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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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

On Anomaly Detection for Transactive Energy Systems with Competitive Market

Two anomaly-detection criteria are proposed for transactive energy systems with competitive markets. Participants of transactive energy systems seek an optimal power allocation through hybrid economic-control methods to facilitate the integration of various types of distributed energy resources to power distribution systems. In transactive energy systems, every participant is assumed to be a rational entity, and consumers have diminishing marginal utility and suppliers have increasing marginal cost. With the first proposed anomaly-detection criterion, the monotonicity of marginal cost and marginal utility are examined. The impact of line flow constraints is also taken into consideration. Then, the second anomaly-detection criterion is proposed for TESs with marginal cost and marginal utility which change faster than a certain rate. The second criterion is more accurate than the first one for TES with marginal cost and marginal utility which change faster than a certain rate, but it requires the knowledge of that rate. Neither criteria requires more data than those necessary to find the optimal power allocation and the market-clear price in a transactive energy system. Therefore, the proposed criteria do not disclose any more data than necessary. As the monotonicity of marginal cost and marginal utility in a TES with competitive markets results in convex objective functions in an optimization problem and strongly convex ones when marginal cost and marginal utility changes faster than a certain rate, the two detection criteria are also applicable to anomaly detection of general convex optimization problems. Simulations are carried out to show the efficacy of the proposed criteria to detect anomalies caused by cyberattacks.

Wang, Peng↗
DOE OSTI2025

Linear Discriminant Analysis-Based Machine Learning and All-Atom Molecular Dynamics Simulations for Probing Electro-Osmotic Transport in Cationic-Polyelectrolyte-Brush-Grafted Nanochannels

Deciphering the correct mechanisms governing certain phenomena in polyelectrolyte (PE) brush grafted systems, revealed through atomistic simulations, is an extremely challenging problem. In a recent study, our all-atom molecular dynamics (MD) simulations revealed a non-linearly large electroosmotic (EOS) flow (in the presence of an applied electric field) in nanochannels grafted with PMETAC [Poly(2-(methacryloyloxy)ethyl trimethylammonium chloride] brushes. Given the lack of any formal procedure that would have directed us to identify the correct factors responsible for such an occurrence, we needed to spend several months and devote significant analyses to unravel the involved mechanisms. In this paper, we propose a Linear Discriminant Analysis (LDA) based Machine Learning (ML) approach to address this gap. At first, we obtain data on certain basic features from the all-atom MD data. These basic features represent the number of atoms of certain species around one atom of another (or same) species. Here, we obtain such data on basic features for a reference case (case of an EOS flow in PMETAC-brush-grafted nanochannels with a smaller electric field) and a perturbed case (case of an EOS flow in PMETAC-brush-grafted nanochannels with a larger electric field) in bins in which the nanochannel half height has been divided into. These datasets are high-dimensional dataset, to which the LDA is applied. This leads to the projection of the data (between the reference and the perturbed states) in a highly separated form on a 1D line. From such LDA calculations, we are able to identify the relative importance of the different basic features in ensuring this separation of the data (between the reference and the perturbed states) on the 1D line. This relative importance of the different basic features is quantified as “importance scores” for the different features, which in turn tell us what to study and where to study. Such knowledge enables us to rapidly identify the key factors responsible for the non-linearly large EOS transport in PMETAC-brush-grafted nanochannels.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Noisy quantum trees: infinite protection without correction

We study quantum networks with tree structures, in which information propagates from a root to leaves. At each node in the network, the received qubit unitarily interacts with fresh ancilla qubits, after which each qubit is sent through a noisy channel to a different node in the next level. Therefore, as the tree depth grows, there is a competition between the irreversible effect of noise and the protection against such noise achieved by the delocalization of information. In the classical setting, where each node simply copies the input bit into multiple output bits, this model has been studied as the broadcasting or reconstruction problem on trees, which has broad applications. In this work, we study the quantum version of this problem. We consider a Clifford encoder at each node that encodes the input qubit in a stabilizer code, along with a single qubit Pauli noise channel at each edge. Such noisy quantum trees describe a scenario in which one has access to a stream of fresh (low-entropy) ancilla qubits, but cannot perform error correction. Therefore, they provide a different perspective on quantum fault tolerance. Furthermore, they provide a useful model for describing the effect of noise within the encoders of concatenated codes. We prove that above certain noise thresholds, which depend on the properties of the code such as its distance, as well as the properties of the encoder, information decays exponentially with the depth of the tree. On the other hand, by studying certain efficient decoders, we prove that for codes with distance d ≥ 2 and for sufficiently small (but non-zero) noise, classical information and entanglement propagate over a noisy tree with infinite depth. Indeed, we find that this remains true even for binary trees with certain 2-qubit encoders at each node, which encodes the received qubit in the binary repetition code with distance d = 1.

Quantum information↗
DOE OSTI2021

Predictability as a probe of manifest and latent physics: The case of atomic scale structural, chemical, and polarization behaviors in multiferroic Sm-doped BiFeO 3

The predictability of a certain effect or phenomenon is often equated with the knowledge of relevant physical laws, typically understood as a functional or numerically derived relationship between the observations and known states of the system. Correspondingly, observations inconsistent with prior knowledge can be used to derive new knowledge on the nature of the system or indicate the presence of yet unknown mechanisms. In this work, we explore the applicability of Gaussian processes (GP) to establish predictability and uncertainty of local behaviors from multimodal observations, providing an alternative to this classical paradigm. Using atomic resolution scanning transmission electron microscopy (STEM) of multiferroic Sm-doped BiFeO 3 across a broad composition range, we directly visualize the atomic structure and structural, physical, and chemical order parameter fields for the material. GP regression is used to establish the predictability of the local polarization field from different groups of parameters, including the adjacent polarization values and several combinations of physical and chemical descriptors, including lattice parameters, column intensities, etc. We observe that certain elements of microstructure, including charged and uncharged domain walls and interfaces with the substrate, are best predicted with specific combinations of descriptors, and this predictability and associated uncertainties are consistent across the composition series. The associated generative physical mechanisms are discussed. It is also found that certain parameter combinations tend to predict the orthorhombic phase in the cases where rhombohedral phase is observed, suggesting a potential role of clamping and confinement phenomena in phase equilibrium in Sm-BiFeO 3 system close to morphotropic phase boundary. We argue that predictability and uncertainty in observational data offer a new pathway to probe the physics of condensed matter systems from multimodal local observations.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2023

What is flat ΛCDM, and may we choose it?

The Universe is neither homogeneous nor isotropic, but it is close enough that we can reasonably approximate it as such on suitably large scales. The inflationary-Λ-Cold Dark Matter (ΛCDM) concordance cosmology builds on these assumptions to describe the origin and evolution of fluctuations. With standard assumptions about stress-energy sources, this system is specified by just seven phenomenological parameters, whose precise relations to underlying fundamental theories are complicated and may depend on details of those fields. Nevertheless, it is common practice to set the parameter that characterizes the spatial curvature, Ω K , exactly to zero. This parameter-fixed ΛCDM is awarded distinguished status as separate model, "flat ΛCDM." Ipso facto this places the onus on proponents of "curved ΛCDM" to present sufficient evidence that Ω K ≠ 0, and is needed as a parameter. While certain inflationary model Lagrangians, with certain values of their parameters, and certain initial conditions, will lead to a present-day universe well-described as containing zero curvature, this does not justify distinguishing that subset of Lagrangians, parameters and initial conditions into a separate model. Absent any theoretical arguments, we cannot use observations that suggest small Ω K to enforce Ω K = 0. Our track record in picking inflationary models and their parameters a priori makes such a choice dubious, and concerns about tensions in cosmological parameters and large-angle cosmic-microwave-background anomalies strengthens arguments against this choice. We argue that Ω K must not be set to zero, and that ΛCDM remains a phenomenological model with at least 7 parameters.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

“Quantum Geometric Nesting” and Solvable Model Flat-Band Systems

We introduce the concept of “quantum geometric nesting” (QGN) to characterize the idealized ordering tendencies of certain flat-band systems implicit in the geometric structure of the flat-band subspace. Perfect QGN implies the existence of an infinite class of local interactions that can be explicitly constructed and give rise to solvable ground states with various forms of possible fermion bilinear order, including flavor ferromagnetism, density waves, and superconductivity. For the ideal Hamiltonians constructed in this way, we show that certain aspects of the low-energy spectrum can also be exactly computed including, in the superconducting case, the phase stiffness. Examples of perfect QGN include flat bands with certain symmetries (e.g., chiral or time reversal) and non-symmetry-related cases exemplified with an engineered model for pair-density wave. Extending this approach, we obtain exact superconducting ground states with nontrivial pairing symmetry. Published by the American Physical Society 2024

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Dynamics of inspiraling dark energy

Abstract We investigate the dynamics of a multifield dark energy model, which arises from certain rapid-turning and inspiraling trajectories in field space. We find the speed of sound $$c_s$$ c s of the dark energy perturbations around the background and show that $$c_s$$ c s is monotonically decreasing with time. Furthermore, it has a positive-definite lower bound that implies a certain clustering scale. We also extend the previously known background solution for dark energy to an exact solution that includes matter. This allows us to address the implications of our model for two cosmological tensions. More precisely, we argue that the $$\sigma _8$$ σ 8 tension can be alleviated generically, while reducing the Hubble tension requires certain constraints on the parameter space of the model. Notably, a necessary condition for alleviating the Hubble tension is that the transition from matter domination to the dark energy epoch begins earlier than in $$\Lambda $$ Λ CDM.

Anguelova, Lilia↗
DOE OSTI2021

Survey of Advanced Generation IV Reactor Parameters for Integrated Energy System Modeling Capabilities

This paper is to categorize and characterize advanced Generation IV nuclear reactors that are not water based according to the key parameters for the Integrated Energy Systems (IES) project[1]. A key goal of the IES project is to assess the economic viability of including an advanced nuclear reactor into an IES. The landscape of advanced reactors is heterogeneous: several unique designs with different heat profiles are currently being studied and designed. This report provides a high-level perspective of advanced reactor concepts, including primary loop nuclear output temperatures and operation schemes that could affect secondary side power production or storage. For example, this report addresses the type of thermodynamic system (Brayton, supercritical CO 2 [S-CO 2 ], etc.) and the various pressures and temperatures on the secondary side that are expected to be provided by the various primary side reactor types. Based on these characteristics, it may be possible to determine which type of nuclear power plant could be the most economical for analysis in the IES framework for power production coupled with energy storage and waste heat utilization. The results of this effort will serve as input for model development of various secondary side components, concentrating on off-the-shelf capabilities for current and near future secondary side power generation and heat rejection as simulated in TRANSFORM. Generic models can be created which are agnostic of the nuclear energy supply without violating intellectual property rights of the developer. These models can be used to make informed decisions on the economics of integrating various energy system components with nuclear plant designs. Once complete, conclusions may be drawn regarding (1) the economics, efficiency, and reliability of certain nuclear reactors, (2) certain industry processes and energy storage systems, and (3) prioritization of certain reactor types with the studied industry types.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

MONTI: A Multi-Omics Non-negative Tensor Decomposition Framework for Gene-Level Integrative Analysis

Multi-omics data is frequently measured to enrich the comprehension of biological mechanisms underlying certain phenotypes. However, due to the complex relations and high dimension of multi-omics data, it is difficult to associate omics features to certain biological traits of interest. For example, the clinically valuable breast cancer subtypes are well-defined at the molecular level, but are poorly classified using gene expression data. Here, we propose a multi-omics analysis method called MONTI (Multi-Omics Non-negative Tensor decomposition for Integrative analysis), which goal is to select multi-omics features that are able to represent trait specific characteristics. Here, we demonstrate the strength of multi-omics integrated analysis in terms of cancer subtyping. The multi-omics data are first integrated in a biologically meaningful manner to form a three dimensional tensor, which is then decomposed using a non-negative tensor decomposition method. From the result, MONTI selects highly informative subtype specific multi-omics features. MONTI was applied to three case studies of 597 breast cancer, 314 colon cancer, and 305 stomach cancer cohorts. For all the case studies, we found that the subtype classification accuracy significantly improved when utilizing all available multi-omics data. MONTI was able to detect subtype specific gene sets that showed to be strongly regulated by certain omics, from which correlation between omics types could be inferred. Furthermore, various clinical attributes of nine cancer types were analyzed using MONTI, which showed that some clinical attributes could be well explained using multi-omics data. We demonstrated that integrating multi-omics data in a gene centric manner improves detecting cancer subtype specific features and other clinical features, which may be used to further understand the molecular characteristics of interest. The software and data used in this study are available at: https://github.com/inukj/MONTI.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Effect of Intermediate Semiconducting TiOx Thin Films on Nanoparticle-Mediated Electron Transfer: Electrooxidation of CO

The concept of nanoparticle-mediated electron transfer (eT) across insulating thin films was elucidated theoretically by Allongue and Chazalviel in 2011. In their model, metal nanoparticles (NPs) are immobilized atop passivating, self-assembled monolayers (SAMs). They found that under certain conditions, related to the thickness of the SAM and the size of the NPs, efficient faradaic oxidation and reduction reactions could proceed at the NP surface. In the absence of NPs, however, eT was suppressed by the insulating SAM thin films. Allongue and Chazalviel concluded that, within certain bounds, eT is mediated by fast tunneling between the conductive electrode and the metal NPs, while the kinetics of the redox reaction are controlled by the NPs. This understanding has been confirmed using a variety of experimental models. The theory is based on electron tunneling; therefore, the nature of the intervening medium (the insulator in prior studies) should not affect the eT rate. In the present manuscript, however, we show that the theory breaks down under certain electrochemical conditions when the medium between conductors is an n-type semiconductor. Specifically, we find that in the presence of either Au or Pt NPs immobilized on a thin film of TiOx, CO electrooxidation does not proceed. In contrast, the exact same systems lead to the efficient reduction of oxygen. At present, we are unable to explain this finding within the context of the model of Allongue and Chazalviel.

Galyamova, Aigerim↗
DOE OSTI2022

Associations between Frequency of Culinary Herb Use and Gut Microbiota

While evidence suggests that culinary herbs have the potential to modulate gut microbiota, much of the current research investigating the interactions between diet and the human gut microbiome either largely excludes culinary herbs or does not assess use in standard culinary settings. As such, the primary objective of this study was to evaluate how the frequency of culinary herb use is related to microbiome diversity and the abundance of certain taxa, measured at the phylum level. In this secondary data analysis of the INCLD Health cohort, we examined survey responses assessing frequency of culinary herb use and microbiome analysis of collected stool samples. We did not observe any associations between frequency of culinary herb use and Shannon Index, a measure of alpha diversity. Regarding the abundance of certain taxa, the frequency of use of polyphenol-rich herbs and herbs with certain quantities of antibacterial compounds was positively associated with Firmicutes abundance, and negatively associated with Proteobacteria abundance. Additionally, the total number of herbs used with high frequency, defined as over three times per week, was also positively associated with Firmicutes abundance, independent of adjustments, and negatively associated with Proteobacteria abundance, after adjusting for dietary factors. Frequency of culinary herb use was not associated with Bacteroidota or Actinobacteria abundance.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

A VOI Web Application for Distinct Geothermal Domains: Statistical Evaluation of Different Data Types within the Great Basin

The Great Basin region contains different domains that have different structural and hydrothermal flow patterns. Depending on the characteristics of these patterns, certain data types may be more successful at detecting hidden geothermal resources. In this paper, we quantitatively evaluate if certain data types are more successful in certain domains. Given different aquifer, strain and structural conditions, we explore which data types statistically reveal positively labeled geothermal sites. We utilize value of information (VOI) metrics to help quantify the reliability of data types to discriminate against "positive" and "negative" labeled geothermal sites. We also evaluate how kernel density estimation can help generalize the statistics that inform VOI, which is necessary given the limited data in geothermal exploration. Except for the Carbonate Aquifer, the highest ranking of the Vimperfect is the Local Structural Setting. Next, the slip and dilation tendency is first for Carbonate Aquifer and second for Central Nevada Seismic Belt and Western Great Basin. For the Carbonate Aquifer, heat flow is has the lowest Vimperfect value compared to the other three domains, which is consistent with the understanding of how heat flow measurements are masked by regional groundwater flow.

Bayesian analysis↗
DOE OSTI2024

Identifying Potential Sub-Synchronous Oscillations Using Impedance Scan Approach: Preprint

This paper presents an impedance scan study on the real-world power system in Australia that has observed 17 to 20 Hz intermittent subsynchronous oscillations. Through the impedance scan of each of the IBRs, both individually and collectively, potential resonance modes are identified. The impedance scans were carried out using electromagnetic transient PSCAD models of the network. The network comprises of site-specific, black-boxed models of IBRs supplied by the generators. The impedance scan approach was divided in three major steps: (1) the first step is to identify IBRs where impedance analysis needs to be performed based on the magnitude of oscillations observed at their points of interconnection (POIs); (2) the second step focuses on performing impedance scans at selected IBRs in single-machine infinite-bus (SMIB) configuration to identify internal resonance modes of an IBR and to evaluate if any of these modes become unstable under certain grid conditions; (3) the third step performs impedance scans at a few selected IBRs while connected to the wider network model to obtain the impedance response of both the IBR and the grid - this step evaluates control interactions among IBRs. The impedance scan study found that a few IBRs have an underdamped resonance mode at around 17 Hz, which becomes unstable under a certain operating condition. Another interesting finding was that certain IBRs increase the effective grid impedance seen by another IBR in proximity for a particular operating condition. Under this operating condition, the resulting subsynchronous oscillation mode is more pronounced due to a combination of control interactions among IBRs through the transmission network under study and a resonance mode inside the IBRs.

control interactions↗
DOE OSTI2025

Open Architecture for Cost Savings in Advanced Nuclear Reactors

Recently, nuclear power plant build projects in the West have run over budget due to high capital costs and schedule overruns. Compared to other sources of energy, nuclear power plants have higher capital costs. Reactors are often different at every site, resulting in a lack of standardization. Nuclear is expected to compete with other low carbon sources of energy which have lower capital costs making it essential for nuclear to develop ways of reducing costs. Strategies such as standardization, learning rates, modularization, and schedule reduction in advanced reactors can reduce nuclear costs by about 40%. Standardization as a way of cutting capital costs has been explored even in large nuclear power plants. Standardization of certain plant components can result in lower component and installation costs and higher learning from experience. Standardization can be achieved by adopting a criterion of key performance indicators and general design principles for a specific system or component such as the balance of plant. Modularization allows the construction of certain components of SMRs in a factory, which saves time, increases productivity, and encourages higher learning rates. Production learning decreases the time and the cost related to an activity. The potential for modularized components of advanced reactors to be manufactured in factories makes it conducive to achieving higher learning rates. Developing large-capacity nuclear programs through sequential builds cultivates a higher learning rate, which in effect may reduce schedule overruns. Open architecture has been identified as a way to drive standardization among advanced reactor designs and result in cost savings. Open architecture (OA) is defined as a design enabling a diverse supply chain by defining and publishing requirements of systems or equipment in functional and/or interface terms, utilizing technical standards in widespread use. Currently, the nuclear industry’s approach is to use closed architecture, making most designs proprietary. However, collaboration between various advanced reactor vendors and suppliers utilizing the concept of open architecture can result in modular and standardized architecture of subsystems or subcomponents of a nuclear power plant. Completely standardizing nuclear power plants may be impossible, however, certain common subsystems amongst the various reactor designs could be standardized and/or access a wider supply chain and leverage existing learning from other sectors. Open architecture will save time and allocate resources to the parts of the plants that have the most unique features. A key advantage of open architecture is its ability to improve production learning across advanced reactors (AR) types in the industry, by providing and utilizing the same kind of component. Sodium fast reactor (SFR), High Temperature Gas Reactor (HTGR) and Molten Salt Reactor (MSR) are the advanced reactors considered for this project. This paper aims to determine the cost savings in advanced reactor programs due to open architecture learning rate. This work is an extension of work done on light water reactor small modular reactors; the cost methodology was utilized to investigate the impact of open architecture on advanced reactors with a particular focus on sodium fast reactors. The cost data on sodium fast reactors used in the model presented the most adequate information required for the analysis.

Advanced Nuclear Reactors↗
DOE OSTI2020

TiO2 Nanocrystal-Framed Li 2 TiSiO 5 Platelets for Low-Voltage Lithium Battery Anode

Titanium-based anode materials are attracting considerable attention for use in high-performance lithium-ion batteries, but the compromised energy density caused by high voltage plateaus and unsatisfactory capacities severely retards their practical applications. Herein, a molten-salt synthesis of Li 2 TiSiO 5 crystalline platelets and a subsequent selective facet modification by in situ growth of TiO 2 nanocrystal frames are facilely achieved. The discharge voltage plateau at around 0.5 V renders the Li 2 TiSiO 5 anode safe compared with graphite and confers a high energy density compared with zero-strain Li 4 Ti 5 O 12 anode. With the optimized size, structure, and content of modified TiO2 nanocrystals associated with the exposed (001) plane of Li 2 TiSiO 5 , the Li 2 TiSiO 5 -based anodes can deliver a capacity of above 300 mAh g -1 , enhanced rate performance, and a capacity retention of 66% after 10 000 cycles. In situ X-ray diffraction and ex situ transmission electron microscopy have demonstrated the structural stability of the anodes upon charge/discharge. Further theoretical calculation reveals 3D migration paths of Li + ions in Li 2 TiSiO 5 . The selective modification of in situ grown TiO2 nanocrystals on certain facets of crystallites opens a new door for the development of electrode materials possessing superior electrochemical properties.

anodes↗
DOE OSTI2021

IMEX Runge-Kutta Parareal for Non-diffusive Equations

Parareal is a widely studied parallel-in-time method that can achieve meaningful speedup on certain problems. However, it is well known that the method typically performs poorly on non-diffusive equations. This paper analyzes linear stability and convergence for IMEX Runge-Kutta Parareal methods on non-diffusive equations. By combining standard linear stability analysis with a simple convergence analysis, we find that certain Parareal configurations can achieve parallel speedup on non-diffusive equations. These stable configurations all posses low iteration counts, large block sizes, and a large number of processors. Numerical examples using the nonlinear Schrodinger equation demonstrate the analytical conclusions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Positive geometry, local triangulations, and the dual of the Amplituhedron

We initiate the systematic study of local positive spaces which arise in the context of the Amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory. We show that all local positive spaces relevant for one-loop MHV amplitudes are characterized by certain sign-flip conditions and are associated with surprisingly simple logarithmic forms. In the maximal sign-flip case they are finite one-loop octagons. Particular combinations of sign-flip spaces can be glued into new local positive geometries. These correspond to local pentagon integrands that appear in the local expansion of the MHV one-loop amplitude. We show that, geometrically, these pentagons do not triangulate the original Amplituhedron space but rather its twin “Amplituhedron-Prime”. This new geometry has the same boundary structure as the Amplituhedron (and therefore the same logarithmic form) but differs in the bulk as a geometric space. On certain two-dimensional boundaries, where the Amplituhedron geometry reduces to a polygon, we check that both spaces map to the same dual polygon. Interestingly, we find that the pentagons internally triangulate that dual space. This gives a direct evidence that the chiral pentagons are natural building blocks for a yet-to-be discovered dual Amplituhedron.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Large deformations of Tr(Φ 3 ) and the world at infinity

The amplitudes of the non-linear sigma model can be obtained from those of Tr(Φ 3 ) theory by sending the kinematic (Mandelstam) variables to infinity in a certain direction. In this paper we characterize the behavior of Tr(Φ 3 ) amplitudes under a general class of large kinematic shifts called g-vector shifts. The objects that live in this world at infinity retain certain key amplitude-like properties, most notably factorization, and admit descriptions in terms of polytopes, but they are not generally amplitudes of any cognizable theory. We identify particular g-vector shifts that lead at infinity to mixed amplitudes involving two pions and any number of scalars, allowing us to provide polytopal descriptions of these amplitudes.

effective field theories↗