove that the qubit cost to represent physical states and the gate count for real-time simulation scale polynomially with the lattice size, energy, time, accuracy, and Hamiltonian parameters in lattice units. The gate cost here for implementing the time evolution of the gauge field is reduced at least by a factor on the order of 10 8 for modest lattice size and accuracy level compared with the previous work.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Spatial and Temporal Variability of Vertical Velocity under Shallow Cumulus

Vertical velocity distribution below cloud is one of the key determinants of cloud life cycle, but observations of this variable are extremely sparse in space. Doppler lidar retrievals and large-eddy simulations at the U.S. Department of Energy’s Atmospheric Radiation Measurement User Facility Southern Great Plains site are used to determine whether vertical velocity statistics from temporally dense profiles at a single location can be substituted for spatial vertical velocity statistics. We show that even a small number (five) of widely distributed [ O (1°) latitude/longitude spacing] lidars is sufficient sampling to reconstruct domainwide spatial vertical velocity variance, but not higher moments of the vertical velocity distribution. Spatial and temporal vertical velocity variances in the Doppler lidar observations are nearly interchangeable as long as the spatial variance is temporally averaged and the temporal variance is averaged across lidars. This is true even though the dominant spatial scales of vertical velocity variability are ≲ 3 km, more than an order of magnitude smaller than the spacing between the lidars. Further, in the limit where the temporal variance does not vary across a spatial domain (e.g., if the meteorological and surface forcing of the atmospheric turbulence is homogeneous across the domain) and the domain-mean vertical velocity is zero, the commonly available retrieval of temporal vertical velocity variance at one site is equivalent to the spatial variance over the domain. We use an updraft parcel model to show that substituting temporal for spatial vertical velocity statistics will have a relatively minor effect on cloud droplet number concentrations.

54 ENVIRONMENTAL SCIENCES↗
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↗
DOE OSTI2026

A Robotic High-Throughput Grid-Search Platform for Mapping Phase Behavior in Triblock Copolymer–Homopolymer Blends

We present a high-throughput experimental investigation of the phase behavior in triblock copolymers (PS-b-PB-b-PS and PS-b-PI-b-PS) and polystyrene (PS) homopolymer blends as a function of homopolymer molecular weight (MW) and blend ratio. Using a robotic thin-film processing platform (NOVA) integrated with Grazing Incidence Small-Angle X-ray Scattering (GISAXS) and Atomic Force Microscopy (AFM), we systematically mapped the order–disorder transition (ODT) boundaries and domain spacing evolution across a broad MW range (4.0–101.3 kDa) with varying homopolymer loadings (10% to 90%). The results reveal three distinct regimes: low-MW homopolymers, corresponding to the wet-brush regime produced only gradual domain swelling before disordering at high blend ratios (weight fraction); medium-MW homopolymers, corresponding to thedry-brush regime induced significant domain spacing increase up to 80% followed by earlier disordering, while high-MW homopolymers led to macrophase separation with minimal changes in domain spacing. Additionally, coarse-grained molecular dynamics simulations confirmed our experimental finding that in the low-MW region, the PS homopolymer uniformly distributed in the PS domain. These findings demonstrate that homopolymer molecular weight critically governs both the extent of domain swelling and the onset of disorder in triblock copolymer systems. This high-throughput platform enables the rapid mapping of composition–morphology relationships and can be integrated with AI/ML tools for designing next-generation nanostructured polymers.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Accelerating the design of lattice structures using machine learning

Lattices remain an attractive class of structures due to their design versatility; however, rapidly designing lattice structures with tailored or optimal mechanical properties remains a significant challenge. With each added design variable, the design space quickly becomes intractable. To address this challenge, research efforts have sought to combine computational approaches with machine learning (ML)-based approaches to reduce the computational cost of the design process and accelerate mechanical design. While these efforts have made substantial progress, significant challenges remain in (1) building and interpreting the ML-based surrogate models and (2) iteratively and efficiently curating training datasets for optimization tasks. Here, we address the first challenge by combining ML-based surrogate modeling and Shapley additive explanation (SHAP) analysis to interpret the impact of each design variable. We find that our ML-based surrogate models achieve excellent prediction capabilities (R 2 > 0.95) and SHAP values aid in uncovering design variables influencing performance. We address the second challenge by utilizing active learning-based methods, such as Bayesian optimization, to explore the design space and report a 5 × reduction in simulations relative to grid-based search. Collectively, these results underscore the value of building intelligent design systems that leverage ML-based methods for uncovering key design variables and accelerating design.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Fate of supernova progenitors in massive binary systems

ABSTRACT How massive stars end their lives depends on the core mass, core angular momentum, and hydrogen envelopes at death. However, these key physical facets of stellar evolution can be severely affected by binary interactions. In turn, the effectiveness of binary interactions itself varies greatly depending on the initial conditions of the binaries, making the situation much more complex. We investigate systematically how binary interactions influence core–collapse progenitors and their fates. Binary evolution simulations are performed to survey the parameter space of supernova progenitors in solar metallicity binary systems and to delineate major evolutionary paths. We first study fixed binary mass ratios ($q=M_2/M_1$ = 0.5, 0.7, and 0.9) to elucidate the impacts of initial mass and initial separation on the outcomes, treating separately Type Ibc supernova, Type II supernova, accretion-induced collapse (AIC), rapidly rotating supernova (Ibc-R), black hole formation, and long gamma ray burst (long GRB). We then conduct 12 binary population synthesis model calculations, varying the initial condition distributions and binary evolution parameters, to estimate various supernova fractions. We obtain a Milky Way supernova rate $R_{\rm SN} = (1.78$–$2.47) \times 10^{-2} \, {\rm yr}^{-1}$ which is consistent with observations. We find the rates of AIC, Ibc-R, and long GRB to be $\sim 1/100$ the rate of regular supernovae. Our estimated long GRB rates are higher than the observed long GRB rate and close to the low luminosity GRB rate, although care must be taken considering our models are computed with solar metallicity. Furthering binary modelling and improving the inputs one by one will enable more detailed studies of these and other transients associated with massive stars.

Astronomy & Astrophysics↗
DOE OSTI2024

Weak-Form Latent Space Dynamics Identification

This software showcases the enhanced capabilities of the Latent Space Dynamics Identification (LaSDI) algorithm through the application of the weak form, resulting in WLaSDI. WLaSDI first compresses the data, then projects it onto test functions, and subsequently learns the local latent space models. Notably, WLaSDI demonstrates significantly improved robustness to noise. Using weak-form equation learning techniques, WLaSDI achieves local latent space modeling. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form ensures robust and precise latent space recovery, enabling fast, robust, and accurate simulations. We demonstrate the efficacy of WLaSDI against LaSDI using several common benchmark examples, including viscid and inviscid Burgers', radial advection, and heat conduction. For instance, in 1D inviscid Burgers' simulations with up to 100% Gaussian white noise, WLaSDI maintains relative errors consistently below 6%, whereas LaSDI errors can exceed 10,000%. Similarly, in radial advection simulations, WLaSDI keeps relative errors below 16%, compared to potential errors of up to 10,000% with LaSDI. Additionally, WLaSDI achieves significant speedups, such as a 140X speedup in 1D Burgers' simulations compared to the corresponding full order model.

Choi, Youngsoo↗
DOE OSTI2026

Ion Trapping Studies and Mitigation Strategies for the EIC ERL-Based Strong Hadron Cooler

An Energy Recovery Linac based strong hadron cooler was previously considered for the Electron-Ion Collider. The required electron beam parameters for variable-energy strong hadron cooling place significant constraints on ion trapping and collective effects. This paper presents initial studies of these constraints through a combination of analytical modelling and numerical simulations of ion production, trapping behaviour, and mitigation strategies. A multi-bunch tracking framework based on ELEGANT with the ionEffects module is used to simulate machine operation over millisecond time scales, corresponding to more than 3 × 10^5 electron bunches. The simulations include modelling of ionisation processes together with transverse electron–ion dynamics, allowing the evolution and accumulation of ions to be investigated. Analytical expressions based on Gaussian beam distributions are used to estimate ion trapping conditions and benchmark the simulation results. A bi-periodic bunch spacing scheme is also investigated as a possible mitigation method by detuning the ion oscillation frequency. These studies provide an initial assessment of ion trapping in the strong hadron cooler and demonstrate possible approaches for reducing beam–ion effects.

Bi, R. [Lancaster University, Cockcroft Institute]↗
DOE OSTI2024

Multigrid Reduction in Time for Chaotic and Hyperbolic Problems (Final Report)

The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations (PDEs), which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures. The chosen methodology for solving these problems parallel-in-time is multigrid, because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. Multigrid is already commonly used in many DOE simulations for scalably and optimally solving space-only PDE problems. The areas of hyperbolic and chaotic problems are chosen because of their relevance to problems of programmatic interest to DOE. However, these problems are also well-known to be difficult for parallel-in-time methods, with the most common method, parareal, diverging in many cases. The current state of-the-art for parallel-in-time at LLNL is the multigrid reduction in time (MGRIT) XBraid package, which also struggles for such problems, while still showing some improvement over parareal. In summary, new methods are needed for an efficient parallel-in-time scheme for hyperbolic and chaotic problems, and this work shall research promising new multigrid methods in this area. In particular, this work shall continue researching the directions from the current collaboration with Dr. Falgout, which are laid out in the work Toward Parallel in Time for Chaotic Dynamical Systems and showed the first known results of a parallel-in-time speedup for a chaotic problem. This work outlines two key improvements to XBraid for chaotic problems, the so-called “theta” and “delta-correction” methods. Here, these two improvements will be further researched and improved (including with a new relaxation method inspired by on Least Squares Shadowing (LSS)) and explored for more complicated problems.

97 MATHEMATICS AND COMPUTING↗