Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “matrices”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2010

Multicanonical sampling of rare events in random matrices

A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble, sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of {approx}10{sup -200}, a region where simple random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events characterized by any statistics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

What is the gradient of a scalar function defined on a subspace of square matrices?

We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), https://doi.org/10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. Here, we examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 x 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Dynamics of disordered mechanical systems with large connectivity, free probability theory, and quasi-Hermitian random matrices

Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual ions in a crystal are subjected essentially to nearest-neighbor interactions. In contrast, the systems studied in this paper have all their degrees of freedom coupled to each other. Thus, the problem of linearized small oscillations of such systems involves two full positive-definite and non-commuting matrices, as opposed to the sparse matrices associated with disordered crystals. Consequently, the familiar methods for determining the averaged vibrational spectra of disordered crystals, introduced many years ago by Dyson and Schmidt, are inapplicable for highly connected disordered systems. In this paper we apply random matrix theory (RMT) to calculate the averaged vibrational spectra of such systems, in the limit of infinitely large system size. At the heart of our analysis lies a calculation of the average spectrum of the product of two positive definite random matrices by means of free probability theory techniques. We also show that this problem is intimately related with quasi-hermitian random matrix theory (QHRMT), which means that the ‘hamiltonian’ matrix is hermitian with respect to a non-trivial metric. This extends ordinary hermitian matrices, for which the metric is simply the unit matrix. The analytical results we obtain for the spectrum agree well with our numerical results. The latter also exhibit oscillations at the high-frequency band edge, which fit well the Airy kernel pattern. We also compute inverse participation ratios of the corresponding amplitude eigenvectors and demonstrate that they are all extended, in contrast with conventional disordered crystals. Finally, we compute the thermodynamic properties of the system from its spectrum of vibrations. In addition to matrix model analysis, we also study the vibrational spectra of various multi-segmented disordered pendula, as concrete realizations of highly connected mechanical systems. A universal feature of the density of vibration modes, common to both pendula and the matrix model, is that it tends to a non-zero constant at vanishing frequency.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

An Evaluation of Actinide Reactivity with CO 2 , O 2 , and O 2 /He Gases using Inductively Coupled Plasma Tandem Mass Spectrometry: Application to Simultaneous Measurement of 241 Am/ 241 Pu Ratios in Unseparated Complex Matrices

Accurate actinide measurements are critical within the field of nuclear science. Traditional methods for actinide quantification require time-consuming sample processing prior to analysis. There is a need for rapid analytical techniques that still maintain a high degree of accuracy. In this work, actinide reactivity was assessed for multiple oxygen-containing reaction gases using quadrupole inductively coupled plasma tandem mass spectrometry (Q-ICP-MS/MS) to evaluate actinide analysis in complex sample matrices without analyte-matrix separation. A novel method was developed to measure 241 Am/ 241 Pu in complex sample matrices using O 2 /He reaction gas with no matrix removal or analyte pre-concentration. This inline method reduces matrix-derived polyatomic interferences that complicate traditional ICP-MS analyses by mass-shifting to 241 Am 16 O + and 241 Pu 16 O 2 + , allowing Am and Pu to be mass separated for simultaneous analysis. While mass shifting is efficient, a small portion of Am + (<1.3%) and Pu + (<1.4%) react to from AmO 2 + and PuO + , respectively. Therefore, a mass balance approach was used, in combination with reactivity determined from 242 Pu and 243 Am standard solutions, to correct for residual 241 PuO + and 241 AmO 2 + . The method was validated by measuring 241 Am/ 241 Pu in Pu isotope standards CRM-136 and CRM-137 (separated in March/April 1970 and February 2022, respectively) in both neat solutions and complex matrices containing diluted soil (NIST SRM 2711a, >1000 µg·g -1 ). Method detection limits of 15.9 and 9.6 fg·g -1 were determined for 241 Am and 241 Pu, respectively, and 241 Am/ 241 Pu ratios were measured with accuracies within <3.5%. In conclusion, this work presents the first direct analysis of 241 Am/ 241 Pu in unseparated complex matrices, advancing capabilities for rapid actinide measurements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

In Situ Evaluation of Calcium Phosphate Nucleation Kinetics and Pathways during Intra- and Extrafibrillar Mineralization of Collagen Matrices

Calcium phosphate (CaP) nanocrystals nucleate and grow in intrafibrillar and/or extrafibrillar spaces of collagen fibrils during the mineralization of bones and teeth. Little is known about the early stages of CaP nucleation and distribution in fibrillar matrices, despite their significant influence on the physical and chemical structures of tissue-level constructs. Using in situ small angle X-ray scattering (SAXS), we examined the nucleation and growth of CaP within collagen matrices and elucidated how a nucleation inhibitor, polyaspartic acid (pAsp), governs mineralization kinetics and pathways at multiple length scales. In situ SAXS analysis clearly revealed that nucleation sites, kinetically-controlled by the nucleation inhibitor, determined the pathways of CaP morphological transformation. Mineralization with pAsp led to intrafibrillar CaP plates with a spatial distribution gradient through the depth of the matrix. Mineralization without pAsp led initially to spherical aggregates of CaP in the entire extrafibrillar spaces. With time, the spherical aggregates transformed into plates at the outermost surface of the collagen matrix, preventing intrafibrillar mineralization inside. The results illuminate mineral nucleation kinetics and real-time nanoparticle distributions within organic matrices in solutions containing body fluid components. Because the macroscale mechanical properties of collagen matrices depend on their mineral content, phase, and arrangement at the nanoscale, this study contributes to better design and fabrication of biomaterials for regenerative medicine.

Kim, Doyoon↗
DOE OSTI2025

Determining the Ensemble N -Representability of Reduced Density Matrices

The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy that embeds an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both simple models as well as molecular systems at finite temperature, demonstrating its robustness.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Ultra-low binder content 3D printed calcium phosphate graphene scaffolds as resorbable, osteoinductive matrices that support bone formation in vivo

Bone regenerative engineering could replace autografts; however, no synthetic material fulfills all design criteria. Nanocarbons incorporated into three-dimensional printed (3DP) matrices can improve properties, but incorporation is constrained to low wt%. Further, unmodified nanocarbons have limited osteogenic potential. Functionalization to calcium phosphate graphene (CaPG) imparts osteoinductivity and osteoconductivity, but loading into matrices remained limited. This work presents ultra-high content (90%), 3DP-CaPG matrices. 3DP-CaPG matrices are highly porous (95%), moderately stiff (3 MPa), and mechanically robust. In vitro, they are cytocompatible and induce osteogenic differentiation of human mesenchymal stem cells (hMSCs), indicated by alkaline phosphatase, mineralization, and COL1α1 expression. In vivo, bone regeneration was studied using a transgenic fluorescent-reporter mouse non-union calvarial defect model. 3DP-CaPG stimulates cellular ingrowth, retains donor cells, and induces osteogenic differentiation. Histology shows TRAP staining around struts, suggesting potential osteoclast activity. Apparent resorption of 3DP-CaPG was observed and presented no toxicity. 3DP-CaPG represents an advancement towards a synthetic bone regeneration matrix.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2025

Esterification of 3-quinuclidinol, a marker for the incapacitant BZ, for analysis by EI-GC-MS in OPCW test matrices

The analysis by EI-GC-MS of 3-quinuclidinol (3Q), a marker for the riot control and incapacitating agent quinuclinidyl benzilate (BZ), using several, new acylation strategies is described. After evaluating and optimizing conditions for the acylations, these were tested on their ability to successfully derivatize 3Q in three different matrices featured in the 44th Organisation for the Prohibition of Chemical Weapons (OPCW) proficiency test (PT). As 3Q is a highly polar compound, the work here describes acylation strategies that generate 3Q analogs with superior gas chromatographic profiles relative to the underivatized material. The acyl groups studied in this work included acetyl, benzoyl, pentafluorobenzoyl and the bis(3,5-trifluoromethyl)benzoyl. The acylated 3Q products provide sharp, detectable peaks with significantly different retention times from that of the unmodified 3Q. Thus, the retention times for acetyl-, benzoyl-, pentafluorobenzoyl- and 3,5-bis(trifluoromethyl)benzoyl-3Q were determined to be 15.9, 24.9, 22.8 and 22.1 min respectively, in stark contrast to the unmodified 3Q which provides a sharp peak with a retention time centered at ~ 13.5 min only at high concentrations (> 10 µg/mL or µg/g) while a broad peak (RT ~ 15–15.5 min) at low concentrations (< 5 µg/mL or µg/g). The developed protocol was used to derivatize 3Q in three separate matrices. The first two matrices were liquid samples featured during the 44th OPCW PT at two separate concentrations in each (5 and 50 µg/mL). The last matrix was a soil sample featured in the same 44th OPCW PT wherein the 3Q had been spiked at a 12 µg/g concentration. The approach involves the extraction of 3Q from all matrices followed by its acylation resulting in a second reportable analytical method for an OPCW PT. The presented derivatization strategies should find wide applicability particularly in laboratories involved with the analysis of this chemical weapon agent degradation product and those participating in the yearly OPCW PTs.

3-quinuclidinol↗
DOE OSTI2014

Fluorescence spectroscopy of UV-MALDI matrices and implications of ionization mechanisms

Matrix-assisted laser desorption ionization (MALDI) has been widely used in the mass analysis of biomolecules; however, there are a lot of debates about the ionization mechanisms. Previous studies have indicated that S{sub 1}-S{sub 1} annihilation might be a key process in the generation of primary ions. This study investigates S{sub 1}-S{sub 1} annihilation by examining the time-resolved fluorescence spectra of 12 matrices. No S{sub 1}-S{sub 1} annihilation was observed in six of these matrices (3-hydroxy-picolinic acid, 6-aza-2-thiothymine, 2,4-dihydroxy-acetophenone, 2,6-dihydroxy-acetophenone, 2,4,6-trihydroxy-acetophenone, and ferulic acid). We observed two matrix molecules reacting in an electronically excited state (S{sub 1}) in five of these matrices (2,5-dihydroxybenzoic acid, α-cyano-4-hydroxycinnamic acid, 2,5-dihydroxy-acetophenone, 2,3-dihydroxybenzoic acid, and 2,6-dihydroxybenzoic acid), and S{sub 1}-S{sub 1} annihilation was a possible reaction. Among these five matrices, no S{sub 1}-S{sub 1} annihilation was observed for 2,3-dihydroxybenzoic acid in typical peak power region of nanosecond laser pulses in MALDI, but a very small value of reaction rate constant was observed only in the high peak power region. The excited-state lifetime of sinapinic acid was too short to determine whether the molecules reacted in an electronically excited state. No correlation was observed between the ion generation efficiency of MALDI and S{sub 1}-S{sub 1} annihilation. The results indicate that the proposal of S{sub 1}-S{sub 1} annihilation is unnecessary in MALDI and energy pooling model for MALDI ionization mechanism has to be modified.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Large covariance matrices: accurate models without mocks

Abstract Covariance matrix estimation is a persistent challenge for cosmology. We focus on a class of model covariance matrices that can be generated with high accuracy and precision, using a tiny fraction of the computational resources that would be required to achieve comparably precise covariance matrices using mock catalogues. In previous work, the free parameters in these models were determined using sample covariance matrices computed using a large number of mocks, but we demonstrate that those parameters can be estimated consistently and with good precision by applying jackknife methods to a single survey volume. This enables model covariance matrices that are calibrated from data alone, with no reference to mocks.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Improved Evaluation of Large Network Matrices for Linear Power Flow Within Optimization Problems

This work presents methods for evaluating the Power Transfer Distribution Factor (PTDF) and Line Outage Distribution Factor (LODF) matrices by employing sparse linear algebra for large-scale computing applications. These matrices play a critical role in many power system applications, such as the Unit Commitment Problem (UC), pre- and post-contingency power flow analysis, and transmission expansion. These matrices are typically dense, which means they require a significant amount of time and memory to be computed for large networks. However, by analyzing the structure of the matrices and their computation method, it is possible to use reduced memory methods based on sparse matrix operations. This paper shows that sparse linear algebra algorithms are faster and require less memory and time than traditional dense approaches. Additionally, we explore the effect of matrix sparsification by eliminating trailing digits on power flow calculations.

large scale↗
DOE OSTI2024

What is the gradient of a scalar function defined on a subspace of square matrices?

We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), doi:10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. We examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 × 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1974

Commutative semigroups of real and complex matrices

The computation of divergence is studied. Covariance matrices to be analyzed admit a common diagonalization, or even triangulation. Sufficient conditions are given for such phenomena to take place, the arguments cover both real and complex matrices, and are not restricted to Hermotian or other special forms. Specifically, it is shown to be sufficient that the matrices in question commute in order to admit a common triangulation. Several results hold in the case that the matrices in question form a closed and bounded set, rather than only in the finite case.

Brown, D. R.↗
NASA NTRS1977

An automated procedure for calculating system matrices from perturbation data generated by an EAI Pacer and 100 hybrid computer system

Techniques are presented for determining the elements of the A, B, C, and D state variable matrices for systems simulated on an EAI Pacer 100 hybrid computer. An automated procedure systematically generates disturbance data necessary to linearize the simulation model and stores these data on a floppy disk. A separate digital program verifies this data, calculates the elements of the system matrices, and prints these matrices appropriately labeled. The partial derivatives forming the elements of the state variable matrices are approximated by finite difference calculations.

Milner, E. J.↗
NASA NTRS1979

Higher dimensional Hadamard matrices

The paper defines higher dimensional Hadamard matrices and enumerates on some of the simplest three-, four-, and five-dimensional cases and procedures for generating them. Special emphasis is given to proper matrices that have a dimensional hierarchy of orthogonalities. It is determined that this property lends itself primarily to the application of higher dimensional Hadamard matrices to error-correcting codes. A list of derived statements for n-dimensional Hadamard matrices are given, as well as a definition of Hadamard matrix families, such as minimal, Petrie polygon, antipodal (n-2)-dimensional sections, and double proximity shells.

Schlichta, P. J.↗
NASA NTRS1982

Numerical computation of exponential matrices using the Cayley-Hamilton theorem

A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.

Walden, H.↗
NASA NTRS1984

Microwave bistatic reflectivity dependence on the moisture content and matric potential of bare soil

Results are presented of an experimental program to determine the functional dependence of the microwave reflectivity of nonvegetated soil surfaces upon volumetric soil moisture and matric potential. A combination evaporation-drainage field experiment was conducted on a bare Captina slit loam with reflectivity, soil moisture content, and matric potential monitored for extended time periods. Results show that for a restricted pressure range (approximately -0.05 to -0.75 bar) there is excellent linear correlation between the log of bistatic reflectivity and both volumetric moisture content and matric potential. Layering effects due to steep moisture content (and matric potential) gradients in the profile are demonstrated to have two distinct and significant effects on the reflectivity response. At near saturation of rough surfaces a very thin dry surface layer appears to modify the effective roughness. This leads to a saturation of reflectivity at high moisture contents. As the surface proceeds to dry further, deeper layers produce coherent interference patterns in the reflectivity response, particularly at the higher frequencies.

Waite, W. P.↗
NASA NTRS1985

Complex symmetric matrices with strongly stable iterates

Complex-valued symmetric matrices are studied. A simple expression for the spectral norm of such matrices is obtained, by utilizing a unitarily congruent invariant form. A sharp criterion is provided for identifying those symmetric matrices whose spectral norm is not exceeding one: such strongly stable matrices are usually sought in connection with convergent difference approximations to partial differential equations. As an example, the derived criterion is applied to conclude the strong stability of a Lax-Wendroff scheme.

Tadmor, E.↗