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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 OSTI2024

Landau Singularities Revisited: Computational Algebraic Geometry for Feynman Integrals

We reformulate the analysis of singularities of Feynman integrals in a way that can be practically applied to perturbative computations in the standard model in dimensional regularization. After highlighting issues in the textbook treatment of Landau singularities, we develop an algorithm for classifying and computing them using techniques from computational algebraic geometry. We introduce an algebraic variety called the principal Landau determinant, which captures the singularities even in the presence of massless particles or UV/IR divergences. We illustrate this for 114 example diagrams, including a cutting-edge 2-loop 5-point nonplanar QCD process with multiple mass scales. Published by the American Physical Society 2024

Physics↗
DOE OSTI2025

Randomized algorithms for accelerating linear algebraic computations

The project supported the development of new methodologies for performing matrix computations that form key building blocks in modern scientific computing, such as low rank approximation of matrices, and efficient representations of global operators that arise in simulations of physical phenomena.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Electromagnetic and two-photon transition form factors of the pseudoscalar mesons: An algebraic model computation

We compute electromagnetic and two-photon transition form factors of ground-state pseudoscalar mesons: π , K , η c , η b . To this end, we employ an algebraic model based upon the coupled formalism of Schwinger-Dyson and Bethe-Salpeter equations. Within this approach, the dressed quark propagator and the relevant Bethe-Salpeter amplitude encode the internal structure of the corresponding meson. Electromagnetic properties of the meson are probed via the quark-photon interaction. The algebraic model employed by us unifies the treatment of all ground-state pseudoscalar mesons. Its parameters are carefully fitted performing a global analysis of existing experimental data including the knowledge of the charge radii of the mesons studied. We then compute and predict electromagnetic and two-photon transition form factors for a wide range of probing photon momentum-squared which is of direct relevance to the experimental observations carried out thus far or planned at different hadron physics facilities such as the Thomas Jefferson National Accelerator Facility (JLab) and the forthcoming Electron-Ion Collider. We also present comparisons with other theoretical models and approaches and lattice quantum chromodynamics. Published by the American Physical Society 2024

Higuera-Angulo, I. M. (ORCID:0000000256008875)↗
DOE OSTI2023

Rigorous Analytic Combinatorics in Several Variables in SageMath

We introduce the new [sage_acsv] package for the SageMath computer algebra system, allowing users to rigorously compute asymptotics for a large variety of multivariate sequences with rational generating functions. Using Sage's support for exact computations over the algebraic number field, this package provides the first rigorous implementation of algorithms from the theory of analytic combinatorics in several variables.

Hackl, Benjamin↗
DOE OSTI2024

Convexity, Squeezing, and the Elekes-Szabó Theorem

Here, this paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szabó Theorem in order to give new information. Namely, if we let $A \subset \mathbb R$, we prove that there exist $a,a' \in A$ such that\[\left | \frac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right | \gtrsim |A|^{31/12}.\]We are also able to prove that\[\max \{|A+ A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\]Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

SymPy: symbolic computing in Python

SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Moments of the Boltzmann collision operator for Coulomb interactions

Exact moments of the Boltzmann collision operator are calculated in the irreducible Hermitian moment expansion written in terms of the random-velocity variable of each species. The formulas are presented in closed, algebraic form and can be straightforwardly implemented in computer algebra systems. They are valid for two arbitrary masses, temperatures, and flow velocities, and hence include all other existing results derived for distribution functions expanded with respect to reference states of one temperature and flow velocity. In comparison, the Landau collisional moments are good approximations for large Coulomb logarithm and small relative flow velocity, but they fail to predict the correct behavior of most collisional moments for large relative flow even for weakly coupled plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Structures Computable in Polynomial Time. II

We consider a new approach to investigating categoricity of structures computable in polynomial time. The approach is based on studying polynomially computable stable relations. It is shown that this categoricity is equivalent to the usual computable categoricity for computable Boolean algebras with computable set of atoms, and for computable linear orderings with computable set of adjacent pairs. Examples are constructed which show that this does not always hold. We establish a connection between dimensions based on computable and polynomially computable stable relations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

On squares of representations of compact Lie algebras

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

MAGMA: Enabling exascale performance with accelerated BLAS and LAPACK for diverse GPU architectures

MAGMA (Matrix Algebra for GPU and Multicore Architectures) is a pivotal open-source library in the landscape of GPU-enabled dense and sparse linear algebra computations. With a repertoire of approximately 750 numerical routines across four precisions, MAGMA is deeply ingrained in the DOE software stack, playing a crucial role in high-performance computing. Notable projects such as ExaConstit, HiOP, MARBL, and STRUMPACK, among others, directly harness the capabilities of MAGMA. In addition, the MAGMA development team has been acknowledged multiple times for contributing to the vendors’ numerical software stacks. Looking back over the time of the Exascale Computing Project (ECP), we highlight how MAGMA has adapted to recent changes in modern HPC systems, especially the growing gap between CPU and GPU compute capabilities, as well as the introduction of low precision arithmetic in modern GPUs. We also describe MAGMA’s direct impact on several ECP projects. Maintaining portable performance across NVIDIA and AMD GPUs, and with current efforts toward supporting Intel GPUs, MAGMA ensures its adaptability and relevance in the ever-evolving landscape of GPU architectures.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Gradient Coding With Iterative Block Leverage Score Sampling

Gradient coding is a method for mitigating straggling servers in a centralized computing network that uses erasure-coding techniques to distributively carry out first-order optimization methods. Randomized numerical linear algebra uses randomization to develop improved algorithms for large-scale linear algebra computations. In this study, we propose a method for distributed optimization that combines gradient coding and randomized numerical linear algebra. The proposed method uses a randomized ℓ 2 -subspace embedding and a gradient coding technique to distribute blocks of data to the computational nodes of a centralized network, and at each iteration the central server only requires a small number of computations to obtain the steepest descent update. The novelty of our approach is that the data is replicated according to importance scores, called block leverage scores, in contrast to most gradient coding approaches that uniformly replicate the data blocks. Furthermore, we do not require a decoding step at each iteration, avoiding a bottleneck in previous gradient coding schemes. We show that our approach results in a valid ℓ 2 -subspace embedding, and that our resulting approximation converges to the optimal solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms

In recent three-loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short S-sums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the S-sums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation with respect to the external summation index and different multi-argument relations, for the compactification of S-sum expressions. Finally, we calculate algebraic relations for infinite S-sums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package HarmonicSums.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Twisted Supergravity and Koszul Duality: A Case Study in AdS 3

In this note, we study a simplified variant of the familiar holographic duality between supergravity on AdS 3 × S 3 × T 4 and the SCFT (on the moduli space of) the symmetric orbifold theory Sym N (T 4 ) as N → ∞. This variant arises conjecturally from a twist proposed by the first author and Si Li. We recover a number of results concerning protected subsectors of the original duality working directly in the twisted bulk theory. Moreover, we identify the symmetry algebra arising in the N → ∞ limit of the twisted gravitational theory. Further, we emphasize the role of Koszul duality—a ubiquitous mathematical notion to which we provide a friendly introduction—in field theory and string theory. After illustrating the appearance of Koszul duality in the “toy” example of holomorphic Chern-Simons theory, we describe how (a deformation of) Koszul duality relates bulk and boundary operators in our twisted setup, and explain how one can compute algebra OPEs diagrammatically using this notion. Further details, results, and computations will appear in a companion paper.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Algebraic model to study the internal structure of pseudoscalar mesons with heavy-light quark content

The internal structure of all lowest-lying pseudoscalar mesons with heavy-light quark content is studied in detail using an algebraic model that has been applied recently, and successfully, to the same physical observables of pseudoscalar and vector mesons with hidden-flavor quark content, from light to heavy quark sectors. The algebraic model consists on constructing simple and evidence-based of the meson’s Bethe-Salpeter amplitude (BSA) and quark’s propagator in such a way that the Bethe-Salpeter wave function (BSWF) can then be readily computed algebraically. Its subsequent projection onto the light front yields the light front wave function (LFWF) whose form allows us a simple access to the valence-quark parton distribution amplitude (PDA) by integrating over the transverse momentum squared. We exploit our current knowledge of the PDAs of lowest-lying pseudoscalar heavy-light mesons to compute their generalized parton distributions (GPDs) through the overlap representation of LFWFs. From these three dimensional knowledge, different limits/projections lead us to deduce the related parton distribution functions (PDFs), electromagnetic form factors (EFFs), and impact parameter space GPDs (IPS-GPDs). When possible, we make explicit comparisons with available experimental results and earlier theoretical predictions. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Genten: Software for Generalized Tensor Decompositions v. 1.0.0

Tensors, or multidimensional arrays, are a powerful mathematical means of describing multiway data. This software provides computational means for decomposing or approximating a given tensor in terms of smaller tensors of lower dimension, focusing on decomposition of large, sparse tensors. These techniques have applications in many scientific areas, including signal processing, linear algebra, computer vision, numerical analysis, data mining, graph analysis, neuroscience and more. The software is designed to take advantage of parallelism present emerging computer architectures such has multi-core CPUs, many-core accelerators such as the Intel Xeon Phi, and computation-oriented GPUs to enable efficient processing of large tensors.

Phipps, EricT.↗
DOE OSTI2019

Differential equations from unitarity cuts: nonplanar hexa-box integrals

We compute ϵ-factorized differential equations for all dimensionally-regularized integrals of the nonplanar hexa-box topology, which contribute for instance to 2-loop 5-point QCD amplitudes. A full set of pure integrals is presented. For 5-point planar topologies, Gram determinants which vanish in 4 dimensions are used to build compact expressions for pure integrals. Using unitarity cuts and computational algebraic geometry, we obtain a compact IBP system which can be solved in 8 hours on a single CPU core, overcoming a major bottleneck for deriving the differential equations. Alternatively, assuming prior knowledge of the alphabet of the nonplanar hexa-box, we reconstruct analytic differential equations from 30 numerical phase-space points, making the computation almost trivial with current techniques. We solve the differential equations to obtain the values of the master integrals at the symbol level. Full results for the differential equations and solutions are included as supplementary material.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

tapir: A tool for topologies, amplitudes, partial fraction decomposition and input for reductions

The demand for precision predictions in the field of high energy physics has dramatically increased over recent years. Experiments conducted at the LHC, as well as precision measurements at the intensity frontier such as Belle II require equally precise theoretical predictions to make full use of the acquired data. To match the experimental precision, second-, third- and, for certain quantities, even higher-order calculations in perturbative quantum field theory are required. To facilitate such calculations, computer software automating as many steps as possible is required. Yet, each calculation poses different challenges and thus, a high level of configurability is required. In this context we present tapir: a tool for identification, manipulation and minimization of Feynman integral families. It is designed to integrate in toolchains based on the computer algebra system FORM, the use of which is common practice in the field. tapir can be used to reduce the complexity of multi-loop problems with cut-filters, topology mapping, partial fraction decomposition and alike. Program Title:tapir CPC Library link to program files:https://doi.org/10.17632/ptc9t46xyn.1 Developer's repository link:https://gitlab.com/tapir-devs/tapir Licensing provisions: GPLv3 Programming language:python 3, C++ Nature of problem: Multi-loop computations require the automatization of a large number of different tasks related to Feynman integral topologies. Among them are the identification and minimization of integral topologies, partial fraction decomposition of topologies in the case of linearly dependent propagators as well as mapping scalar products of loop momenta to scalar functions. Solution method: The minimization of topologies is performed by comparison of their respective Nickel indices [1], even further minimization utilizes Pak's algorithm [2]. To efficiently map scalar products of loop momenta to scalar functions FORM [3] code is generated. Additional comments including restrictions and unusual features: Minimization based on Pak's algorithm slows down for many lines and scales. A coarser minimization using the Nickel indices, however, is still possible. [1]B. Nickel, D. Meiron, G.A.J. Baker, Compilation of 2-pt and 4-pt graphs for continuous spin model, Report, University of Guelph, 1977.[2]A. Pak, J. Phys. Conf. Ser. 368 (2012) 012049, https://doi.org/10.1088/1742-6596/368/1/012049, arXiv:1111.0868.[3]B. Ruijl, T. Ueda, J. Vermaseren, FORM version 4.2, arXiv:1707.06453, 7 2017.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2019

Generalized Bose–Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy

Highlights: • Hot topic: physics with non-Hermitian Hamiltonians. • Computer algebra used to construct new, next-to-BH models. • Degeneracy verification via variable-precision arithmetics. • Unfolding patterns at N>5 (recent problem solved). - Abstract: The method of construction of the tridiagonal and symmetric complex-matrix Hamiltonians H{sup (N)}(z) exhibiting an exceptional-point (EP) degeneracy of the Nth (i.e., maximal) order at a preselected parameter z=z{sup (EPN)}=1 is proposed and tested. In general, the implementation of the method requires the use of computer-assisted symbolic manipulations, especially at the larger matrix dimensions N. The well known PT−symmetric N-by-N-matrix Bose–Hubbard Hamiltonians as well as their recent N≤5 non-Bose–Hubbard alternatives are obtained as special solutions. Several other N≥6 non-Bose–Hubbard models H{sup (N)}(z) exhibiting the maximal EPN degeneracy are also constructed and analyzed in detail. In particular, their z−dependent, real or complex spectra of energies are displayed and discussed near as well as far from z{sup (EPN)}.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗