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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
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↗
NASA NTRS1982

Study of sensor spectral responses and data processing algorithms and architectures for onboard feature identification

A computational model of the deterministic and stochastic processes involved in remote sensing is used to study spectral feature identification techniques for real-time onboard processing of data acquired with advanced earth-resources sensors. Preliminary results indicate that: Narrow spectral responses are advantageous; signal normalization improves mean-square distance (MSD) classification accuracy but tends to degrade maximum-likelihood (MLH) classification accuracy; and MSD classification of normalized signals performs better than the computationally more complex MLH classification when imaging conditions change appreciably from those conditions during which reference data were acquired. The results also indicate that autonomous categorization of TM signals into vegetation, bare land, water, snow and clouds can be accomplished with adequate reliability for many applications over a reasonably wide range of imaging conditions. However, further analysis is required to develop computationally efficient boundary approximation algorithms for such categorization.

Huck, F. O.↗
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↗
NASA NTRS1996

A Frequency-Domain Substructure System Identification Algorithm

A new frequency-domain system identification algorithm is presented for system identification of substructures, such as payloads to be flown aboard the Space Shuttle. In the vibration test, all interface degrees of freedom where the substructure is connected to the carrier structure are either subjected to active excitation or are supported by a test stand with the reaction forces measured. The measured frequency-response data is used to obtain a linear, viscous-damped model with all interface-degree of freedom entries included. This model can then be used to validate analytical substructure models. This procedure makes it possible to obtain not only the fixed-interface modal data associated with a Craig-Bampton substructure model, but also the data associated with constraint modes. With this proposed algorithm, multiple-boundary-condition tests are not required, and test-stand dynamics is accounted for without requiring a separate modal test or finite element modeling of the test stand. Numerical simulations are used in examining the algorithm's ability to estimate valid reduced-order structural models. The algorithm's performance when frequency-response data covering narrow and broad frequency bandwidths is used as input is explored. Its performance when noise is added to the frequency-response data and the use of different least squares solution techniques are also examined. The identified reduced-order models are also compared for accuracy with other test-analysis models and a formulation for a Craig-Bampton test-analysis model is also presented.

Blades, Eric L.↗
NASA NTRS2004

Magnetohydrodynamic Convection in the Outer Core and its Geodynamic Consequences

The Earth's fluid outer core is in vigorous convection through much of the Earth's history. In addition to generating and maintaining Earth s time-varying magnetic field (geodynamo), the core convection also generates mass redistribution in the core and a dynamical pressure field on the core-mantle boundary (CMB). All these shall result in various core-mantle interactions, and contribute to surface geodynamic observables. For example, electromagnetic core-mantle coupling arises from finite electrically conducting lower mantle; gravitational interaction occurs between the cores and the heterogeneous mantle; mechanical coupling may also occur when the CMB topography is aspherical. Besides changing the mantle rotation via the coupling torques, the mass-redistribution in the core shall produce a spatial-temporal gravity anomaly. Numerical modeling of the core dynamical processes contributes in several geophysical disciplines. It helps explain the physical causes of surface geodynamic observables via space geodetic techniques and other means, e.g. Earth's rotation variation on decadal time scales, and secular time-variable gravity. Conversely, identification of the sources of the observables can provide additional insights on the dynamics of the fluid core, leading to better constraints on the physics in the numerical modeling. In the past few years, our core dynamics modeling efforts, with respect to our MoSST model, have made significant progress in understanding individual geophysical consequences. However, integrated studies are desirable, not only because of more mature numerical core dynamics models, but also because of inter-correlation among the geophysical phenomena, e.g. mass redistribution in the outer core produces not only time-variable gravity, but also gravitational core-mantle coupling and thus the Earth's rotation variation. They are expected to further facilitate multidisciplinary studies of core dynamics and interactions of the core with other components of the Earth.

Kuang, Weijia↗
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↗
NASA NTRS1993

Numerical modeling of physical vapor transport under microgravity conditions: Effect of thermal creep and stress

One of the most promising applications of microgravity (micro-g) environments is the manufacture of exotic and high-quality crystals in closed cylindrical ampoules using physical vapor transport (PVT) processes. The quality enhancements are believed to be due to the absence of buoyant convection in the weightless environment - resulting in diffusion-limited transport of the vapor. In a typical experiment, solid-phase sample material is initially contained at one end of the ampoule. The sample is made to sublime into the vapor phase and deposit onto the opposite end by maintaining the source at an elevated temperature with respect to the deposit. Identification of the physical factors governing both the rates and uniformity of crystal growth, and the optimization of the micro-g technology, will require an accurate modeling of the vapor transport within the ampoule. Previous micro-g modeling efforts have approached the problem from a 'classical' convective/diffusion formulation, in which convection is driven by the action of buoyancy on thermal and solutal density differences. The general conclusion of these works have been that in low gravity environments the effect of buoyancy on vapor transport is negligible, and vapor transport occurs in a diffusion-limited mode. However, it has been recently recognized than in the non-isothermal (and often low total pressure) conditions encountered in ampoules, the commonly-assumed no-slip boundary condition to the differential equations governing fluid motion can be grossly unrepresentative of the actual situation. Specifically, the temperature gradients can give rise to thermal creep flows at the ampoule side walls. In addition, temperature gradients in the vapor itself can, through the action of thermal stress, lead to bulk fluid convection.

Mackowski, Daniel W.↗
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↗
NASA NTRS2017

Fatigue Life and Damage Tolerance of Postbuckled Composite Stiffened Structures with Indentation Damage

The fatigue life and damage tolerance of composite stiffened panels with indentation damage are investigated experimentally using single-stringer compression specimens. The indentation damage was induced to one of the two flanges of the stringer of every panel. The advantages of indentation compared to impact are the simplicity of application, less dependence on boundary conditions, better controllability, and repeatability of the imparted damage. The tests were conducted using advanced instrumentation, including digital image correlation, passive thermography, and in situ ultrasonic scanning. Specimens with initial indentation damage ranging between 32 and 56 mm in length were tested quasi-statically and in fatigue, and the effects of cyclic load amplitude and damage size were studied. A means of comparison of the damage propagation rates and collapse loads based on a stress intensity measure and the Paris law is proposed. The stress intensity measure provides the means to compare the collapse loads of specimens with different damage types and damage sizes, while the Paris law is used to compare the damage propagation rates in specimens subjected to different cyclic loads. This approach enables a comparison of different tests and the potential identification of the effects that influence the fatigue lives and damage tolerance of postbuckled structures with defects.

Davila, Carlos G.↗
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↗
NASA NTRS2019

Boundary Layer Stability and Laminar-Turbulent Transition Analysis with Thermochemical Nonequilibrium Applied to Martian Atmospheric Entry

As Martian atmospheric entry vehicles increase in size to accommodate larger payloads, transitional ow may need to be taken into account in the design of the heat shield in order to reduce heat shield mass. The mass of the Thermal Protection System (TPS) comprises a significant portion of the vehicle mass, and a reduction of this mass would result in fuel savings. The current techniques used to design entry shields generally assume fully turbulent flow when the vehicle is large enough to expect transitional flow, and while this worst-case scenario provides a greater factor of safety it may also result in overdesigned TPS and unnecessarily high vehicle mass. Greater accuracy in the prediction of transition would also reduce uncertainty in the thermal and aerodynamic loads. Stability analysis, using e(sup⁡ N) -based methods including Linear Stability Theory (LST) and the Parabolized Stability Equations (PSE), offers a physics-based method of transition prediction that has been thoroughly studied and applied in perfect gas flows, and to a more limited extent in reacting and nonequilibrium flows. These methods predict the amplification of a known disturbance frequency and allow identification of the most unstable frequency. Transition is predicted to occur at a critical amplification or N Factor, frequently determined through experiment and empirical correlations. The LAngley Stability and TRansition Analysis Code (LASTRAC), with modifications for thermochemically reacting flows and arbitrary gas mixtures, will be presented with LST results on a simulation of a high enthalpy CO2 gas wind tunnel test relevant to Martian atmospheric entry. The results indicate transition caused by modified Tollmien-Schlichting waves on the leeward side, which are predicted to be more stable and cause transition slightly downstream when thermochemical nonequilibrium is included in the stability analysis for the same mean flow solution.

Transition↗
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↗