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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 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

MO-C-17A-04: Forecasting Longitudinal Changes in Oropharyngeal Tumor Morphology Throughout the Course of Head and Neck Radiation Therapy

Purpose: To generate, evaluate, and compare models that predict longitudinal changes in tumor morphology throughout the course of radiation therapy. Methods: Two morphology feature vectors were used to describe the size, shape, and position of 35 oropharyngeal GTVs at each treatment fraction during intensity-modulated radiation therapy. The feature vectors comprised the coordinates of the GTV centroids and one of two shape descriptors. One shape descriptor was based on radial distances between the GTV centroid and 614 GTV surface landmarks. The other was based on a spherical harmonic decomposition of these distances. Feature vectors over the course of therapy were described using static, linear, and mean models. The error of these models in forecasting GTV morphology was evaluated with leave-one-out cross-validation, and their accuracy was compared using Wilcoxon signed-rank tests. The effect of adjusting model parameters at 1, 2, 3, or 5 time points (adjustment points) was also evaluated. Results: The addition of a single adjustment point to the static model decreased the median error in forecasting the position of GTV surface landmarks by 1.2 mm (p<0.001). Additional adjustment points further decreased forecast error by about 0.4 mm each. The linear model decreased forecast error compared to the static model for feature vectors based on both shape descriptors (0.2 mm), while the mean model did so only for those based on the inter-landmark distances (0.2 mm). The decrease in forecast error due to adding adjustment points was greater than that due to model selection. Both effects diminished with subsequent adjustment points. Conclusion: Models of tumor morphology that include information from prior patients and/or prior treatment fractions are able to predict the tumor surface at each treatment fraction during radiation therapy. The predicted tumor morphology can be compared with patient anatomy or dose distributions, opening the possibility of anticipatory re-planning. American Legion Auxiliary Fellowship; The University of Texas Graduate School of Biomedical Sciences at Houston.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2018

Testing the possibility of magnetic domain imaging based on circular & linear dichroism using photoemission electron microscopy

In recent years, an increasing number of memory and spintronic devices have been developed exploiting the combination of ferromagnetic (FM) and anti-ferromagnetic (aFM) materials. Consequently, magnetic imaging based on continuous-wave (CW) ultraviolet (UV) (λ = 266nm and longer wavelength) photoemission electron microscopy (PEEM) is gaining considerable attention due to the possibility of determining magnetizations for FM and aFM materials with 10 nm lateral resolution at video rate image acquisition. This PEEM-based approach exploits the polarization-dependent photoemission yield, which is subject to the polarization vector and the FM or aFM magnetization direction. Because of this unique attribute, magnetic imaging using PEEM when coupled to a laser with multiple illumination geometries allows for characterizing in-plane and out-of-plane magnetizations. This concept, however, has not been tested using a deep-UV laser (λ = 213nm), which has a much broader application space than the longer wavelength excitation used in previous reports. The purpose of this project in FY17 was to show the proof-of-concept of magnetic circular dichroism (MCD)-PEEM imaging using a λ = 210nm pulsed laser. Our results demonstrated the feasibility of in-plane and out-of-plane magnetic imaging with the limitations in the lateral resolution, data acquisition time, and signal-to-noise ratio anticipated for using a pulsed laser of moderate power. The project goal for FY18 is to construct the automated polarization-controlled data acquisition, and to establish the new lab facility in anticipation of acquiring a state-of-the-art high-power 213nm CW laser, planned to be installed in FY19. We successfully demonstrate the former by measuring dielectric stacks with polarization-dependent photoemission yield. Extrapolating from our result, we conclude that the capability of PEEM-based magnetic imaging using a CW deep UV laser could be a potential game-changer for scientific investigations and technological developments of magnetic materials and spintronic devices. In addition, polarization controlled PEEM imaging shows the potential for ellipsometry imaging of embedded nanomaterials exploiting their subtle differences in optical constants with respect to their surrounding dielectrics.

36 MATERIALS SCIENCE↗
DOE OSTI2020

A generalized Boltzmann kinetic theory for strongly magnetized plasmas with application to friction

Coulomb collisions in plasmas are typically modeled using the Boltzmann collision operator, or its variants, which apply to weakly magnetized plasmas in which the typical gyroradius of particles significantly exceeds the Debye length. Conversely, O’Neil has developed a kinetic theory to treat plasmas that are so strongly magnetized that the typical gyroradius of particles is much smaller than the distance of the closest approach in a binary collision. Here, we develop a generalized collision operator that applies across the full range of magnetization strength. Since there is no closed-form solution for the scattering cross section when plasma is strongly magnetized, the input to the collision operator is obtained by numerically calculating particle trajectories. To demonstrate novel physics associated with strong magnetization, it is used to compute the friction force on a massive test charge. In addition to the traditional stopping power component, this is found to exhibit a transverse component that is perpendicular to both the velocity and Lorentz force vectors in the strongly magnetized regime, as was predicted recently using linear response theory. As a result, good agreement is found between the collision theory and linear response theory in the regime in which both apply, but the new collision theory also applies to stronger magnetization strength regimes than the linear response theory is expected to apply in.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Electrical design of the flexible imaging diffraction diagnostic for laser experiments (FIDDLE) at the National Ignition Facility (NIF)—Requirements, design, and performance

The Flexible Imaging Diffraction Diagnostic for Laser Experiments (FIDDLE) is a newly developed diagnostic for imaging time resolved diffraction in experiments at the National Ignition Facility (NIF). It builds on the successes of its predecessor, the Gated Diffraction Development Diagnostic (G3D). The FIDDLE was designed to support eight Daedalus version 2 sensors (six more hCMOS sensors than any other hCMOS-based diagnostic in NIF to date) and an integrated streak camera. We will review the electrical requirements, design, and performance of the electrical subsystems that were created to support this large number of cameras in the FIDDLE. The analysis of the data that the FIDDLE is intended to collect relies heavily on the accurate and well-understood timing of each sensor. We report camera-to-camera timing jitter of less than 100 ps rms and sensor integration times of 2.2 ns FWHM in 2-2 timing mode. Additionally, diffraction experiments on the NIF produce electric fields (EMI) on the order of 1 kV/m, which have been observed to negatively impact the performance of some electrical components of the FIDDLE. We report on the results of testing hCMOS camera electronics in a similar EMI environment generated in an offline lab. In addition, we also summarize the use of a novel approach to using a vector network analyzer as an EMI leak detector to understand and reduce the negative impacts of EMI on the FIDDLE.

47 OTHER INSTRUMENTATION↗
DOE OSTI2014

The K giant stars from the LAMOST survey data. I. Identification, metallicity, and distance

We present a support vector machine classifier to identify the K giant stars from the LAMOST survey directly using their spectral line features. The completeness of the identification is about 75% for tests based on LAMOST stellar parameters. The contamination in the identified K giant sample is lower than 2.5%. Applying the classification method to about two million LAMOST spectra observed during the pilot survey and the first year survey, we select 298,036 K giant candidates. The metallicities of the sample are also estimated with an uncertainty of 0.13 ∼ 0.29 dex based on the equivalent widths of Mg{sub b} and iron lines. A Bayesian method is then developed to estimate the posterior probability of the distance for the K giant stars, based on the estimated metallicity and 2MASS photometry. The synthetic isochrone-based distance estimates have been calibrated using 7 globular clusters with a wide range of metallicities. The uncertainty of the estimated distance modulus at K = 11 mag, which is the median brightness of the K giant sample, is about 0.6 mag, corresponding to ∼30% in distance. As a scientific verification case, the trailing arm of the Sagittarius stream is clearly identified with the selected K giant sample. Moreover, at about 80 kpc from the Sun, we use our K giant stars to confirm a detection of stream members near the apo-center of the trailing tail. These rediscoveries of the features of the Sagittarius stream illustrate the potential of the LAMOST survey for detecting substructures in the halo of the Milky Way.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2019

Error Analysis of ZFP Compression for Floating-Point Data

Compression of floating-point data will play an important role in high-performance computing as data bandwidth and storage become dominant costs. Lossy compression of floating-point data is powerful, but theoretical results are needed to bound its errors when used to store look-up tables, simulation results, or even the solution state during the computation. In this paper, we analyze the round-off error introduced by ZFP, a lossy compression algorithm. Here, the stopping criteria for ZFP depends on the compression mode specified by the user: fixed rate, fixed accuracy, or fixed precision [P. Lindstrom, ZFP 0.5.3 Documentation, 2018]. While most of our discussion is focused on the fixed precision mode of ZFP, we establish a bound on the error introduced by all three compression modes. In order to tightly capture the error, first we introduce a vector space that allows us to work with binary representations of components. Under this vector space, we define operators that implement each step of the ZFP compression and decompression to establish a bound on the error caused by ZFP. To conclude, numerical tests are provided to demonstrate the accuracy of the established bounds.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

CFD-Enhanced Calibration of a Multihole Probe for Small Uncrewed Aircraft Systems

A robust calibration method with a 9-hole probe (9HP) for inertial wind vector measurements from a small uncrewed aircraft system (sUAS) is presented. Calibration accuracy is improved by using computational fluid dynamics (CFD) to estimate corrections, such as those for test section blockage, that are generally applied in wind tunnel testing. A method for estimating experimental bias is presented to account for flow variations over the pressure taps as the 9HP is repositioned in the test section. Installation effects from flow over the airframe and upwash produced by the wings are estimated from CFD simulations of the entire airframe at the expected cruise speed. An initial CFD analysis has demonstrated a quantifiable linear relationship between the measured and actual angle of attack as measured from the relative wind frame. In conclusion, the objective of these additional steps is to increase the accuracy of 9HP calibration.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Visualizing Electrical Power Systems as Flow Fields

We describe a method for visualizing data flows on large networks. We transform data flow on fixed networks into a vector field, which can be directly visualized using scientific flow visualization techniques. We evaluated the method on power flowing through two transmission power networks: a small, regional, IEEE test system (RTS-96) and a large national-scale system (the Eastern Interconnection). For the larger and more complex transmission system, the method illustrates features of the power flow that are not accessible when visualizing the power transmission with traditional network visualization techniques.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

iPNHOT: a knowledge-based approach for identifying protein-nucleic acid interaction hot spots

The interaction between proteins and nucleic acids plays pivotal roles in various biological processes such as transcription, translation, and gene regulation. Hot spots are a small set of residues that contribute most to the binding affinity of a protein-nucleic acid interaction. Compared to the extensive studies of the hot spots on protein-protein interfaces, the hot spot residues within protein-nucleic acids interfaces remain less well-studied, in part because mutagenesis data for protein-nucleic acids interaction are not as abundant as that for protein-protein interactions. In this study, we built a new computational model, iPNHOT, to effectively predict hot spot residues on protein-nucleic acids interfaces. One training data set and an independent test set were collected from dbAMEPNI and some recent literature, respectively. To build our model, we generated 97 different sequential and structural features and used a two-step strategy to select the relevant features. The final model was built based only on 7 features using a support vector machine (SVM). The features include two unique features such as ΔSASsa 1/2 and esp3, which are newly proposed in this study. Based on the cross validation results, our model gave F1 score and AUROC as 0.725 and 0.807 on the subset collected from ProNIT, respectively, compared to 0.407 and 0.670 of mCSM-NA, a state-of-the art model to predict the thermodynamic effects of protein-nucleic acid interaction. The iPNHOT model was further tested on the independent test set, which showed that our model outperformed other methods. Here, by collecting data from a recently published database dbAMEPNI, we proposed a new model, iPNHOT, to predict hotspots on both protein-DNA and protein-RNA interfaces. The results show that our model outperforms the existing state-of-art models. Our model is available for users through a webserver: http://zhulab.ahu.edu.cn/iPNHOT/.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Testing the conservative character of particle simulations: I. Canonical and noncanonical guiding center model in Boozer coordinates

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating terms containing the covariant component B Ψ P of the magnetic field vector with respect to the poloidal flux function Ψ P . In an unperturbed plasma, B Ψ P can be eliminated via exact coordinate transformations, but, in general, one relies on approximations, assuming that the effect of B Ψ P is small. In this work, we are interested in the question whether Hamiltonian conservation laws are still satisfied when B Ψ P is retained in the presence of fluctuations. Considering fast ions in the presence of a shear Alfvén wave field with fixed amplitude, fixed frequency, and a single toroidal mode number n, we show that simulations using the code ORBIT with and without B Ψ P yield practically the same resonant and nonresonant GC orbits. The numerical results are consistent with theoretical analyses (presented in the appendix), which show that the unabridged GC Lagrangian with B Ψ P retained yields equations of motion that possess two key properties of Hamiltonian flows: (i) phase space conservation and (ii) energy conservation. As counter-examples, we also show cases where energy conservation (ii) or both conservation laws (i) and (ii) are broken by omitting certain small terms. When testing the conservative character of the simulation code, it is found to be beneficial to apply perturbations that do not resemble normal (eigen) modes of the plasma. The deviations are enhanced and, thus, more easily spotted when one inspects wave-particle interactions using nonnormal modes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2015

Continuous Positive Airway Pressure for Motion Management in Stereotactic Body Radiation Therapy to the Lung: A Controlled Pilot Study

Objective: To determine the effect of continuous positive airway pressure (CPAP) on tumor motion, lung volume, and dose to critical organs in patients receiving stereotactic body radiation therapy (SBRT) for lung tumors. Methods and Materials: After institutional review board approval in December 2013, patients with primary or secondary lung tumors referred for SBRT underwent 4-dimensional computed tomographic simulation twice: with free breathing and with CPAP. Tumor excursion was calculated by subtracting the vector of the greatest dimension of the gross tumor volume (GTV) from the internal target volume (ITV). Volumetric and dosimetric determinations were compared with the Wilcoxon signed-rank test. CPAP was used during treatment if judged beneficial. Results: CPAP was tolerated well in 10 of the 11 patients enrolled. Ten patients with 18 lesions were evaluated. The use of CPAP decreased tumor excursion by 0.5 ± 0.8 cm, 0.4 ± 0.7 cm, and 0.6 ± 0.8 cm in the superior–inferior, right–left, and anterior–posterior planes, respectively (P≤.02). Relative to free breathing, the mean ITV reduction was 27% (95% confidence interval [CI] 16%-39%, P<.001). CPAP significantly augmented lung volume, with a mean absolute increase of 915 ± 432 cm{sup 3} and a relative increase of 32% (95% CI 21%-42%, P=.003), contributing to a 22% relative reduction (95% CI 13%-32%, P=.001) in mean lung dose. The use of CPAP was also associated with a relative reduction in mean heart dose by 29% (95% CI 23%-36%, P=.001). Conclusion: In this pilot study, CPAP significantly reduced lung tumor motion compared with free breathing. The smaller ITV, the planning target volume (PTV), and the increase in total lung volume associated with CPAP contributed to a reduction in lung and heart dose. CPAP was well tolerated, reproducible, and simple to implement in the treatment room and should be evaluated further as a novel strategy for motion management in radiation therapy.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2013

INTERPRETING ERUPTIVE BEHAVIOR IN NOAA AR 11158 VIA THE REGION'S MAGNETIC ENERGY AND RELATIVE-HELICITY BUDGETS

In previous works, we introduced a nonlinear force-free method that self-consistently calculates the instantaneous budgets of free magnetic energy and relative magnetic helicity in solar active regions (ARs). Calculation is expedient and practical, using only a single vector magnetogram per computation. We apply this method to a time series of 600 high-cadence vector magnetograms of the eruptive NOAA AR 11158 acquired by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory over a five-day observing interval. Besides testing our method extensively, we use it to interpret the dynamical evolution in the AR, including eruptions. We find that the AR builds large budgets of both free magnetic energy and relative magnetic helicity, sufficient to power many more eruptions than the ones it gave within the interval of interest. For each of these major eruptions, we find eruption-related decreases and subsequent free-energy and helicity budgets that are consistent with the observed eruption (flare and coronal mass ejection (CME)) sizes. In addition, we find that (1) evolution in the AR is consistent with the recently proposed (free) energy-(relative) helicity diagram of solar ARs, (2) eruption-related decreases occur before the flare and the projected CME-launch times, suggesting that CME progenitors precede flares, and (3) self terms of free energy and relative helicity most likely originate from respective mutual terms, following a progressive mutual-to-self conversion pattern that most likely stems from magnetic reconnection. This results in the non-ideal formation of increasingly helical pre-eruption structures and instigates further research on the triggering of solar eruptions with magnetic helicity firmly placed in the eruption cadre.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Viscoelastic material inversion using Sierra-SD and ROL

In this report we derive frequency-domain methods for inverse characterization of the constitutive parameters of viscoelastic materials. The inverse problem is cast in a PDE-constrained optimization framework with efficient computation of gradients and Hessian vector products through matrix free operations. The abstract optimization operators for first and second derivatives are derived from first principles. Various methods from the Rapid Optimization Library (ROL) are tested on the viscoelastic inversion problem. The methods described herein are applied to compute the viscoelastic bulk and shear moduli of a foam block model, which was recently used in experimental testing for viscoelastic property characterization.

36 MATERIALS SCIENCE↗
DOE OSTI2020

A New Capability of E4D For 3D Parallel Joint Inversion of DC Resistivity And Traveltime Data on Unstructured Mesh

A major challenge in interpreting geophysical data is how to derive consistent three-dimensional (3D) earth models of different physical properties from spatially and temporally limited measurements. Joint inversion with cross-gradient constraints is an approach to find such models by imposing structural similarities between different physical parameters. We have developed a parallel distributed-memory joint inversion code for direct-current (DC) resistivity and traveltime data using the cross-gradient constraint on unstructured mesh. The code utilizes existing E4D framework for parallel forward simulation, distributed storage and computation of the Jacobian matrix of forward operator, and parallel execution of matrix-vector multiplication during inversion. Besides, the joint inversion is solved by nonlinear conjugate gradient algorithm parallelized for DC resistivity and traveltime data. The joint inversion capability of E4D was tested using synthetic data from cross-borehole DC resistivity and traveltime data. The results indicate that the shape and size of the anomalies from the joint inversion are more reliable than those from separate inversions.

58 GEOSCIENCES↗
DOE OSTI2016

Testing and validation of multi‐lidar scanning strategies for wind energy applications

Abstract Several factors cause lidars to measure different values of turbulence than an anemometer on a tower, including volume averaging, instrument noise and the use of a scanning circle to estimate the wind field. One way to avoid the use of a scanning circle is to deploy multiple scanning lidars and point them toward the same volume in space to collect velocity measurements and extract high‐resolution turbulence information. This paper explores the use of two multi‐lidar scanning strategies, the tri‐Doppler technique and the virtual tower technique, for measuring 3‐D turbulence. In summer 2013, a vertically profiling Leosphere WindCube lidar and three Halo Photonics Streamline lidars were operated at the Southern Great Plains Atmospheric Radiation Measurement site to test these multi‐lidar scanning strategies. During the first half of the field campaign, all three scanning lidars were pointed at approximately the same point in space and a tri‐Doppler analysis was completed to calculate the three‐dimensional wind vector every second. Next, all three scanning lidars were used to build a ‘virtual tower’ above the WindCube lidar. Results indicate that the tri‐Doppler technique measures higher values of horizontal turbulence than the WindCube lidar under stable atmospheric conditions, reduces variance contamination under unstable conditions and can measure high‐resolution profiles of mean wind speed and direction. The virtual tower technique provides adequate turbulence information under stable conditions but cannot capture the full temporal variability of turbulence experienced under unstable conditions because of the time needed to readjust the scans. Copyright © 2016 John Wiley & Sons, Ltd.

Newman, Jennifer F.↗
DOE OSTI2018

Evaluation of dose recalculation vs dose deformation in a commercial platform for deformable image registration with a computational phantom

Deformable image registration (DIR) is an important tool for mapping both dose and contours of a new set of planning images when recurrent, that is, adaptive radiotherapy, or further treatment, that is, re-treatment, is required. The aim of this study was to evaluate the need for plan recalculation in deformed anatomies and to develop a reliable workflow for validating the DIR algorithm to be applied to dose warping such as dose accumulation (DA) for adaptive radiation therapy, and dose summation (DS) in the case of re-treatment. A set of 3 computational phantoms was developed to validate the application of B-Spline-based registrations for dose mapping among the various computed tomography image data sets. Two different versions were defined for each phantom to simulate clinical needs: adaptive radiotherapy and re-treatment; a DIR was performed to obtain a displacement vector field (DVF) for dose applications. Comparison of calculated and deformed doses was carried out by means of known markers inside the virtual phantoms. The differences were evaluated using a 3% dose index as acceptance criteria. A paired Wilcoxon signed-rank test was carried out to test the statistical significance of differences within the markers. Significant differences were only observed for the deformed dose in the DA test; no significant differences were observed for recalculated dose values in the DA and DS tests. The dose index is in accordance with these results. The warping dose process obtained by registering various sets of images was validated; a recalculation approach seems to be more accurate for DA purposes and image-guided adaptive radiation therapy (IGART) applications.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2020

In Situ Micro-Pillar Compression to Examine Radiation-Induced Hardening Mechanisms of FeCrAl Alloys

Here, the effects of 5 MeV Fe 2+ ion irradiation at 300°C on the microstructure evolution and deformation behavior of a FeCrAl C26M alloy are presented. It has been found that dislocation loop density increases an order of magnitude from 1 dpa to 16 dpa irradiations, whereas, the dislocation loop size saturates with increasing damage. Micropillars, 600 nm in diameter and 1.3 µm in height, were fabricated and compressed inside grains with <001>, <011> and <111> crystallographic orientations, respectively. {112} <111> has been identified as the primary slip system in both unirradiated and irradiated alloy. The increase in yield stress after irradiation is observed with measurable variation along <001> and <011> vs. along <111>. By applying the Orowan dispersed barrier model, the increase of yield stress is found mainly due to the slip resistance of radiation generated defect loops. Detailed transmission electron microscopy (TEM) studies were performed to quantify the Burgers vector and the distribution of irradiation induced dislocations at elevated strains. It is revealed that localized shear instability is caused by avalanche slip events of ½<111> dislocations gliding out of tested pillars. Simultaneously, a large number of sessile/immobile <100> dislocations formed in the vicinity of slip band, leading to the hardening at elevated strains.

36 MATERIALS SCIENCE↗
DOE OSTI2014

Practical auxiliary basis implementation of Rung 3.5 functionals

Approximate exchange-correlation functionals for Kohn-Sham density functional theory often benefit from incorporating exact exchange. Exact exchange is constructed from the noninteracting reference system's nonlocal one-particle density matrix γ(r{sup -vector},r{sup -vector}′). Rung 3.5 functionals attempt to balance the strengths and limitations of exact exchange using a new ingredient, a projection of γ(r{sup -vector},r{sup -vector} ′) onto a semilocal model density matrix γ{sub SL}(ρ(r{sup -vector}),∇ρ(r{sup -vector}),r{sup -vector}−r{sup -vector} ′). γ{sub SL} depends on the electron density ρ(r{sup -vector}) at reference point r{sup -vector}, and is closely related to semilocal model exchange holes. We present a practical implementation of Rung 3.5 functionals, expanding the r{sup -vector}−r{sup -vector} ′ dependence of γ{sub SL} in an auxiliary basis set. Energies and energy derivatives are obtained from 3D numerical integration as in standard semilocal functionals. We also present numerical tests of a range of properties, including molecular thermochemistry and kinetics, geometries and vibrational frequencies, and bandgaps and excitation energies. Rung 3.5 functionals typically provide accuracy intermediate between semilocal and hybrid approximations. Nonlocal potential contributions from γ{sub SL} yield interesting successes and failures for band structures and excitation energies. The results enable and motivate continued exploration of Rung 3.5 functional forms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗