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Results for “EUCLIDEAN SPACE”

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 NTRS1994

Mimicking within Euclidean space a cosmological time dilation of gamma-ray burst durations

If gamma-ray burst sources are cosmological in origin, then the time dilation at large z can correlate a burst's duration with its peak flux. Detection of this effect is thought by many to be strong evidence for a cosmological burst origin. In this Letter I show that an apparent time distortion--either a dilation or contraction--is generally expected for an ensemble of bursts that is spatially limited within Euclidean space. The appearance of this effect is correlated with the falling away of the log N-log P curve from a -3/2 slope line. An example of this effect is provided by the relativistic bulk motion model, which produces a strong time dilation when spatially limited in Euclidean space. As a consequence, envidence that weak bursts have longer durations than strong bursts is not evidence of a cosmological burst origin.

Brainerd, J. J.↗
NASA NTRS1980

Numerical generation of two-dimensional orthogonal curvilinear coordinates in an Euclidean space

A noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves was developed. The basic generating equation is the Gaussian equation for an Euclidean space which is solved analytically. The method is applied in many cases and these test results demonstrate that the proposed method can be readily applied to a wide variety of problems. The method can also be used for simply connected regions only by obtaining the solution of the linear equation under the changed boundary conditions.

Warsi, Z. U. A.↗
DOE OSTI2025

Generalized fiducial inference on differentiable manifolds

We introduce a novel approach to inference on parameters that take values in a Riemannian manifold embedded in a Euclidean space. Parameter spaces of this form are ubiquitous across many fields, including chemistry, physics, computer graphics, and geology. Here, this new approach uses generalized fiducial inference (GFI) to obtain a posterior-like distribution on the manifold, without needing to know local parameterizations that map to the constrained space from an unconstrained Euclidean space. Using mathematical tools from Riemannian geometry, we construct a constrained generalized fiducial distribution (CGFD). A Bernstein-von Mises-type result for the CGFD, which provides intuition for how the desirable asymptotic qualities of the unconstrained generalized fiducial distribution are inherited by the CGFD, is provided. To illustrate the practical use of the CGFD, we provide a proof-of-concept example in the context of a linear logspline density estimation problem, and demonstrate that CGFD-based confidence sets exhibit desirable coverage properties via simulation. As an application, we fit a CGFD to COVID-19 case count data from North Carolina, USA.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1989

Extension of Euler's theorem to n-dimensional spaces

Euler's theorem states that any sequence of finite rotations of a rigid body can be described as a single rotation of the body about a fixed axis in three-dimensional Euclidean space. The usual statement of the theorem in the literature cannot be extended to Euclidean spaces of other dimensions. Equivalent formulations of the theorem are given and proved in a way which does not limit them to the three-dimensional Euclidean space. Thus, the equivalent theorems hold in other dimensions. The proof of one formulation presents an algorithm which shows how to compute an angular-difference matrix that represents a single rotation which is equivalent to the sequence of rotations that have generated the final n-D orientation. This algorithm results also in a constant angular velocity which, when applied to the initial orientation, eventually yields the final orientation regardless of what angular velocity generated the latter. The extension of the theorem is demonstrated in a four-dimensional numerical example.

Bar-Itzhack, Itzhack Y.↗
NASA NTRS1989

Extension to Eulers's theorem to n-dimensional spaces

Euler's theorem states that any sequence of finite rotations of a rigid body can be described as a single rotation of the body about a fixed axis in three-dimensional Euclidean space. The usual statement of the theorem in the literature cannot be extended to Euclidean spaces of other dimensions. Equivalent formulations of the theorem are given in this paper and proven in a way which does not limit them to the three-dimensional Euclidean space. Thus, the equivalent theorems hold in other dimensions. The proof of one formulation presents an algorithm which shows how to compute an angular-difference matrix that represents a single rotation which is equivalent to the sequence of rotations that have generated the final n-D orientation. This algorithm results also in a constant angular-velocity which, when applied to the initial orientation, yields eventually the final orientation regardless of what angular velocity generated the latter. Finally, the extension of the theorem is demonstrated in a four-dimensional numerical example.

Bar-Itzhack, Itzhack Y.↗
NASA NTRS1985

Generation of Surface Coordinates by Elliptic Partial Differential Equations

The problem of generating spatial coordinates by numerical methods through carefully selected mathematical models is of current interest both in mechanics and physics. The problem of generation of a desired system of coordinates in a given surface was considered, which essentially is an effort directed to the problem of grid generation in a two-dimensional non-Euclidean space. The mathematical model selected for this purpose is based on the formulae of Gauss for a surface. The proposed equations can be used to generate a new coordinate system from the data of an already given coordinate system in a surface. If the coefficients of the first and second fundamental forms have been given, then the proposed equations can be used to generate a surface satisfying the given data (surface fitting). The proposed equations can also be used to generate surfaces in the space between two arbitrary given surfaces, thus providing 3D grids in an Euclidean space.

Warsi, Z. U. A.↗
NASA NTRS1989

Querying databases of trajectories of differential equations: Data structures for trajectories

One approach to qualitative reasoning about dynamical systems is to extract qualitative information by searching or making queries on databases containing very large numbers of trajectories. The efficiency of such queries depends crucially upon finding an appropriate data structure for trajectories of dynamical systems. Suppose that a large number of parameterized trajectories gamma of a dynamical system evolving in R sup N are stored in a database. Let Eta is contained in set R sup N denote a parameterized path in Euclidean Space, and let the Euclidean Norm denote a norm on the space of paths. A data structure is defined to represent trajectories of dynamical systems, and an algorithm is sketched which answers queries.

Grossman, Robert↗
DOE OSTI2024

Riemannian Optimization Applied to AC Optimal Power Flow: Preprint

The nonlinear, nonconvex AC optimal power flow problem is of growing importance as the nature of the power grid evolves. This problem can be difficult to solve for interior point methods. However, the advent of optimization algorithms over smooth Riemannian manifolds presents an alternative approach. The nonlinear, nonconvex constraints in the AC power flow problem form an embedded submanifold of Euclidean space. In this paper, the authors explore the performance of Riemannian optimization algorithms for the ACOPF problem where the optimization is performed directly on the AC power flow manifold. They demonstrate that these are viable computational alternatives to interior point methods. This is done by using Julia and the packages PowerModels.jl and Manopt.jl.

manifold optimization↗
NASA NTRS2000

The Intensity Distribution of Faint Gamma-Ray Bursts Detected with BATSE

We have recently completed a search of six years of archival Burst and Transient Source Experiment (BATSE) data for gamma-ray bursts (GRBs) that were too faint to activate the real-time burst detection system running on board the spacecraft. These 'nontriggered' bursts can be combined with the 'triggered' bursts detected on board to produce a GRB intensity distribution that reaches peak fluxes a factor of approximately two lower than could be studied previously. The value of the statistic (in Euclidean space) for the bursts we detect is 0.177 +/- 0.006. This surprisingly low value is obtained because we detected very few bursts on the 4.096 s and 8.192 s timescales (where most bursts have their highest signal-to-noise ratio) that were not already detected on the 1.024 s timescale. If allowance is made for a power-law distribution of intrinsic peak luminosities, the extended peak flux distribution is consistent with models in which the redshift distribution of the gamma-ray burst rate approximately traces the star formation history of the universe. We argue that this class of models is preferred over those in which the burst rate is independent of redshift. We use the peak flux distribution to derive a limit of 10% (99% confidence) on the fraction of the total burst rate that could be contributed by a spatially homogeneous (in Euclidean space) subpopulation of burst sources, such as type Ib/c supernovae. These results lend support to the conclusions of previous studies predicting that relatively few faint 'classical' GRBs will be found below the BATSE onboard detection threshold.

Kommers, Jefferson M.↗
NASA NTRS2000

The Intensity Distribution of Faint Gamma-Ray Bursts Detected with BATSE

We have recently completed a search of 6 years of archival BATSE data for gamma-ray bursts (GRBS) that were too faint to activate the real-time burst detection system running on board the spacecraft. These "nontriggered" bursts can be combined with the "triggered" bursts detected on board to produce a GRB intensity distribution that reaches peak fluxes a factor of approximately 2 lower than could be studied previously. The value of the statistic (in Euclidean space) for the bursts we detect is 0.177 plus or minus 0.006. This surprisingly low value is obtained because we detected very few bursts on the 4.096 s and 8.192 s timescales (where most bursts have their highest signal-to-noise ratio) that were not already detected on the 1.024 s timescale. If allowance is made for a power-law distribution of intrinsic peak luminosities, the extended peak flux distribution is consistent with models in which the redshift distribution of the gamma-ray burst rate approximately traces the star formation history of the universe. We argue that this class of models is preferred over those in which the burst rate is independent of redshift. We use the peak flux distribution to derive a limit of 10% (99% confidence) on the fraction of the total burst rate that could be contributed by a spatially homogeneous (in Euclidean space) subpopulation of burst sources, such as type lb/c supernovae. These results lend support to the conclusions of previous studies predicting that relatively few faint "classical" GRBs will be found below the BATSE onboard detection threshold.

Kommers, Jefferson M.↗
NASA NTRS1985

A Factorization Approach to the Linear Regulator Quadratic Cost Problem

A factorization approach to the linear regulator quadratic cost problem is developed. This approach makes some new connections between optimal control, factorization, Riccati equations and certain Wiener-Hopf operator equations. Applications of the theory to systems describable by evolution equations in Hilbert space and differential delay equations in Euclidean space are presented.

Milman, M. H.↗
NASA NTRS2021

Development of an SE(3)-based Rigid Body Pose Estimation Scheme for Unknown Moments of Inertia in a Spacecraft

In real-world applications, the consideration of state and parameter uncertainties is an important part of any navigation and control system. Uncertainties can arise because of the instruments, communication systems, or external disturbances. Parameter uncertainty can be due to uncertainties in the data or the calibration process used. The objective of this research is to develop a rigid body pose estimation scheme on nonlinear manifolds of rigid body motion groups for unknown mass properties moments of inertia with application in space launch. An unscented Kalman filter is developed on special Euclidean space and their tangent bundle to address uncertainties in the states and inertia properties. This estimator accounts for uncertainties, while considering rotational-translational coupling, and avoids singularity or non-uniqueness issues. Since this estimator is based on the nonlinear manifolds on which rigid body motion evolves, it is reliable and is expected to result in high level of accuracy. The estimators are validated on a generic model of Logistic Module (LM) and a cuboid spacecraft model. The algorithms can then be tested on Dragon XL, HTV-X, and possibly other spacecraft.

Moments of Inertia↗
DOE OSTI2025

A Class of Sparse Johnson–Lindenstrauss Transforms and Analysis of their Extreme Singular Values

The Johnson–Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be projected into lower dimensions while approximately preserving pairwise Euclidean distances. Random matrices satisfying this lemma are called JL transforms (JLTs). Inspired by existing $s$-hashing JLTs with exactly $s$ nonzero elements on each column, the present work introduces an ensemble of sparse matrices encompassing so-called $s$-hashing-like matrices whose expected number of nonzero elements on each column is $s$. The independence of the sub-Gaussian entries of these matrices and the knowledge of their exact distribution play an important role in their analyses. Using properties of independent sub-Gaussian random variables, these matrices are demonstrated to be JLTs, and their smallest nontrivial singular values and largest singular values are estimated nonasymptotically using a technique from geometric functional analysis. As the dimensions of the matrix grow to infinity, these singular values are proved to converge almost surely to fixed quantities (by using the universal Bai–Yin law) and in distribution to the Gaussian orthogonal ensemble Tracy–Widom law after proper rescalings. Understanding the behaviors of extreme singular values is important in general because they are often used to define a measure of stability of matrix algorithms. For example, JLTs were recently used in derivative-free optimization algorithmic frameworks to select random subspaces in which are constructed random models or poll directions to achieve scalability, and hence estimating their smallest singular value in particular helps determine the dimension of these subspaces.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1972

Davidon-Broyden rank-one minimization methods in Hilbert space with application to optimal control problems

The Davidon-Broyden class of rank one, quasi-Newton minimization methods is extended from Euclidean spaces to infinite-dimensional, real Hilbert spaces. For several techniques of choosing the step size, conditions are found which assure convergence of the associated iterates to the location of the minimum of a positive definite quadratic functional. For those techniques, convergence is achieved without the problem of the computation of a one-dimensional minimum at each iteration. The application of this class of minimization methods for the direct computation of the solution of an optimal control problem is outlined. The performance of various members of the class are compared by solving a sample optimal control problem. Finally, the sample problem is solved by other known gradient methods, and the results are compared with those obtained with the rank one quasi-Newton methods.

Straeter, T. A.↗
DOE OSTI2025

Growth functions of periodic space tessellations

This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.

Chemistry↗
NASA NTRS1981

Cosmic plasma

Attention is given to experimental and theoretical approaches to plasma physics, plasma phenomena in laboratory and space, field and particle aspects of plasmas, the present state of the classical theory, boundary conditions and circuit dependence, and cosmology. Electric currents in space plasmas are considered, taking into account dualism in physics, particle-related phenomena in plasma physics, magnetic field lines, filaments, local plasma properties and the circuit, electric double layers, field-aligned currents as 'cables', an expanding circuit, different types of plasma regions, the cellular structure of space, and the fine structure of active plasma regions. Other topics discussed are related to circuits, the theory of cosmic plasmas, the origin of the solar system, the coexistence of matter and antimatter, annihilation as a source of energy, the Hubble expansion in a Euclidean space, and a model for the evolution of the Metagalaxy.

Alfven, H.↗