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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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NASA NTRS1999

Membrane-Based Functions in the Origin of Cellular Life

Our objective is to help explain how the earliest ancestors of contemporary cells (protocells) performed their essential functions employing only the molecules available in the protobiological milieu. Our hypothesis is that vesicles, built of amphiphilic, membrane-forming materials, emerged early in protobiological evolution and served as precursors to protocells. We further assume that the cellular functions associated with contemporary membranes, such as capturing and, transducing of energy, signaling, or sequestering organic molecules and ions, evolved in these membrane environments. An alternative hypothesis is that these functions evolved in different environments and were incorporated into membrane-bound structures at some later stage of evolution. We focus on the application of the fundamental principles of physics and chemistry to determine how they apply to the formation of a primitive, functional cell. Rather than attempting to develop specific models for cellular functions and to identify the origin of the molecules which perform these functions, our goal is to define the structural and energetic conditions that any successful model must fulfill, therefore providing physico-chemical boundaries for these models. We do this by carrying out large-scale, molecular level computer simulations on systems of interest.

Chipot, Christophe↗

DOE OSTI2020

Possibility of 14 C cluster as a building block of medium-mass nuclei

The possibility of the 14 C cluster being a basic building block of medium-mass nuclei is discussed. Although α cluster structures have been widely discussed in the light N ≈ Z mass region, the neutron-to-proton ratio deviates from unity in the nuclei near β-stability line and in neutron-rich nuclei. Thus, more neutron-rich objects with N > Z could become the building blocks of cluster structures in such nuclei. The 14 C nucleus is strongly bound and can be regarded as such a candidate. In addition, the path to the lowest shell-model configuration at short relative distances is closed for the 14 C+ 14 C structure, contrary to the case of the 12 C+ 12 C structure; this allows an appreciable separation distance between the 14 C clusters. The recent development of the antisymmetrized quasicluster model (AQCM) allows us to utilize a j j-coupling shell-model wave function for each cluster in a simplified way. The AQCM results for the 14 C+ 14 C structure in 28 Mg are compared with the ones of cranked relativistic mean field (CRMF) calculations. Although theoretical frameworks of these two models are quite different, they give similar results for the nucleonic densities and rotational properties of the structure under investigation. The existence of a linear chain three- 14 C cluster structure in 42 Ar has also been predicted in AQCM. Furthermore, these results confirm the role of the 14 C cluster as a possible building block of cluster structures in medium-mass nuclei.

20 ≤ A ≤ 38↗

NASA NTRS1987

Collision detection for spacecraft proximity operations

Collision Detection for Spacecraft Proximity Operations This thesis describes the development of a new collision detection algorithm to be used when two spacecraft are operating in the same vicinity. The two spacecraft are modelled as unions of convex polyhedra, where the polyhedron resulting from the union may be either convex or nonconvex. The relative motion of the two spacecraft is assumed to be such that one vehicle is moving with constant linear and angular velocity with respect to the other. The algorithm determines if a collision is possible and, if so, predicts the time when the collision will take place. The theoretical basis for the new collision detection algorithm is the C-function formulation of the configuration space approach recently introduced by researchers in robotics. Three different types of C-functions are defined that model the contacts between the vertices, edges, and faces of the polyhedra representing the two spacecraft. These C-functions are used to formulate three "collision" conditions. The first of these conditions limits the points representing potential collisions to the zeros of the C-functions. The new algorithm is fundamentally a search for the smallest zero of any C-function that satisfies the second and third collision conditions. The C-functions are shown to be transcendental functions of time for the assumed trajectory of the moving spacecraft. The zeros of these functions cannot be expressed in dosed form. Therefore, numerical search procedures are developed to find aLl of the zeros of a C-function in specified bounded intervals of time. These bounded intervals of time are found by examining the second and third collision conditions. The capabilities of the new algorithm are demonstrated for several example cases. These include examples of collisions determined by zeros of each of the three different types of C-functions. In addition to predicting the time of first contact of the polyhedra, the algorithm identifies the features of the two polyhedra that are touching at this time. The new collision detection algorithm is the first such algorithm that is capable of solving the collision detection problem exactly for the case where the moving object has constant linear and angular velocities. This is a significant improvement on previous collision detection algorithms described in the literature. In particular, the ability to handle constant angular velocity represents a more realistic type of rotational motion than those which have been used in other algorithms.

Robin M Vaughan↗

DOE OSTI2023

Search for Z' bosons decaying to pairs of heavy Majorana neutrinos in proton-proton collisions at $ \sqrt{s} $ = 13 TeV

A search for the production of pairs of heavy Majorana neutrinos (N ℓ ) from the decays of Z' bosons is performed using the CMS detector at the LHC. The data were collected in proton-proton collisions at a center-of-mass energy of $\sqrt s$ = 13 TeV, with an integrated luminosity of 138 fb –1 . The signature for the search is an excess in the invariant mass distribution of the final-state objects, two same-flavor leptons (e or μ) and at least two jets. No significant excess of events beyond the expected background is observed. Upper limits at 95% confidence level are set on the product of the Z' production cross section and its branching fraction to a pair of N ℓ , as functions of N ℓ and Z' boson masses (m N ℓ and m Z' , respectively) for m Z' from 0.4 to 4.6 TeV and m N ℓ from 0.1 TeV to m Z' /2. In the theoretical framework of a left-right symmetric model, exclusion bounds in the m N ℓ -m Z' plane are presented in both the electron and muon channels. The observed upper limit on m Z' reaches up to 4.42 TeV. These are the most restrictive limits to date on the mass of N ℓ as a function of the Z' boson mass.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1988

A knowledge-based approach to identification and adaptation in dynamical systems control

Artificial intelligence techniques are applied to the problems of model form and parameter identification of large-scale dynamic systems. The object-oriented knowledge representation is discussed in the context of causal modeling and qualitative reasoning. Structured sets of rules are used for implementing qualitative component simulations, for catching qualitative discrepancies and quantitative bound violations, and for making reconfiguration and control decisions that affect the physical system. These decisions are executed by backward-chaining through a knowledge base of control action tasks. This approach was implemented for two examples: a triple quadrupole mass spectrometer and a two-phase thermal testbed. Results of tests with both of these systems demonstrate that the software replicates some or most of the functionality of a human operator, thereby reducing the need for a human-in-the-loop in the lower levels of control of these complex systems.

Glass, B. J.↗

NASA NTRS2011

The Dolinar Receiver in an Information Theoretic Framework

Optical communication at the quantum limit requires that measurements on the optical field be maximally informative, but devising physical measurements that accomplish this objective has proven challenging. The Dolinar receiver exemplifies a rare instance of success in distinguishing between two coherent states: an adaptive local oscillator is mixed with the signal prior to photodetection, which yields an error probability that meets the Helstrom lower bound with equality. Here we apply the same local-oscillator-based architecture with aninformation-theoretic optimization criterion. We begin with analysis of this receiver in a general framework for an arbitrary coherent-state modulation alphabet, and then we concentrate on two relevant examples. First, we study a binary antipodal alphabet and show that the Dolinar receiver's feedback function not only minimizes the probability of error, but also maximizes the mutual information. Next, we study ternary modulation consistingof antipodal coherent states and the vacuum state. We derive an analytic expression for a near-optimal local oscillator feedback function, and, via simulation, we determine its photon information efficiency (PIE). We provide the PIE versus dimensional information efficiency (DIE) trade-off curve and show that this modulation and the our receiver combination performs universally better than (generalized) on-off keying plus photoncounting, although, the advantage asymptotically vanishes as the bits-per-photon diverges towards infinity.

optical communications↗

DOE OSTI2023

IDAES-PSE 2.0 Release

The Institute for the Design of Advanced Energy Systems (IDAES) Integrated Platform is a versatile computational environment offering extensive process systems engineering (PSE) capabilities for optimizing the design and operation of complex, interacting technologies and systems. IDAES enables users to efficiently search vast, complex design spaces to discover the lowest cost, most environmentally sustainable solutions while supporting the full process modeling lifecycle, from conceptual design to dynamic optimization and control. The extensible, open platform empowers users to create models of novel processes and rapidly develop custom analyses, workflows, and end-user applications. IDAES-PSE 2.0.0 Release Highlights Removal of deprecated features from IDAES v1 Update to Pyomo v6.5 – this required a number of updates to support the new NL solver writer and to address some changes in Pyomo Creation of new testing suite for backward compatibility, model robustness and verification More general implementation of the Helmholtz EoS. This brings some new features like standard property diagrams, choice of mass or mole basis, and new state variable options Standardizing names in Heat Exchanger models (breaking change from v2.0.0a2): Control Volumes named hot_side and cold_side Ports names hot_side_inlet, hot_side_outlet, cold_side_inlet and cold_side_outlet Config Blocks names hot_side_config and cold_side_config Config arguments for user provided names for each side: hot_side_name and cold_side_name. Updating Keras surrogate tool to use v1.1 of OMLT New prototype API for model initialization (idaes.core.initialization) The new API uses "Model Initializer" objects instead of class methods, allowing for the definition of multiple initialization routines for a single model A number of common, model agnostic initialization routines have also been defined, including initialization from data, block-decomposition and a general hierarchical approach equivalent to the existing method for common unit models New metadata for thermophysical properties – valid_range This can be used to record the range of values over which a property value can be trusted, such as the range of experimental data used to regress parameters A number of new utility functions have been added to check for properties with values outside the valid range and to set bounds based on this metadata Updated construction of balance expressions in Control Volumes to remove unneeded terms In the past, unneeded terms were added as a constant 0 term, however they will now be dropped entirely from the expression This was necessary due to more strict unit checking in the new Pyomo solver writer which no longer ignores 0 terms Updates to metadata for thermophysical properties to better define known properties and units of measurement This results in more strict enforcement of standard naming for thermophysical and reaction properties Users can still define custom properties, but these must be done explicitly using the define_custom_properties() method instead of being implicitly created by add_property() Updated convergence tester utility tool to support definition of benchmark files (JSON format) and comparison of performance to benchmarks Set default iteration limit for IPOPT in IDAES config to 200 iterations Update scaling of example models to work with new Pyomo NL solver writer Improve testing of extensions and examples infrastructure to avoid need for downloading files Updated distillation column to centralize common functionality and remove a number of Pyomo warnings

IDAES↗

DOE OSTI2025

QCD Constraints on Isospin-Dense Matter and the Nuclear Equation of State

Understanding the behavior of dense hadronic matter is a central goal in nuclear physics as it governs the nature and dynamics of astrophysical objects such as supernovae and neutron stars. Because of the nonperturbative nature of quantum chromodynamics (QCD), little is known rigorously about hadronic matter in these extreme conditions. Here, lattice QCD calculations are used to compute thermodynamic quantities and the equation of state of QCD over a wide range of isospin chemical potentials with controlled systematic uncertainties. Agreement is seen with chiral perturbation theory when the chemical potential is small. Comparison to perturbative QCD at large chemical potential allows for an estimate of the gap in the superconducting phase, and this quantity is seen to agree with perturbative determinations. Since the partition function for an isospin chemical potential μ I bounds the partition function for a baryon chemical potential μ B = 3 μ I / 2 , these calculations also provide rigorous nonperturbative QCD bounds on the symmetric nuclear matter equation of state over a wide range of baryon densities for the first time. Published by the American Physical Society 2025

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1992

Visualization and modeling of factors influencing visibility in computer-aided crewstation design

We have developed two modules for use in computer-aided design (CAD) of crewstation environments that enhance the designer's appreciation of factors influencing the pilot's vision and visual processing capacity. The Binocular Optics Module (BOM) is an interactive tool for visualizing geometric aspects of (1) how retinal imagery of the environment changes on the pilot's retinas under conditions of eye and object motion, and (2) how visual capabilities that can be modeled as regions or contours on the retinas, affect spatial perception of the environment. The Visual Performance Module (VPM) contains a signal processing model of human visual discrimination that quantitatively predicts visual discrimination performance. The outputs of the VPM are retinal contours that represent performance probabilities. These contours may be used as inputs to the BOM for visualizing those volumes of space within the crewstation that bound different levels of the pilot's of visual discrimination capability. Used together, the BOM and VPM provide the designer with the opportunity to interactively explore relationships between environmental retinal imagery and visual function, and the ability to factor the pilot's visual capabilities into the earliest phases of crewstation CAD.

Arditi, Aries↗

DOE OSTI2026

Augmenting subspace optimization methods with linear bandits

In this work, we consider the framework of methods for unconstrained minimization that are, in each iteration, restricted to a model that is only a valid approximation to the objective function on some affine subspace containing an incumbent point. These methods are of practical interest in computational settings where derivative information is either expensive or impossible to obtain. Recent attention has been paid in the literature to employing randomized matrix sketching for generating the affine subspaces within this framework. We consider a relatively straightforward, deterministic augmentation of such a generic subspace optimization method. In particular, we consider a sequential optimization framework where actions consist of one-dimensional linear subspaces and rewards consist of (approximations to) the magnitudes of directional derivatives computed in the direction of the action subspace. Reward maximization in this context is consistent with maximizing lower bounds on descent guaranteed by first-order Taylor models. This sequential optimization problem can be analysed through the lens of dynamic regret. We modify an existing linear upper confidence bound (UCB) bandit method and prove sublinear dynamic regret in the subspace optimization setting. We demonstrate the efficacy of employing this linear UCB method in a setting where forward-mode algorithmic differentiation can provide directional derivatives in arbitrary directions and in a derivative-free setting. For the derivative-free setting, we propose SS-POUNDers, an extension of the derivative-free optimization method POUNDers that employs the linear UCB mechanism to identify promising subspaces. Our numerical experiments suggest a preference, in either computational setting, for employing a linear UCB mechanism within a subspace optimization method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Evaluating Space Object Conjunction Probabilities Using Characteristic Function Inversion

This report discusses an approach to computing the probability of a conjunction between two space objects in the short-term encounter scenario. A conjunction is defined here as an event where the miss distance between the objects is less than some specified value. The scenario assumptions are that the motion of the objects is linear, their positions are Gaussian distributed, and their velocities are known and constant. Under these assumptions, the squared-miss distance is shown to have the generalized chi-square distribution. An established statistical technique called characteristic function inversion is employed to evaluate the distribution and obtain conjunction probabilities. The method is closely related to a recent approach based on moment generating function inversion, and a qualitative comparison of the approaches is provided. Last, the method is tested on several benchmark test cases where it agrees with numerical integration on the cases with conjunction probabilities above 10 –12 . However, the exact probability in these cases is usually not needed and this probability can be bounded above using an independent Gaussian approximation. Overall, the report shows how to compute conjunction probabilities using a standard statistical method, though numerical integration seems to perform equally well.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2008

Space Transportation System Availability Requirements and Its Influencing Attributes Relationships

It is essential that management and engineering understand the need for an availability requirement for the customer's space transportation system as it enables the meeting of his needs, goal, and objectives. There are three types of availability, e.g., operational availability, achieved availability, or inherent availability. The basic definition of availability is equal to the mean uptime divided by the sum of the mean uptime plus the mean downtime. The major difference is the inclusiveness of the functions within the mean downtime and the mean uptime. This paper will address tIe inherent availability which only addresses the mean downtime as that mean time to repair or the time to determine the failed article, remove it, install a replacement article and verify the functionality of the repaired system. The definitions of operational availability include the replacement hardware supply or maintenance delays and other non-design factors in the mean downtime. Also with inherent availability the mean uptime will only consider the mean time between failures (other availability definitions consider this as mean time between maintenance - preventive and corrective maintenance) that requires the repair of the system to be functional. It is also essential that management and engineering understand all influencing attributes relationships to each other and to the resultant inherent availability requirement. This visibility will provide the decision makers with the understanding necessary to place constraints on the design definition for the major drivers that will determine the inherent availability, safety, reliability, maintainability, and the life cycle cost of the fielded system provided the customer. This inherent availability requirement may be driven by the need to use a multiple launch approach to placing humans on the moon or the desire to control the number of spare parts required to support long stays in either orbit or on the surface of the moon or mars. It is the intent of this paper to provide the visibility of relationships of these major attribute drivers (variables) to each other and the resultant system inherent availability, but also provide the capability to bound the variables providing engineering the insight required to control the system's engineering solution. An example of this visibility will be the need to provide integration of similar discipline functions to allow control of the total parts count of the space transportation system. Also the relationship visibility of selecting a reliability requirement will place a constraint on parts count to achieve a given inherent availability requirement or accepting a larger parts count with the resulting higher reliability requirement. This paper will provide an understanding for the relationship of mean repair time (mean downtime) to maintainability, e.g., accessibility for repair, and both mean time between failure, e.g., reliability of hardware and the system inherent availability. Having an understanding of these relationships and resulting requirements before starting the architectural design concept definition will avoid considerable time and money required to iterate the design to meet the redesign and assessment process required to achieve the results required of the customer's space transportation system. In fact the impact to the schedule to being able to deliver the system that meets the customer's needs, goals, and objectives may cause the customer to compromise his desired operational goal and objectives resulting in considerable increased life cycle cost of the fielded space transportation system.

Rhodes, Russel E.↗

NASA NTRS2004

Trajectory Specification for High-Capacity Air Traffic Control

In the current air traffic management system, the fundamental limitation on airspace capacity is the cognitive ability of human air traffic controllers to maintain safe separation with high reliability. The doubling or tripling of airspace capacity that will be needed over the next couple of decades will require that tactical separation be at least partially automated. Standardized conflict-free four-dimensional trajectory assignment will be needed to accomplish that objective. A trajectory specification format based on the Extensible Markup Language is proposed for that purpose. This format can be used to downlink a trajectory request, which can then be checked on the ground for conflicts and approved or modified, if necessary, then uplinked as the assigned trajectory. The horizontal path is specified as a series of geodetic waypoints connected by great circles, and the great-circle segments are connected by turns of specified radius. Vertical profiles for climb and descent are specified as low-order polynomial functions of along-track position, which is itself specified as a function of time. Flight technical error tolerances in the along-track, cross-track, and vertical axes define a bounding space around the reference trajectory, and conformance will guarantee the required separation for a period of time known as the conflict time horizon. An important safety benefit of this regimen is that the traffic will be able to fly free of conflicts for at least several minutes even if all ground systems and the entire communication infrastructure fail. Periodic updates in the along-track axis will adjust for errors in the predicted along-track winds.

Paielli, Russell A.↗

DOE OSTI2021

Scalable, In-situ Data Clustering Data Analysis for Extreme Scale Scientific Computing (Final Report)

The objective of this project is to address challenges in the design and development of scalable in-situ data clustering and analytics algorithms and software. Our goal is to develop parallel software consisting of a set of spatio-temporal data clustering and anomaly detection functions, both of which are very important for large-scale analysis and have wide applicability for in-situ runs as well as post-processing analysis. Our design principles for in-situ analysis consider the following: (1) identify parts of the computation can be done close to the data within the nodes, while it is still in memory; (2) extract analysis components can (and should) be performed in remote staging and analysis nodes; (3) develop error-bound approximation methods for applications tolerable for small errors; (4) identify the type of derived distributions and statistics, for spatio-temporal data, that can be kept locally in order to both accelerate computations and meet energy constraints in subsequent iterations and phases; (5) use a self-describing data format so that data can be consistent and understood among local storage (memory and SSDs) and at staging and analysis nodes, thereby providing portability and flexibility; (6) develop service-oriented functions that can schedule in-situ and post-hoc analysis tasks based on the dynamic requirements of applications. Our development focus is to produce the parallel data analysis software/library that will be scalable, reusable, extensible, and generic for applications in different disciplines. The software will be able to run in-situ with the simulations as well as post-hoc analysis. This approach will satisfy many synergistic requirements for data intensive applications executed on data coming from instruments and experiments. In particular, the proposed multilevel approach is directly applicable to perform design tradeoffs for running part of the algorithms near the instruments and the rest on remote (analysis) systems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

gC1qR/C1qBP/HABP-1: Structural Analysis of the Trimeric Core Region, Interactions With a Novel Panel of Monoclonal Antibodies, and Their Influence on Binding to FXII

The protein gC1qR/C1qBP/HABP-1 plays an essential role in mitochondrial biogenesis, but becomes localized at the cellular surface in numerous pathophysiological states. When this occurs on endothelial cells, surface-exposed gC1qR activates the classical pathway of complement. It also promotes assembly of a multi-protein complex comprised of coagulation factor XII (FXII), pre-kallikrein (PK), and high-molecular weight kininogen (HMWK) that activates the contact system and the kinin-generating system. Since surface-exposed gC1qR triggers intravascular inflammatory pathways, there is interest in identifying molecules that block gC1qR function. Here we further that objective by reporting the outcome of a structure/function investigation of gC1qR, its interactions with FXII, and the impact of a panel of monoclonal anti-gC1qR antibodies on FXII binding to gC1qR. Although deletion mutants have been used extensively to assess gC1qR function, none of these proteins have been characterized structurally. To that end, we determined a 2.2 Å resolution crystal structure of a gC1qR mutant lacking both of its acidic loops, but which retained nanomolar-affinity binding to FXII and FXIIa. This structure revealed that the trimeric gC1qR assembly was maintained despite loss of roughly thirty residues. Characterization of a novel panel of anti-gC1qR monoclonal antibodies identified several with biochemical properties distinct from previously described antibodies, as well as one which bound to the first acidic loop of gC1qR. Intriguingly, we found that each of these antibodies could partly inhibit binding of FXII and FXIIa to gC1qR. Based on these results and previously published studies, we offer new perspectives for developing gC1qR inhibitors.

60 APPLIED LIFE SCIENCES↗

NASA NTRS1999

A complex ligase ribozyme evolved in vitro from a group I ribozyme domain

Like most proteins, complex RNA molecules often are modular objects made up of distinct structural and functional domains. The component domains of a protein can associate in alternative combinations to form molecules with different functions. These observations raise the possibility that complex RNAs also can be assembled from preexisting structural and functional domains. To test this hypothesis, an in vitro evolution procedure was used to isolate a previously undescribed class of complex ligase ribozymes, starting from a pool of 10(16) different RNA molecules that contained a constant region derived from a large structural domain that occurs within self-splicing group I ribozymes. Attached to this constant region were three hypervariable regions, totaling 85 nucleotides, that gave rise to the catalytic motif within the evolved catalysts. The ligase ribozymes catalyze formation of a 3',5'-phosphodiester linkage between adjacent template-bound oligonucleotides, one bearing a 3' hydroxyl and the other a 5' triphosphate. Ligation occurs in the context of a Watson-Crick duplex, with a catalytic rate of 0.26 min(-1) under optimal conditions. The constant region is essential for catalytic activity and appears to retain the tertiary structure of the group I ribozyme. This work demonstrates that complex RNA molecules, like their protein counterparts, can share common structural domains while exhibiting distinct catalytic functions.

NASA Discipline Exobiology↗

DOE OSTI2025

Multistage economic MPC for systems with a cyclic steady state: A gas network case study

Multistage model predictive control (MPC) provides a robust control strategy for dynamic systems with uncertainties and a setpoint tracking objective. Moreover, extending MPC to minimize an economic cost instead of tracking a pre-calculated optimal setpoint improves controller performance. This paper presents a novel multistage economic nonlinear model predictive control (E-NMPC) framework for dynamic systems operating under uncertainty, with specific application to natural gas transmission networks. A key innovation lies in the integration of cyclic steady-state (CSS) constraints within the multistage MPC formulation, enabling the controller to manage periodic operating conditions commonly observed in energy systems. A Lyapunov-based descent condition is enforced to ensure robust stability of the controller. The multistage economic MPC framework is validated on two gas pipeline case studies, where it successfully minimizes net energy consumption, respects operational constraints under uncertain demand profiles, and guides the network to optimal cyclic operation. The Lyapunov function remains bounded in both case studies, validating the robust stability of multistage E-NMPC.

03 NATURAL GAS↗

DOE OSTI2025

Five-point functions and the permutation group 𝑆 5

Five-point functions and five-body wave functions play an important role in many areas of nuclear and particle physics, e.g., in 2 →3 scattering processes, in the five-gluon vertex, or in the study of pentaquarks. In this work we consider the permutation group 𝑆 5 to facilitate the description of such objects. We work out the multiplets transforming under irreducible representations of 𝑆 5 and provide compact formulas allowing one to cast the permutations of an object 𝑓 12345 into combinations with definite permutation symmetry. We also give the explicit expressions for the irreducible multiplet products. We consider several practical applications as examples: We arrange the four-momenta and Lorentz invariants of a five-point function into the multiplet structure, we work out the color tensors of the five-gluon vertex in the multiplet notation, and we discuss applications for five-body wave functions like those of pentaquarks.

Bethe-Salpeter equation↗