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

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

A symmetry algebra in double-scaled SYK

The double-scaled limit of the Sachdev-Ye-Kitaev (SYK) model takes the number of fermions and their interaction number to infinity in a coordinated way. In this limit, two entangled copies of the SYK model have a bulk description of sorts known as the “chord Hilbert space”. We analyze a symmetry algebra acting on this Hilbert space, generated by the two Hamiltonians together with a two-sided operator known as the chord number. This algebra is a deformation of the JT gravitational algebra, and it contains a subalgebra that is a deformation of the \mathfrak{sl}_2 𝔰 𝔩 2 near-horizon symmetries. The subalgebra has finite-dimensional unitary representations corresponding to matter moving around in a discrete Einstein-Rosen bridge. In a semiclassical limit the discreteness disappears and the subalgebra simplifies to \mathfrak{sl}_2 𝔰 𝔩 2 , but with a non-standard action on the boundary time coordinate. One can make the action of \mathfrak{sl}_2 𝔰 𝔩 2 algebra more standard at the cost of extending the boundary circle to include some “fake” portions. Such fake portions also accommodate certain subtle states that survive the semi-classical limit, despite oscillating on the scale of discreteness. We discuss applications of this algebra, including sub-maximal chaos, the traversable wormhole protocol, and a two-sided OPE.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1989

Computer algebra and operators

The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.

Fateman, Richard↗
NASA NTRS1989

A spatial operator algebra for manipulator modeling and control

A recently developed spatial operator algebra, useful for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanizaton of specific algorithms is greatly simplified. The analytical formulation of the operator algebra, as well as its implementation in the Ada programming language are discussed.

Rodriguez, G.↗
NASA NTRS1993

The algebra of supertraces for 2+1 super de Sitter gravity

The algebra of the observables for 2+1 super de Sitter gravity, for one genus of the spatial surface is calculated. The algebra turns out to be an infinite Lie algebra subject to non-linear constraints. The constraints are solved explicitly in terms of five independent complex supertraces. These variables are the true degrees of freedom of the system and their quantized algebra generates a new structure which is referred to as a 'central extension' of the quantum algebra SU(2)q.

Urrutia, L. F.↗
NASA NTRS1993

Structured adaptive grid generation using algebraic methods

The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration of points to the grid redistribution scheme. The evaluation of the weighting mesh is accomplished by utilizing the weight function representing the solution variation and the equidistribution law. The selection of the weight function plays a key role in grid adaptation. A new weight function utilizing a properly weighted boolean sum of various flowfield characteristics is defined. The redistribution scheme is developed utilizing Non-Uniform Rational B-Splines (NURBS) representation. The application of NURBS representation results in a well distributed smooth grid by maintaining the fidelity of the geometry associated with boundary curves. Several algebraic methods are applied to smooth and/or nearly orthogonalize the grid lines. An elliptic solver is utilized to smooth the grid lines if there are grid crossings. Various computational examples of practical interest are presented to demonstrate the success of these methods.

Yang, Jiann-Cherng↗
NASA NTRS2017

Color Algebras

A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.

color constancy↗
DOE OSTI2023

Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras

Lie-Trotter-Suzuki product formulas are ubiquitous in quantum mechanics, computing, and simulations. Approximating exponentiated sums with such formulas are investigated in the JB-algebraic setting. We show that the Suzuki type approximation for exponentiated sums holds in JB-algebras, we give explicit estimation formulas, and we deduce three generalizations of Lie-Trotter formulas for arbitrary number elements in such algebras. In conclusion, we also extended the Lie-Trotter formulas in a Jordan Banach algebra from three elements to an arbitrary number of elements.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Deformation of Lie–Poisson algebras and chirality

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, -λ and $\barλ$ (hence, -$\barλ$) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra s o ( 3 ) , yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Algebraic branch points at all loop orders from positive kinematics and wall crossing

A bstract There is a remarkable connection between the boundary structure of the positive kinematic region and branch points of integrated amplitudes in planar $$ \mathcal{N} $$ N = 4 SYM. A long-standing question has been precisely how algebraic branch points emerge from this picture. We use wall crossing and scattering diagrams to systematically study the boundary structure of the positive kinematic regions associated with MHV amplitudes. The notion of asymptotic chambers in the scattering diagram naturally explains the appearance of algebraic branch points. Furthermore, the scattering diagram construction also motivates a new coordinate system for kinematic space that rationalizes the relations between algebraic letters in the symbol alphabet. As a direct application, we conjecture a complete list of all algebraic letters that could appear in the symbol alphabet of the 8-point MHV amplitude.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

In this paper we present a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of the solidification dynamics.

42 ENGINEERING↗
DOE OSTI2020

LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory

In this paper, we present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. The basic procedure is unchanged—it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible representations are included in the supplementary material for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Data assimilation in operator algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

PyAMG: Algebraic Multigrid Solvers in Python

PyAMG is a Python package of algebraic multigrid (AMG) solvers and supporting tools for approximating the solution to large, sparse linear systems of algebraic equations, Ax = b, where A is an n × n sparse matrix. Sparse linear systems arise in a range of problems in science, from fluid flows to solid mechanics to data analysis. While the direct solvers available in SciPy’s sparse linear algebra package (scipy.sparse.linalg) are highly efficient, in many cases iterative methods are preferred due to overall complexity. However, the iterative methods in SciPy, such as CG and GMRES, often require an efficient preconditioner in order to achieve a lower complexity. Preconditioning is a powerful tool whereby the conditioning of the linear system and convergence rate of the iterative method are both dramatically improved. PyAMG constructs multigrid solvers for use as a preconditioner in this setting. A summary of multigrid and algebraic multigrid solvers can be found in Olson (2015a), in Olson (2015b), and in Falgout (2006); a detailed description can be found in Briggs et al. (2000) and Trottenberg et al. (2001).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Algebraic expressions for estimating the impact depths of a surface barrier over a homogeneous soil

Engineered surface barriers are used to isolate subsurface contaminants for effective long-term containment of municipal solid waste, other nonhazardous solid and liquid waste, hazardous and toxic wastes, and radioactive waste. The impact depths of a surface barrier are affected by the pre-barrier recharge rate and the properties of the soil beneath the barrier. In this paper, the pore-size-specific (PSS) water velocity is defined and an algebraic expression of PSS velocity is derived based on the stream tube concept and the Brooks and Corey hydraulic retention model. Algebraic expressions are developed to estimate drainage velocities and barrier impact depths after the emplacement of a surface barrier. Four impact depth terms are used to convey the protective effect: drainage front, average drainage, the location with 50% impact, and drainage tail. The drainage front depth is the deepest point at which the barrier has a detectable impact at a specific time (also called the near zero-impact depth). The average-impact depth is the depth at which average drainage occurs. At the 50% impact depth, the water flux rate is reduced by half because of the surface barrier. Lastly, the drainage tail depth (also called the full-impact depth) is the deepest depth at which the water conditions above it are in equilibrium with the barrier. The algebraic expressions show that the average-impact depth is no more than 1/3 of near zero-impact depth, while the 50% impact depth is slightly larger than 1/2 of the near zero-impact depth. The full-impact depth, depending on the final recharge rate from the surface barrier, is usually much smaller than the other impact depths. These differences lead to a very large transition zone beneath a surface barrier. Numerical simulations were conducted to replicate the same conditions. The numerical results corroborated the analytical models by predicting very similar water content profiles and near zero-, average-, 50%, and full-impact depths. The algebraic expressions provided in this paper are useful for quickly identifying sites where the depth of the existing contaminants could be beyond the protection of a surface barrier.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

TAMM: Tensor algebra for many-body methods

Tensor algebra operations such as contractions in computational chemistry consume a significant fraction of the computing time on large-scale computing platforms. The widespread use of tensor contractions between large multi-dimensional tensors in describing electronic structure theory has motivated the development of multiple tensor algebra frameworks targeting heterogeneous computing platforms. In this paper, we present Tensor Algebra for Many-body Methods (TAMM), a framework for productive and performance-portable development of scalable computational chemistry methods. TAMM decouples the specification of the computation from the execution of these operations on available high-performance computing systems. With this design choice, the scientific application developers (domain scientists) can focus on the algorithmic requirements using the tensor algebra interface provided by TAMM, whereas high-performance computing developers can direct their attention to various optimizations on the underlying constructs, such as efficient data distribution, optimized scheduling algorithms, and efficient use of intra-node resources (e.g., graphics processing units). The modular structure of TAMM allows it to support different hardware architectures and incorporate new algorithmic advances. We describe the TAMM framework and our approach to the sustainable development of scalable ground- and excited-state electronic structure methods. We present case studies highlighting the ease of use, including the performance and productivity gains compared to other frameworks.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Quantum state reduction: Generalized bipartitions from algebras of observables

Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. In our work, we demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Asymptotic symmetry algebra of N = 8 supergravity

The asymptotic symmetry algebra of N = 1 supergravity was recently constructed using the well-known two-dimensional celestial conformal field (CFT) theory technique [A. Fotopoulos , ]. In this paper, we extend the construction to the maximally supersymmetric four-dimensional N = 8 supergravity theory in asymptotically flat spacetime and construct the extended asymptotic symmetry algebra, which we call N = 8 s b m s 4 . We use the celestial CFT technique to find the appropriate currents for extensions of N = 8 super-Poincaré and SU ( 8 ) R R-symmetry current algebra on the celestial sphere C S 2 . We generalize the definition of shadow transformations and show that there is infinite dimensional extension of the global SU ( 8 ) R algebra in the theory. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
DOE OSTI2024

Algebraic structures in closed superstring field theory, homotopy transfer, and effective actions

A consistent action for heterotic and type II superstring field theory was recently proposed by Sen. We give an algebraic formulation of this action in terms of certain twisted L ∞ algebra. We further show that Sen’s Wilsonian effective superstring field action can be obtained using homotopy transfer and the effective theory also possesses the algebraic structure of a twisted L ∞ algebra. Published by the American Physical Society 2024

Singh, Ranveer Kumar↗