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Results for “Algebraic Geometry”

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 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1977

Applications of Algebraic Geometry to Systems Theory

Basic theorems of algebraic geometry are applied to prove some pole-placement theorems, including an improved version of pole placement with output feedback. Examples are given which show the limitations of the algebro-geometric theorems and their potential value for systems theory. This paper and those to follow might contribute towards making the powerful theorems of modern algebraic geometry accessible and applicable to problems of engineering.

Hermann, Robert↗
NASA NTRS1976

A new application of algebraic geometry to systems theory

Following an introduction to algebraic geometry, the dominant morphism theorem is stated, and the application of this theorem to systems-theoretic problems, such as the feedback problem, is discussed. The Gaussian elimination method used for solving linear equations is shown to be an example of a dominant morphism.

Martin, C. F.↗
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↗
NASA NTRS1979

On the algebraic geometry of the output feedback pole placement map

Pole placement by output feedback is based on a concept that a transfer function can be considered as a curve in a Grassmanian manifold, so that pole placement consists of finding a special hypersurface in the manifold which intersects the curve at a prescribed set of points. An equation is derived for the number of different gains which yield the same set of poles in the case where the number of output feedback gains equals the number of poles of the system.

Brockett, R. W.↗
NASA NTRS1983

On families of systems and deformations

Until recently, there has not been any systematic effort towards a theory of systems with parameter variations. It is presently suggested that the concept of families of systems is basic to such an effort, and entails the techniques of Lie theory, differential geometry, and algebraic geometry. Attention is given to the geometric characterization of certain families of systems that appear in control and identification problems. The ways in which families of systems degenerate as parameter variations become large are isolated, using the topological rather than algebraic geometric case.

Krishnaprasad, P. S.↗
DOE OSTI2021

Machine learning and algebraic approaches towards complete matter spectra in 4d F-theory

Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To quantify jumps of these cohomologies, we first generate 1.8 million pairs of line bundles and curves embedded in dP 3 , for which we compute the cohomologies. A white-box machine learning approach trained on this data provides intuition for jumps due to curve splittings, which we use to construct additional vector-like Higgs-pairs in an F-Theory toy model. We also find that, in order to explain quantitatively the full dataset, further tools from algebraic geometry, in particular Brill-Noether theory, are required. Using these ingredients, we introduce a diagrammatic way to express cohomology jumps across the parameter space of each family of matter curves, which reflects a stratification of the F-theory complex structure moduli space in terms of the vector-like spectrum. Furthermore, these insights provide an algorithmically efficient way to estimate the possible cohomology dimensions across the entire parameter space.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1977

Linear systems with structure group and their feedback invariants

A general method described by Hermann and Martin (1976) for the study of the feedback invariants of linear systems is considered. It is shown that this method, which makes use of ideas of topology and algebraic geometry, is very useful in the investigation of feedback problems for which the classical methods are not suitable. The transfer function as a curve in the Grassmanian is examined. The general concepts studied in the context of specific systems and applications are organized in terms of the theory of Lie groups and algebraic geometry. Attention is given to linear systems which have a structure group, linear mechanical systems, and feedback invariants. The investigation shows that Lie group techniques are powerful and useful tools for analysis of the feedback structure of linear systems.

Martin, C.↗
NASA NTRS1993

On curve veering and flutter of rotating blades

The eigenvalues of rotating blades usually change with rotation speed according to the Stodola-Southwell criterion. Under certain circumstances, the loci of eigenvalues belonging to two distinct modes of vibration approach each other very closely, and it may appear as if the loci cross each other. However, our study indicates that the observable frequency loci of an undamped rotating blade do not cross, but must either repel each other (leading to 'curve veering'), or attract each other (leading to 'frequency coalescence'). Our results are reached by using standard arguments from algebraic geometry--the theory of algebraic curves and catastrophe theory. We conclude that it is important to resolve an apparent crossing of eigenvalue loci into either a frequency coalescence or a curve veering, because frequency coalescence is dangerous since it leads to flutter, whereas curve veering does not precipitate flutter and is, therefore, harmless with respect to elastic stability.

Afolabi, Dare↗
NASA NTRS2022

A Survey of Mathematical Structures for Lunar Networks

To sustain the current and increasing accessibility of space, a scalable communications infrastructure (i.e. the Solar System Internet, SSI) is necessary. The goal of this paper is to begin the discovery of the fundamental underlying mathematical structure of space networks to help the research community harness these structures for algorithm development and optimization. To ensure the applicability of the research, the approaches are considered through the lens of simulated scenarios inspired by the Artemis Back-to-the-Moon mission set for 2024. We note that any approach to an SSI must fit under the umbrella of Delay Tolerant Networking (DTN), due to celestial mobility, high link latencies, high variance in link latencies, disconnections, lack of end-to-end paths, and so on. These difficulties are exacerbated by the fact that the underlying structure of a space network is a time-evolving network and may experience multiple discontinuities in its topology. In this paper we propose several novel approaches to a mathematical foundation for Delay Tolerant Networking Theory that fall outside the traditional scope of temporal network theory. These techniques include methods from Topological Data Analysis, Dynamic Graph Analysis, Applied Algebraic Geometry, Probability Theory, and Game Theory. Some of these methods include tools adapted to the study of dynamic metric spaces, such as zigzag persistent homology and their higher parameter analogs. We find that several of these methods target desired engineering outcomes such as discovery and automatic sub-netting. While each approach is theoretical, they are also algorithmic in nature and offer immediate practical applications. The paper concludes with comparisons of the various methods along with suggestions for future work.

Delay tolerant networking↗
NASA NTRSDate not supplied

Advances in Modeling Solar System Internet Structures and their Data Flows

With an ever-increasing presence in space, there is also an increasing burden on existing communications infrastructure. We are heading towards an inflection point where the traditional approach of scheduled, single-path communications for space will no longer be viable. One answer is Delay Tolerant Networking (DTN), which takes the once disparate system of point-to-point links and unifies them in a networked architecture, thereby making communications more scalable. However, much work remains for discovering and harnessing the underlying theory of DTN. For example, in the terrestrial setting the interplay between routing domains is well-understood, however this is not the case in DTNs. In this paper, we build up the fundamental foundations of DTN, with an emphasis on modeling time varying networks and data flows across them, with examples of cross-domain routing in a DTN. A lofty goal of DTN is to enable the so-called Solar System Internet (SSI), which implies a standardized and robust suite of protocols. These protocols include routing across disconnected networks using store, carry, and forward mechanisms, which is necessary due to the disconnections, delays, and mobility intrinsic to space networks. Due to these factors, each of which generalize traditional networking, there is a deep and rich theory of DTNs. Here we build off of past successes to broaden this theory while striving to keep actionable results a goal for future implementations and operations. The approach includes modeling the unicast, broadcast, and multicast communications using the language of hypergraphs, which capture the geometric properties of such networked communications algebraically. Also inherent to these networks is their time-varying nature, particularly given mobility, and hence we also cultivate modeling techniques that respect this time dependence. This leads us to develop models using tools from category theory and algebraic geometry, which provide a language well-suited to describing synchronization and optimization over such networks. We also introduce and study a novel generalization of curvature applicable to time-evolving networks, which provides quantitative controls on diffusion processes on the network. Because an interplanetary network would feature links with propagation delays the preclude discovery (feedback) mechanisms, they will always feature a scheduled component. However, it is beneficial to support discovery where possible. While DTNs do not yet have strong definitions for their analogues of autonomous systems or network areas, we show how to join dynamic and schedule-based routing domains, using the language of sheaves, which marks progress towards such definitions. We conclude with a discussion of the progress made, as well as suggestions for future work.

Delay Tolerant Networking↗
NASA NTRS1985

Space Mathematics: A Resource for Secondary School Teachers

A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.

Kastner, Bernice↗
DOE OSTI2024

Effective rationality for local unitary invariants of mixed states of two qubits

Abstract We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.

Physics↗
NASA NTRSDate not supplied

Formal Verification of the Interaction Between Semi-Algebraic Sets and Real Analytic Functions

Semi-algebraic sets and real analytic functions are fundamental concepts in Real Algebraic Geometry and Real Analysis, respectively. These concepts interact in the study of Differential Equations, where the real analytic solution to a differential equation is known to enter or exit a semi-algebraic set in a predicable way. Motivated to enhance the capability to reason about differential equations in the Prototype Verification System (PVS), a formalization of multivariate polynomials, semi-algebraic sets, and real analytic functions is developed. The favorable way that a real analytic function enters and exits a semi-algebraic set is proven. It is further shown that if the function is assumed to be smooth, a slightly weaker assumption than real analytic, these favorable interactions with semi-algebraic sets may fail.

Real analytic functions↗
NASA NTRS1983

On the stabilizability of multivariable systems by minimum order compensation

In this paper, a derivation is provided of the necessary condition, mp equal to or greater than n, for stabilizability by constant gain feedback of the generic degree n, p x m system. This follows from another of the main results, which asserts that generic stabilizability is equivalent to generic solvability of a deadbeat control problem, provided mp equal to or less than n. Taken together, these conclusions make it possible to make some sharp statements concerning minimum order stabilization. The techniques are primarily drawn from decision algebra and classical algebraic geometry and have additional consequences for problems of stabilizability and pole-assignability. Among these are the decidability (by a Sturm test) of the equivalence of generic pole-assignability and generic stabilizability, the semi-algebraic nature of the minimum order, q, of a stabilizing compensator, and the nonexistence of formulae involving rational operations and extraction of square roots for pole-assigning gains when they exist, answering in the negative a question raised by Anderson, Bose, and Jury (1975).

Byrnes, C. I.↗
DOE OSTI2022

Vector boson pair production at one loop: analytic results for the process $q\bar{q}$$ℓ\bar{ℓ}$$ℓ'\bar{ℓ}'g$

We present compact analytic results for the one-loop amplitude for the process 0 → $q\bar{q}$$ℓ\bar{ℓ}$$ℓ'\bar{ℓ}'g$, relevant for both the production of a pair of Z and W-bosons in association with a jet. We focus on the gauge-invariant contribution mediated by a loop of quarks. We explicitly include all effects of the loop-quark mass m, appropriate for the production of a pair of Z-bosons. In the limit m → 0, our results are also applicable to the production of W-boson pairs, mediated by a loop of massless quarks. Implemented in a numerical code, the results are fast. The calculation uses novel advancements in spinor-helicity simplification techniques, for the first time applied beyond five-point massless kinematics. We make use of primary decompositions from algebraic-geometry, which now involve non-radical ideals, and p-adic numbers from number theory. We show how to infer whether numerator polynomials belong to symbolic powers of non-radical ideals through numerical evaluations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1995

Application of Navier-Stokes code PAB3D with kappa-epsilon turbulence model to attached and separated flows

A three-dimensional Navier-Stokes solver was used to determine how accurately computations can predict local and average skin friction coefficients for attached and separated flows for simple experimental geometries. Algebraic and transport equation closures were used to model turbulence. To simulate anisotropic turbulence, the standard two-equation turbulence model was modified by adding nonlinear terms. The effects of both grid density and the turbulence model on the computed flow fields were also investigated and compared with available experimental data for subsonic and supersonic free-stream conditions.

Abdol-Hamid, Khaled S.↗