Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Boundary identification”

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

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

At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Exploring the Relationship of Microstructure and Conductivity in Metal Halide Perovskites via Active Learning-Driven Automated Scanning Probe Microscopy

Electronic transport and hysteresis in metal halide perovskites (MHPs) are key to the applications in photovoltaics, light emitting devices, and light and chemical sensors. These phenomena are strongly affected by the materials microstructure including grain boundaries, ferroic domain walls, and secondary phase inclusions. Here, we demonstrate an active machine learning framework for “driving” an automated scanning probe microscope (SPM) to discover the microstructures responsible for specific aspects of transport behavior in MHPs. In our setup, the microscope can discover the microstructural elements that maximize the onset of conduction, hysteresis, or any other characteristic that can be derived from a set of current–voltage spectra. Furthermore, this approach opens new opportunities for exploring the origins of materials functionality in complex materials by SPM and can be integrated with other characterization techniques either before (prior knowledge) or after (identification of locations of interest for detail studies) functional probing.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

The Making of a Metric: Co-Producing Decision-Relevant Climate Science

Developing decision-relevant science for adaptation requires the identification of climatic parameters that are both actionable for practitioners as well as tractable for modelers. In many sectors, these decision-relevant climatic metrics and the approaches that enable their identification remain largely unknown. “Co-production” of science with scientists and decision-makers is one potential way to identify these metrics, but there is little research describing specific and successful co-production approaches. This paper examines the negotiations and outcomes from Project Hyperion, wherein scientists and water managers jointly developed decision-relevant climatic metrics for adaptive water management. We identify successful co-production strategies by analyzing the project’s numerous back-and-forth engagements and tracing the evolution of the science during these engagements. We found that effective mediation between scientists and managers needed dedicated “boundary spanners” with significant modeling expertise. Translating practitioners’ information needs into tractable climatic metrics required direct and indirect methods of eliciting knowledge. We identified four indirect methods that were particularly salient for extracting tacitly held knowledge and enabling shared learning: developing a hierarchical framework linking management issues with metrics, starting discussions from the planning challenges, collaboratively exploring the planning relevance of new scientific capabilities, and using analogies of other “good” metrics. The decision-relevant metrics we developed provide insights into advancing adaptation-relevant climate science in the water sector. The co-production strategies we identified can be used to design and implement productive scientist–decision-maker interactions. Overall, the approaches and metrics we developed can help climate science to expand in new and more use-inspired directions.

Planning↗
DOE OSTI2022

Identification of a non-axisymmetric mode in laboratory experiments searching for standard magnetorotational instability

Abstract The standard magnetorotational instability (SMRI) is a promising mechanism for turbulence and rapid accretion in astrophysical disks. It is a magnetohydrodynamic (MHD) instability that destabilizes otherwise hydrodynamically stable disk flow. Due to its microscopic nature at astronomical distances and stringent requirements in laboratory experiments, SMRI has remained unconfirmed since its proposal, despite its astrophysical importance. Here we report a nonaxisymmetric MHD instability in a modified Taylor-Couette experiment. To search for SMRI, a uniform magnetic field is imposed along the rotation axis of a swirling liquid-metal flow. The instability initially grows exponentially, becoming prominent only for sufficient flow shear and moderate magnetic field. These conditions for instability are qualitatively consistent with SMRI, but at magnetic Reynolds numbers below the predictions of linear analyses with periodic axial boundaries. Three-dimensional numerical simulations, however, reproduce the observed instability, indicating that it grows linearly from the primary axisymmetric flow modified by the applied magnetic field.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Code Verification of Multiple Physics-Fidelity Models in Hypersonic Aerodynamics

Hypersonic aerodynamics models exist across a range of physics fidelities with associated computational expenses. These models may be run independently or in a multifidelity framework that leverages their complementary strengths of speed for lower-fidelity and accuracy for higher-fidelity models. This work presents applied code verification of two lower-fidelity models contained within the Sandia hypersonic aerodynamics code. Each model has a different form that requires individualized verification approaches, including comparison to analytical solutions as well as manufactured solutions with order-of-accuracy testing. In conclusion, results of this effort include the identification and resolution of code errors and shortcomings, as well as the demonstration of code correctness and consistency for both models.

Aerodynamics↗
DOE OSTI2023

Correlative chemical and elemental nano-imaging of morphology and disorder at the nacre-prismatic region interface in Pinctada margaritifera

Abstract Understanding biomineralization relies on imaging chemically heterogeneous organic–inorganic interfaces across a hierarchy of spatial scales. Further, organic minority phases are often responsible for emergent inorganic structures from the atomic arrangement of different polymorphs, to nano- and micrometer crystal dimensions, up to meter size mollusk shells. The desired simultaneous chemical and elemental imaging to identify sparse organic moieties across a large field-of-view with nanometer spatial resolution has not yet been achieved. Here, we combine nanoscale secondary ion mass spectroscopy (NanoSIMS) with spectroscopic IR s -SNOM imaging for simultaneous chemical, molecular, and elemental nanoimaging. At the example of Pinctada margaritifera mollusk shells we identify and resolve ~ 50 nm interlamellar protein sheets periodically arranged in regular ~ 600 nm intervals. The striations typically appear ~ 15 µm from the nacre-prism boundary at the interface between disordered neonacre to mature nacre. Using the polymorph distinctive IR-vibrational carbonate resonance, the nacre and prismatic regions are consistently identified as aragonite ( $${\overline{\nu }}_{a}=860$$ ν ¯ a = 860 cm −1 ) and calcite ( $${\overline{\nu }}_{c}=880$$ ν ¯ c = 880 cm −1 ), respectively. We observe previously unreported morphological features including aragonite subdomains encapsulated in extensions of the prism-covering organic membrane and regions of irregular nacre tablet formation coincident with dispersed organics. We also identify a ~ 200 nm region in the incipient nacre region with less well-defined crystal structure and integrated organics. These results show with the identification of the interlamellar protein layer how correlative nano-IR chemical and NanoSIMS elemental imaging can help distinguish different models proposed for shell growth in particular, and how organic function may relate to inorganic structure in other biomineralized systems in general.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Entropy-Assisted Quality Pattern Identification in Finance

Short-term patterns in financial time series form the cornerstone of many algorithmic trading strategies, yet extracting these patterns reliably from noisy market data remains a formidable challenge. In this paper, we propose an entropy-assisted framework for identifying high-quality, non-overlapping patterns that exhibit consistent behavior over time. We ground our approach in the premise that historical patterns, when accurately clustered and pruned, can yield substantial predictive power for short-term price movements. To achieve this, we incorporate an entropy-based measure as a proxy for information gain: patterns that lead to high one-sided movements in historical data yet retain low local entropy are more “informative” in signaling future market direction. Compared to conventional clustering techniques such as K-means and Gaussian Mixture Models (GMMs), which often yield biased or unbalanced groupings, our approach emphasizes balance over a forced visual boundary, ensuring that quality patterns are not lost due to over-segmentation. By emphasizing both predictive purity (low local entropy) and historical profitability, our method achieves a balanced representation of Buy and Sell patterns, making it better suited for short-term algorithmic trading strategies. This paper offers an in-depth illustration of our entropy-assisted framework through two case studies on Gold vs. USD and GBPUSD. While these examples demonstrate the method’s potential for extracting high-quality patterns, they do not constitute an exhaustive survey of all possible asset classes.

Physics↗
DOE OSTI2021

Defect detection in atomic-resolution images via unsupervised learning with translational invariance

Abstract Crystallographic defects can now be routinely imaged at atomic resolution with aberration-corrected scanning transmission electron microscopy (STEM) at high speed, with the potential for vast volumes of data to be acquired in relatively short times or through autonomous experiments that can continue over very long periods. Automatic detection and classification of defects in the STEM images are needed in order to handle the data in an efficient way. However, like many other tasks related to object detection and identification in artificial intelligence, it is challenging to detect and identify defects from STEM images. Furthermore, it is difficult to deal with crystal structures that have many atoms and low symmetries. Previous methods used for defect detection and classification were based on supervised learning, which requires human-labeled data. In this work, we develop an approach for defect detection with unsupervised machine learning based on a one-class support vector machine (OCSVM). We introduce two schemes of image segmentation and data preprocessing, both of which involve taking the Patterson function of each segment as inputs. We demonstrate that this method can be applied to various defects, such as point and line defects in 2D materials and twin boundaries in 3D nanocrystals.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Multi-scale Analysis of Surface Heterogeneity Induced Convection on Isentropic Coordinates

This research uses semi-idealized simulations to investigate multi-scale processes induced by the heterogeneity of soil moisture observed during the 2016 Holistic Interactions of Shallow Clouds, Aerosols, and Land-Ecosystems (HI-SCALE) field campaign. The semi-idealized simulations have realistic land heterogeneity, but large-scale winds are removed. Analysis on isentropic coordinates enables the tracking of circulation that transports energy vertically and facilitates the identification of the primary convective processes induced by realistic land heterogeneity. The isentropes associated with upward motion are found to connect the ground characterized by high latent heat flux to cloud bases directly over the ground with high sensible heat flux, while isentropes associated with downward motion connect precipitation to the ground characterized by high sensible heat fluxes. The mixing of dry, warm parcels ascending from the ground with high sensible heat fluxes and moist parcels from high latent heat regions leads to cloud formation. This new mechanism explains how soil moisture heterogeneity provides the key ingredients such as buoyancy and moisture for shallow cloud formation. We also found that the sub-mesoscale dominates upward energy transport in the boundary layer, while mesoscale circulations contribute to vertical energy transport above the boundary layer. Our novel method better illustrates and elucidates the nature of land atmospheric interactions under irregular and realistic soil moisture patterns.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Segmentation of tomography datasets using 3D convolutional neural networks

Dendritic microstructures are ubiquitous in nature and are the primary solidification morphologies in metallic materials. Techniques such as X-ray computed tomography (XCT) have provided new insights into dendritic phase transformation phenomena. However, manual identification of dendritic morphologies in microscopy data can be both labor intensive and potentially ambiguous. The analysis of 3D datasets is particularly challenging due to their large sizes (terabytes) and the presence of artifacts scattered within the imaged volumes. Here, in this study, we trained 3D convolutional neural networks (CNNs) to segment 3D datasets. Three CNN architectures were investigated, including a new version of FCDenseNet which we extended to 3D. We show that using hyperparameter optimization (HPO) and fine-tuning techniques, both 2D and 3D CNN architectures outperform the previous state of the art. The 3D U-Net architecture trained in this study produced the best segmentations according to quantitative metrics (intersection-over-union of 95.56% and a boundary displacement error of 0.58 pixels), while 3D FCDense produced the smoothest boundaries and best segmentations according to visual inspection. The trained 3D CNNs are able to segment entire 852 × 852 × 250 voxel 3D volumes in only ~60 s, thus hastening the progress towards a deeper understanding of phase transformation phenomena such as dendritic solidification.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Identification of Recombination Losses in CdSe/CdTe Solar Cells from Spectroscopic and Microscopic Time-Resolved Photoluminescence

Due to the lowest-cost and best reliability, CdTe solar cells are the leading thin-film photovoltaic technology. Increasing open-circuit voltage by reducing recombination represents the most promising path toward further improvements. Analysis is needed to identify limitations that cause efficiency losses. To achieve this goal for Cu-doped CdSe/CdTe solar cells, time-resolved spectroscopy and microscopy are developed and applied. Recombination lifetimes and radiative efficiency identify that defect-mediated recombination is the dominant voltage loss mechanism. When carrier lifetimes are averaged over many crystalline grains, they increase from 180 to 430 ns when Al2O3 is applied to the back contact. The quasi-Fermi-level splitting correspondingly increases from 880–905 to 906–931 mV, indicating a pathway to overcome the long-standing 900 mV voltage limitation. However, the dominant recombination losses are attributed to the absorber bulk. From microscopic carrier lifetime measurements, it is identified that space charge fields due to charged grain boundaries (GBs) lead to recombination in the CdTe absorber bulk. At high injection, GB space charge fields are screened, but that occurs above 1 Sun excitation conditions. Alloying with selenium in the near-interface CdSeTe absorber region reduces GB losses and is identified as one of the factors leading to high radiative and power conversion efficiency.

14 SOLAR ENERGY↗
DOE OSTI2026

Cloud-Based Demonstration of the Eastern Interconnection Situational Awareness Monitoring System (ESAMS)

This report describes a cloud-based implementation and field demonstration of the Eastern Interconnection Situational Awareness and Monitoring System (ESAMS). ESAMS was developed to support the detection and source localization of forced oscillations using synchrophasor measurements from tie-lines connecting areas served by different reliability coordinators (RCs), so that RCs could better coordinate their response to wide-area events. A previous effort had identified deployment barriers associated with hosting shared situational awareness tools at a single RC. To address these barriers, ESAMS was migrated to Amazon Web Services and evaluated in a six-month field demonstration. ISO New England (ISO-NE) and PJM streamed data to the platform using AWS Direct Connect and a site-to-site VPN, respectively. The resulting multi-utility measurement footprint enabled regional source localization across major portions of the U.S. Eastern Interconnection and supported routine identification of oscillation events. During the final three months of the trial, 24 events above 2 MW/MVAR were detected. The largest detected oscillation approached a 25 MW peak-to-peak amplitude, and the longest persisted intermittently for more than 11 hours. The demonstration also assessed operational considerations—including data transfer volumes, end-to-end latency, and cloud computing costs—and found that network and compute requirements were modest relative to typical cloud capabilities while providing performance comparable to prior on-premises deployments. Overall, the results indicate that cloud hosting can provide a practical path to shared interconnection-wide oscillation monitoring. The cloud ESAMS demonstration establishes a foundation for broader utility participation and for building future wide-area analytics that leverage measurements across organizational boundaries.

Follum, James D.↗
DOE OSTI2020

Tape casting and characterizations of MgO ceramics

Herein we report a high density MgO ceramic substrate produced by the tape casting technology. The tape casting formulation and process produced a uniform tape free of cracking. Y 2 O 3 and SiO 2 were used as the sintering aid for the pressureless sintering of the green tape. X-ray diffraction phase identification indicates that MgO is the main phase, while both Y 2 O 3 and SiO 2 sintering aids react with MgO to form MgY 4 Si 3 O 13 as the second phase. Liquid phase sintering occurs in the temperature range from 1030°C to ~1500°C, which is confirmed by the simultaneous Thermal Gravitation Analysis/ Differential Scanning Calorimeter (TGA/DSC) and the percent linear shrinkage and densification. A 96.5% theoretical density was achieved by presureless sintering at 1650°C for 2 hours, which was further increased to a fully dense structure using hotisostatic- pressing(HIP) at 1650°C and 207 MPa in argon. Scanning electron microscopy (SEM) and energy dispersive(EDS) spectroscopic analysis on the HIP’ed sample show that MgY 4 Si 3 O 13 is located at the MgO grain boundary and the sample has a fully dense structure. The refractive indices and extinction coefficient were measured on the HIP’ed sample along with thermal properties and dielectric properties. Thermal diffusivity and heat capacity were measured to calculate the thermal conductivity.

36 MATERIALS SCIENCE↗
DOE OSTI2023

A review of neutronics and thermal hydraulics–based screening methods applied to accelerated nuclear fuel qualification

This paper reviews the state-of-the-art engineering approach for using thermal hydraulic (TH) and neutronics modeling and simulation (M&S) tools to perform rapid screening studies of novel nuclear fuel concepts within the context of accelerated fuel qualification. Global research efforts have introduced nuclear fuel and material concepts that mark a significant departure from traditional reactor materials. The number of new technologies being considered for development for light water reactors and advanced reactor types has created the need for an accelerated fuel qualification procedure. A key component of this procedure is the rapid identification of the most promising fuel concepts using computational screening studies. Advanced TH and neutronic M&S tools should be leveraged to efficiently determine whether the reactor performance and safety characteristics of a given concept warrant additional studies or whether the concept requires modification or elimination. This paper reviews best practices for performing these TH and neutronics screening studies at various stages during a fuel concept's progression through the qualification procedure. The motivation behind standardizing this approach is to minimize time and resources spent on qualification activities for fuel concepts that could be quickly refined or eliminated from consideration based on their reactor physics and TH characteristics. Adoption of this screening procedure—which focuses primarily on nuclear fuels but may be applicable to other reactor materials—will also help accelerate new material qualification by generating boundary conditions crucial to fuel performance evaluations and highlighting needed areas of separate effects experimentation. This article reviews the motivation behind the introduction of novel nuclear fuel concepts, provides incentive for utilizing TH- and neutronics-based screening studies, describes the screening approach and methodology, and includes discussion on how to interpret screening results to provide recommendations for the continued development of a given concept.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2021

Diagnosing nuclear power plant pipe wall thinning due to flow accelerated corrosion using a passive, thermal non-destructive evaluation method: Feasibility assessment via numerical experiments

Flow accelerated corrosion (FAC) in nuclear power plant pipes is one of the leading causes of accidents, fatalities, damage and outages. Current FAC identification methods employ expensive sensing technology and are “active” methods, where the response of the piping system to an externally-generated thermal, mechanical or optical excitation must be measured. As a result, these techniques require a disruptive and time-consuming setup. Here we propose a method that utilizes pipe surface temperature measurements to passively monitor for FAC-induced pipe wall thinning without the need for expensive equipment or post-installation setup time. This diagnostic method utilizes a simulation data-driven diagnostic model to estimate the amount of thickness reduction in a pipe based on changes in measured steady-state pipe temperatures. In order to reduce the computational burden of generating large, simulation-based datasets, the behavior of the insulation of the pipe was modeled using a suitably calibrated heat transfer boundary parameter. Additionally, global sensitivity analysis was performed to determine system parameter(s), such as the temperature of water flowing inside the pipe, which significantly affect the steady state pipe wall temperature and could cause errors in diagnosis. Two diagnostic models, one using only the change in steady-state temperature as an indicator for FAC-induced pipe wall thinning and the other using water temperature as an additional diagnostic model input were evaluated for their ability to estimate thickness reductions in a pipe using simulated pipe wall temperature data. For the numerical experiments conducted in this work, both models estimated wall thickness with errors within 0.5 mm, indicating that the proposed technique can potentially be used as a low-cost, first-pass method for FAC monitoring.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

Advection-dispersion equations (ADEs) are commonly used to describe transport phenomena in porous media. Even though mature discretization-based numerical methods for ADEs exist, some challenges still remain, especially when it comes to solving advection-dominated forward ADEs and diffusion-dominated backward ADEs. The latter problem usually arises in the source identification context and leads to numerically unstable grid-based solutions that require a form of regularization or should be treated as an inverse problem that is computationally more expensive because it requires solving the forward problem multiple times. In this study, we propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled ADE and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one- and two-dimensional forward ADE problems, where its performance for various P\'{e}clet numbers ($Pe$) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1\% and outperforms some conventional discretization-based methods for $Pe$ larger that 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5\% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50\% in the considered cases) the accuracy of the PINN solution of the backward ADE.

He, Qizhi↗
DOE OSTI2023

The saturation mechanism of thermal instability

The literature on thermal instability (TI) reveals that even for a simple homogeneous plasma, the nonlinear outcome ranges from a gentle reconfiguration of the initial state to an explosive one, depending on whether the condensations that form evolve in an isobaric or nonisobaric manner. After summarizing the recent developments on the linear and nonlinear theory of TI, here we derive several general identities from the evolution equation for entropy that reveal the mechanism by which TI saturates; whenever the boundary of the instability region (the Balbus contour) is crossed, a dynamical change is triggered that causes the comoving time derivative of the pressure to change the sign. This event implies that the gas pressure force reverses direction, slowing the continued growth of condensation. For isobaric evolution, this “pressure reversal” occurs nearly simultaneously for every fluid element in condensation and a steady state is quickly reached. For nonisobaric evolution, the condensation is no longer in mechanical equilibrium and the contracting gas rebounds with greater force during the expansion phase that accompanies the gas reaching the equilibrium curve. The cloud then pulsates because the return to mechanical equilibrium becomes wave mediated. We show that both the contraction rebound event and subsequent pulsation behavior follow analytically from an analysis of the new identities. Our analysis also leads to the identification of an isochoric TI zone and makes it clear that unless this zone intersects the equilibrium curve, isochoric modes can only become unstable if the plasma is in a state of thermal non-equilibrium.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Global quantum phase diagram and non-Abelian chiral spin liquid in a spin- 3 2 square-lattice antiferromagnet

Since strong quantum fluctuations are essential for the emergence of quantum spin liquids, there have been extensive exploration and identification of spin liquid candidates in spin-$\frac{1}{2}$ systems, while such activities are rare in higher spin systems. Here we report an example of non-Abelian chiral spin liquid emerging in a spin-$\frac{3}{2}$ Heisenberg model on a square lattice. By tuning Heisenberg exchange interaction and scalar chirality interaction, we map out a quantum phase diagram enclosing three conventional magnetic orders and a chiral spin liquid based on density-matrix renormalization group studies. The nature of the spin liquid is identified as a long-sought bosonic version of the Read-Rezayi state that supports non-Abelian Fibonacci anyonic statistics, identified by the ground state entanglement spectrum. Significantly, we establish that the non-Abelian chiral spin liquid emerges through the enlarged local degrees of freedom and enhanced quantum fluctuations near the classical phase boundaries of competing magnetic orders. Finally, our numerical discovery of an exotic quantum spin liquid in a spin-$\frac{3}{2}$ system suggests a route for discovering fractionalized quantum phases in frustrated higher spin magnetic compounds.

2-dimensional systems↗
DOE OSTI2024

Classifying photonic topology using the spectral localizer and numerical K -theory

Recently, the spectral localizer framework has emerged as an efficient approach for classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts such as topological solitons and topological dynamics, where a system’s occupation induces a local change in its topology due to nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of C*-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches for classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework’s application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell’s equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical K-theory, two mathematical concepts that form the foundation for the spectral localizer framework. Finally, we provide some mathematically oriented comments on the C*-algebraic origins of this framework, including a discussion of real C*-algebras and graded C*-algebras that are necessary for incorporating physical symmetries. Looking forward, we hope that this tutorial will serve as an approachable starting point for learning the foundations of the spectral localizer framework.

97 MATHEMATICS AND COMPUTING↗