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NASA NTRS1988

Adaptive guidance for an aero-assisted boost vehicle

An adaptive guidance system incorporating dynamic pressure constraint is studied for a single stage to low earth orbit (LEO) aero-assist booster with thrust gimbal angle as the control variable. To derive an adaptive guidance law, cubic spline functions are used to represent the ascent profile. The booster flight to LEO is divided into initial and terminal phases. In the initial phase, the ascent profile is continuously updated to maximize the performance of the boost vehicle enroute. A linear feedback control is used in the terminal phase to guide the aero-assisted booster onto the desired LEO. The computer simulation of the vehicle dynamics considers a rotating spherical earth, inverse square (Newtonian) gravity field and an exponential model for the earth's atmospheric density. This adaptive guidance algorithm is capable of handling large deviations in both atmospheric conditions and modeling uncertainties, while ensuring maximum booster performance.

Pamadi, Bandu N.↗

NASA NTRS2014

Grail Refinements to Lunar Seismic Structure

To probe a planet's interior, seismology provides the most direct constraints on the variables that govern the dynamic properties of the body. However, the GRAIL (Gravity Recovery and Interior Laboratory) mission's high-resolution measurements of the lunar gravity field provide constraints on crustal thickness, mantle structure, core radius and stratification, and core state (solid vs. molten). These data complement seismic investigations, and joint interpretation permits improved constraints on the Moon's internal structure. Joint interpretation of disparate geophysical datasets helps reduce drawbacks that can result from analyzing them individually. The Apollo seismic network was situated on the lunar nearside surface in a roughly equilateral triangle having sides approximately 1000 km long, with stations 12/14 nearly co-located at one corner. Due to this limited geographical extent, near-surface ray coverage from moonquakes is low, but increase with depth. In comparison, gravity surveys and their resulting gravity anomaly maps have traditionally offered optimal resolution at crustal depths. Gravimetric maps and seismic data sets are therefor well suited to joint inversion, since the complementary information reduces inherent model ambiguity. We will perform a joint inversion of Apollo seismic delay times and gravity data collected by GRAIL lunar gravity mission, in order to recover seismic velocity and density as a function of latitude, longitude and depth within the Moon. We will relate density (rho) to seismic velocity (v) using a linear relationship that is allowed to be depth-dependent. The corresponding coefficient (B) can reflect a variety of material properties that vary with depth, including temperature and composition. The inversion seeks to recover the set of rho, v, and B perturbations that minimize (in a least-squares sense) the difference between the observed and calculated data.

Weber, Renee C.↗

NASA NTRS2010

New Results on High Energy Cosmic Ray Electrons Observed with Fermi LAT and Their Implications on the Origin of Cosmic Rays

The Large Area Telescope on-board the Fermi Gamma-Ray Space Telescope has collected more than 10 million cosmic ray electrons with energy above 7 GeV since its science operation on orbit. High energy electrons rapidly lose their energy by synchrotron radiation on Galactic magnetic fields and by inverse Compton scattering on the interstellar radiation field. The typical distance over which a 1 TeV electron loses half its total energy is estimated to be 300-400 pc.This makes them a unique tool for probing nearby Galactic space. Observed spectrum has a harder spectral index than was previously reported and suggests the presence of nearby sources of high energy electrons. One of viable candidates are nearby pulsars, possibly some of recently discovered by Fermi. At the same time the dark matter origin of such sources cannot be ruled out. I will also report our current upper limits on cosmic ray electrons anisotropy which helps to set constraints on their local sources.

Moiseev, Alexander↗

NASA NTRS2002

Aerodynamic Design Study of Advanced Multistage Axial Compressor

As a direct response to the need for further performance gains from current multistage axial compressors, an investigation of advanced aerodynamic design concepts that will lead to compact, high-efficiency, and wide-operability configurations is being pursued. Part I of this report describes the projected level of technical advancement relative to the state of the art and quantifies it in terms of basic aerodynamic technology elements of current design systems. A rational enhancement of these elements is shown to lead to a substantial expansion of the design and operability space. Aerodynamic design considerations for a four-stage core compressor intended to serve as a vehicle to develop, integrate, and demonstrate aerotechnology advancements are discussed. This design is biased toward high efficiency at high loading. Three-dimensional blading and spanwise tailoring of vector diagrams guided by computational fluid dynamics (CFD) are used to manage the aerodynamics of the high-loaded endwall regions. Certain deleterious flow features, such as leakage-vortex-dominated endwall flow and strong shock-boundary-layer interactions, were identified and targeted for improvement. However, the preliminary results were encouraging and the front two stages were extracted for further aerodynamic trimming using a three-dimensional inverse design method described in part II of this report. The benefits of the inverse design method are illustrated by developing an appropriate pressure-loading strategy for transonic blading and applying it to reblade the rotors in the front two stages of the four-stage configuration. Multistage CFD simulations based on the average passage formulation indicated an overall efficiency potential far exceeding current practice for the front two stages. Results of the CFD simulation at the aerodynamic design point are interrogated to identify areas requiring additional development. In spite of the significantly higher aerodynamic loadings, advanced CFD-based tools were able to effectively guide the design of a very efficient axial compressor under state-of-the-art aeromechanical constraints.

Larosiliere, Louis M.↗

NASA NTRS2007

Low Velocity Airdrop Tests of an X-38 Backup Parachute Design

The NASA Johnson Space Center's X-38 program designed a new backup parachute system to recover the 25,000 lb X-38 prototype for the Crew Return Vehicle spacecraft. Due to weight and cost constraints, the main backup parachute design incorporated rapid and low cost fabrication techniques using off-the-shelf materials. Near the vent, the canopy was constructed of continuous ribbons, to provide more damage tolerance. The remainder of the canopy was a constructed with a continuous ringslot design. After cancellation of the X-38 program, the parachute design was resized, built, and drop tested for Natick Soldiers Center's Low Velocity Air Drop (LVAD) program to deliver cargo loads up to 22,000 lbs from altitudes as low as 500 feet above the ground. Drop tests results showed that the 500-foot LVAD parachute deployment conditions cause severe skirt inversion and inflation problems for large parachutes. The bag strip occurred at a high angle of attack, causing skirt inversion before the parachute could inflate. The addition of a short reefing line prevented the skirt inversion. Using a lower porosity in the vent area, than is normally used in large parachutes, improved inflation. The drop testing demonstrated that the parachute design could be refined to meet the requirements for the 500-foot LVAD mission.

Stein, Jenny M.↗

NASA NTRS2021

Application of Radon Transform to Multi-Angle Measurements Made by the Research Scanning Polarimeter: A New Approach to Cloud Tomography. Part I: Theory and Tests on Simulated Data.

The Research Scanning Polarimeter (RSP) is an airborne along-track scanner measuring the polarized and total reflectances in 9 spectral channels. The RSP was a prototype for the Aerosol Polarimetry Sensor (APS) launched on-board the NASA Glory satellite. Currently the retrieval algorithms developed for the RSP are being adopted for the measurements of the space-borne polarimeters on the upcoming NASA’s Plankton, Aerosol, Cloud Ocean Ecosystem (PACE)satellite mission. The RSP’s uniquely high angular resolution coupled with the high frequency of measurements allows for characterization of liquid water cloud droplet sizes using the polarized rainbow structure. It also provides geometric constraints on the cumulus cloud’s 2D cross section yielding the cloud’s geometric shape estimates. In this study we further build on the latter technique to develop a new tomographic approach to retrieval of cloud internal structure from remote sensing measurements. While tomography in the strict definition is a technique based on active measurements yielding a tomogram (directional optical thickness as a function of angle and offset of the view ray), we developed a “semi-tomographic” approach in which tomogram of the cloud is estimated from passive observations instead of being measured directly. This tomogram is then converted into 2D spatial distribution of the extinction coefficient using inverse Radon transform (filtered back projection) which is the standard tomographic procedure used e.g., in medical CT scans. This algorithm is computationally inexpensive compared to techniques relying on highly-multi-dimensional least-square fitting; it does not require iterative 3D RT simulations. The resulting extinction distribution is defined up to an unknown constant factor, so we discuss the ways to calibrate it using additional independent measurements. In the next step we use the profile of the droplet size distribution parameters from the cloud’s side (derived by fitting the polarized rainbows) to convert the 2D extinction distribution into that of the droplet number concentration. We illustrate and validate the proposed technique using 3D-RT-simulatedRSP observations of a LES-generated Cu cloud. Quantitative comparisons between the retrieved and the original optical and microphysical parameters are presented.

clouds↗

NASA NTRS1995

Ozone vertical distribution retrieval from ground-based high resolution infrared solar spectra

A practical procedure for the retrieval of ozone vertical profiles from ground-based high resolution Fourier transform infrared solar spectra has been developed. The analysis is based on a multilayer line-by-line forward model and a semi-empirical version of the optimal estimation inversion method of Rodgers. The 1002.6-1003.2 cm(exp -1) spectral interval has been selected for the analysis on the basis of synthetic spectrum calculations. This interval contains numerous ozone lines covering a range of intensities and providing retrieval sensitivity from ground level to about 35 km. Characterization of the method and an error analysis have been performed. For a spectral resolution of 0.05-0.01 cm(exp -1) and a signal-to-noise ratio greater than or equal to 100 the retrieval is stable with a vertical resolution of approximately 5 km attainable near the surface degrading to approximately 10 km in the stratosphere. Synthetic spectra studies show that the a priori profile and weak constraints selected for the retrievals introduce no significant biases for a wide range of ozone profiles.

Pougatchev, N. S.↗

NASA NTRSDate not supplied

TPSAS-NF1676L-27509-DND

The information about composition of aerosols has a great importance for monitoring and understanding of climate and environment dynamics. Such information can be obtained using situ measurements or chemical transport models. However, in situ sampling has limited spatial and temporal coverage, while estimations have large uncertainties. The present work enables the monitoring of aerosol chemical species from space-borne observations, providing observationally-based results with spatial and temporal coverage. Following the ideas of Schuster et al.(2005, 2009, 2016), we retrieve chemical composition directly from satellite observations without intermediate retrieval of the refractive index (in contrast with Schuster's approach). This approach is expected to reduce the influence of modeling uncertainties, and to provide additional constraints in situations where remote sensing observations do not have enough spectral sensitivity to refractive index. One of principal difficulties is the identification of an adequate model for linking refractive index to chemical composition. Therefore, the initial effort of this work has focused on identifying an optimal physico-chemistry to refractive index conversion model. With that purpose, we first tested the retrieval approach using a simplified volume-weighting model and updated by the Maxwell Garnett mixing model. This concept was incorporated into the GRASP algorithm designed to retrieve an extended set of atmospheric parameters from remote sensing observations. Then a series of sensitivity tests using synthetic data of POLDER/PARASOL were conducted, and followed by inversion of real PARASOL observations over Africa. The sensitivity tests showed that these two models allow the retrieval to distinguish amongst the assumed chemical species. Results obtained from real PARASOL data demonstrated good agreement with the optical characteristics provided by AERONET. In addition, the obtained patterns of chemical component distribution agreed with known physical expectations. We also compared the chemical composition retrievals to GOCART model simulations and obtained good consistencies.

Lei Li↗

NASA NTRS2014

A Decade of Volcanic Observations from Aura and the A-Train

Aura observations have made many seminal contributions to volcanology. Prior to the Aura launch, satellite observations of volcanic degassing (e.g., from TOMS) were mostly restricted to large eruptions. However, the vast majority of volcanic gases are released during quiescent 'passive' degassing between eruptions. The improved sensitivity of Aura OMI permitted the first daily, space-borne measurements of passive volcanic SO2 degassing, providing improved constraints on the source locations and magnitude of global SO2 emissions for input to atmospheric chemistry and climate models. As a result of this unique sensitivity to volcanic activity, OMI data were also the first satellite SO2 measurements to be routinely used for volcano monitoring at several volcano observatories worldwide. Furthermore, the Aura OMI SO2 data also offer unprecedented sensitivity to volcanic clouds in the UTLS, elucidating the transport, fate and lifetime of volcanic SO2 and providing critical input to aviation hazard mitigation efforts. Another major advance has been the improved vertical resolution of volcanic clouds made possible by synergy between Aura and other A-Train instruments (e.g., AIRS, CALIPSO, CloudSat), advanced UV SO2 altitude retrievals, and inverse trajectory modeling of detailed SO2 cloud maps. This altitude information is crucial for climate models and aviation hazards. We will review some of the highlights of a decade of Aura observations of volcanic activity and look ahead to the future of volcanic observations from space.

Aura↗

NASA NTRS2012

Optimal Dynamic Sub-Threshold Technique for Extreme Low Power Consumption for VLSI

For miniaturization of electronics systems, power consumption plays a key role in the realm of constraints. Considering the very large scale integration (VLSI) design aspect, as transistor feature size is decreased to 50 nm and below, there is sizable increase in the number of transistors as more functional building blocks are embedded in the same chip. However, the consequent increase in power consumption (dynamic and leakage) will serve as a key constraint to inhibit the advantages of transistor feature size reduction. Power consumption can be reduced by minimizing the voltage supply (for dynamic power consumption) and/or increasing threshold voltage (V(sub th), for reducing leakage power). When the feature size of the transistor is reduced, supply voltage (V(sub dd)) and threshold voltage (V(sub th)) are also reduced accordingly; then, the leakage current becomes a bigger factor of the total power consumption. To maintain low power consumption, operation of electronics at sub-threshold levels can be a potentially strong contender; however, there are two obstacles to be faced: more leakage current per transistor will cause more leakage power consumption, and slow response time when the transistor is operated in weak inversion region. To enable low power consumption and yet obtain high performance, the CMOS (complementary metal oxide semiconductor) transistor as a basic element is viewed and controlled as a four-terminal device: source, drain, gate, and body, as differentiated from the traditional approach with three terminals: i.e., source and body, drain, and gate. This technique features multiple voltage sources to supply the dynamic control, and uses dynamic control to enable low-threshold voltage when the channel (N or P) is active, for speed response enhancement and high threshold voltage, and when the transistor channel (N or P) is inactive, to reduce the leakage current for low-leakage power consumption.

Duong, Tuan A.↗

NASA NTRS2020

Chandra and XMM-Newton Observations of A2256: Cold Fronts, Merger Shocks, and Constraint on the IC Emission

We present the results of deep Chandra and XMM-Newton observations of a complex merging galaxy cluster Abell 2256 (A2256) that hosts a spectacular radio relic (RR).The temperature and metallicity maps show clear evidence of a merger between the western subcluster (SC) and the primary cluster (PC). We detect five X-ray surface brightness edges. Three of them near the cluster center are cold fronts (CFs): CF1is associated with the infalling SC; CF2 is located in the east of the PC; and CF3 is to the west of the PC core. The other two edges at cluster outskirts are shock fronts(SFs): SF1 near the RR in the NW has Mach numbers derived from the temperature and the density jumps, respectively, of MT=1.62±0.12andMρ=1.23±0.06; SF2 in the SE has MT=1.54±0.05andMρ=1.16±0.13. In the region of the RR, there is no evidence for the correlation between X-ray and radio substructures, from which we estimate an upper limit for the inverse-Compton emission, and therefore set a lower limit on the magnetic field (∼450 kpc from PC center) of B>1.0μG for a single power-law electron spectrum or B>0.4μG for a broken power-law electron spectrum. We propose a merger scenario including a PC, an SC, and a group. Our merger scenario accounts for the X-ray edges, diffuse radio features, and galaxy kinematics, as well as projection effects.

Chong Ge↗

NASA NTRS2020

Chandra and XMM-Newton Observations of A2256: Cold Fronts, Merger Shocks, and Constraint on the IC Emission

We present the results of deep Chandra and XMM-Newton observations of a complex merging galaxy cluster Abell 2256 (A2256) that hosts a spectacular radio relic (RR). The temperature and metallicity maps show clear evidence of a merger between the western subcluster (SC) and the primary cluster (PC). We detect five X-ray surface brightness edges. Three of them near the cluster center are cold fronts (CFs): CF1 is associated with the infalling SC; CF2 is located in the east of the PC; and CF3 is to the west of the PC core. The other two edges at cluster outskirts are shock fronts(SFs): SF1 near the RR in the NW has Mach numbers derived from the temperature and the density jumps, respectively, of MT=1.62±0.12 and Mρ=1.23±0.06; SF2 in the SE has MT=1.54±0.05 and Mρ=1.16±0.13. In the region of the RR, there is no evidence for the correlation between X-ray and radio substructures, from which we estimate an upper limit for the inverse-Compton emission, and therefore set a lower limit on the magnetic field (∼450 kpc from PC center) of B>1.0μG for a single power-law electron spectrum or B>0.4μG for a broken power-law electron spectrum. We propose a merger scenario including a PC, an SC, and a group. Our merger scenario accounts for the X-ray edges, diffuse radio features, and galaxy kinematics, as well as projection effects.

Chong Ge↗

NASA NTRS1988

Solid earth as a recycling systems and the lateral growth of Precambrian North America

If plotted on mass vs time diagrams, geologic entities (for example, continental and oceanic crust, sediments, and mineral resources) display an exponential (power law) relationship, with entity per unit time increasing toward the present. This relationship is consistent with the concept of recycling and can be simulated mathematically. The approach is based on the plate tectonic theory and considers area-age or mass-age distributions of crystalline basement and sediments for major global tectonic realms. Each tectonic realm is characterized by a specific lifespan, which is an inverse function of its recycling rate. The estimated average half-area of half-mass ages are given. The corresponding parameters for continental crust are 690 Ma for K/Ar, and approximately 1200 Ma for Rb/St and U-Th/Pb dating pairs. Tectonic diversity preserved in the geologic record is therefore a function of time, with oceanic tectonic realms, because of their rapid recycling, underrepresented in the rocks older than approximately 300 Ma. The Sm/Nd isotopic systematic of sediments suggest that, for a near steady-state post-Archean sedimentary mass, recycling is approximately 90 + or - 5 percent cannibalistic. This yields an estimated upper limit on crust-mantle exchange via sediment subduction of approximately 1.1 + or - 0.5 x 10 g a(sup -1) considerably less than demanded by isotopic constraints. The discrepancy may indicate the existence of additional loci, such as orogenic belts, for significant crust-mantle interaction.

Veizer, Jan↗

NASA NTRS1993

STEP: Satellite Test of the Equivalence Principle. Report on the phase A study

During Phase A, the STEP Study Team identified three types of experiments that can be accommodated on the STEP satellite within the mission constraints and whose performance is orders of magnitude better than any present or planned future experiment of the same kind on the ground. The scientific objectives of the STEP mission are to: test the Equivalence Principle to one part in 10(exp 17), six orders of magnitude better than has been achieved on the ground; search for a new interaction between quantum-mechanical spin and ordinary matter with a sensitivity of the mass-spin coupling constant g(sub p)g(sub s) = 6 x 10(exp -34) at a range of 1 mm, which represents a seven order-of-magnitude improvement over comparable ground-based measurements; and determine the constant of gravity G with a precision of one part in 10(exp 6) and to test the validity of the inverse square law with the same precision, both two orders of magnitude better than has been achieved on the ground.

Blaser, J. P.↗

NASA NTRS1991

Lateral variation in upper mantle temperature and composition beneath mid-ocean ridges inferred from shear-wave propagation, geoid, and bathymetry

Resolution of both the extent and mechanism of lateral heterogeneity in the upper mantle constraints the nature and scales of mantle convection. Oceanic regions are of particular interest as they are likely to provide the closest glimpse at the patterns of temperature anomalies and convective flow in the upper mantle because of their young age and simple crustal structure relative to continental regions. Lateral variations were determined in the seismic velocity and attenuation structure of the lithosphere and astenosphere beneath the oceans, and these seismological observations were combined with the data and theory of geoid and bathymetry anomalies in order to test and improve current models for seafloor spreading and mantle convection. Variations were determined in mantle properties on a scale of about 1000 km, comparable to the thickness of the upper mantle. Seismic velocity, geoid, and bathymetry anomalies are all sensitive to variations in upper mantle density, and inversions were formulated to combine quantitatively these different data and to search for a common origin. Variations in mantle density can be either of thermal or compositional origin and are related to mantle convection or differentiation.

Sheehan, Anne Francis↗

NASA NTRS2008

A New Technique for Retrieval of Tropospheric and Stratospheric Ozone Profiles using Sky Radiance Measurements at Multiple View Angles: Application to a Brewer Spectrometer

This paper describes and applies a new technique for retrieving diurnal variability in tropospheric ozone vertical distribution using ground-based measurements of ultraviolet sky radiances. The measured radiances are obtained by a polarization-insensitive modified Brewer double spectrometer located at Goddard Space Flight Center, in Greenbelt, Maryland, USA. Results demonstrate that the Brewer angular (0-72deg viewing zenith angle) and spectral (303-320 nm) measurements of sky radiance in the solar principal plane provide sufficient information to derive tropospheric ozone diurnal variability. In addition, the Brewer measurements provide stratospheric ozone vertical distributions at least twice per day near sunrise and sunset. Frequent measurements of total column ozone amounts from direct-sun observations are used as constraints in the retrieval. The vertical ozone profile resolution is shown in terms of averaging kernels to yield at least four points in the troposphere-low stratosphere, including good information in Umkehr layer 0 (0-5 km). The focus of this paper is on the derivation of stratospheric and tropospheric ozone profiles using both simulated and measured radiances. We briefly discuss the necessary modifications of the Brewer spectrometer that were used to eliminate instrumental polarization sensitivity so that accurate sky radiances can be obtained in the presence of strong Rayleigh scattering and aerosols. The results demonstrate that including a site-specific and time-dependent aerosol correction, based on Brewer direct-sun observations of aerosol optical thickness, is critical to minimize the sky radiance residuals as a function of observing angle in the optimal estimation inversion algorithm and improve the accuracy of the retrieved ozone profile.

Tzortziou, Maria↗

NASA NTRS2004

Path following using visual odometry for Mars Rover in high-slip environments.

An architecture for autonomous operation of Mars rovers in high slip environments has been designed, implemented, and tested. This architecture is composed of several key technologies that enable the rover to accurately follow a designated path, compensate for slippage, and reach intended goals independent of the terrain over which it is traversing (within the mechanical constraints of the mobility system). These technologies include: visual odometry, full vehicle kinematics, a Kalman filter, and a slop compensation/path follower. Visual odometry tracks distinctive scene features in stereo imagery to estimate rover motion between successively acquired stereo image pairs using a maximum likelihood motion estimation algorithm. The full vehicle kinematics for a rocker-bogie suspension system estimates motion, with a no-slip assumption, by measuring wheel, rates, and rocker, bogie, and steering angles. The Kalman filter merges data from an Inertial Measurement Unit (IMU) and visual odometry. This merged estimate is then compared to the kinematic estimate to determining (tracking into account estimate uncertainties) if and how much slippage has occurred then a slip vector is calculated by differencing the current Kalman filter estimate from the kinematic estimate. This slip vector is then used, in conjunction with the inverse kinematics, to determine the necessary wheel velocities and steering angles to compensate for slip and follow the desired path.

Roumeliotis, Stergios I.↗

NASA NTRS2017

Map Projection Induced Variations in Locations of Polygon Geofence Edges

This Paper under-estimates answers to the following question under various constraints: If a geofencing algorithm uses a map projection to determine whether a position is inside/outside a polygon region, how far outside/inside the polygon can the point be and the algorithm determine that it is inside/outside (the opposite and therefore incorrect answer)? Geofencing systems for unmanned aircraft systems (UAS) often model stay-in and stay-out regions using 2D polygons with minimum and maximum altitudes. The vertices of the polygons are typically input as latitude-longitude pairs, and the edges as paths between adjacent vertices. There are numerous ways to generate these paths, resulting in numerous potential locations for the edges of stay-in and stay-out regions. These paths may be geodesics on a spherical model of the earth or geodesics on the WGS84 reference ellipsoid. In geofencing applications that use map projections, these paths are inverse images of straight lines in the projected plane. This projected plane may be a projection of a spherical earth model onto a tangent plane, called an orthographic projection. Alternatively, it may be a projection where the straight lines in the projected plane correspond to straight lines in the latitudelongitude coordinate system, also called a Plate Carr´ee projection. This paper estimates distances between different edge paths and an oracle path, which is a geodesic on either the spherical earth or the WGS84 ellipsoidal earth. This paper therefore estimates how far apart different edge paths can be rather than comparing their path lengths, which are not considered. Rather, the comparision is between the actual locations of the edges between vertices. For edges drawn using orthographic projections, this maximum distance increases as the distance from the polygon vertices to the projection point increases. For edges drawn using Plate Carr´ee projections, this maximum distance increases as the vertices become further from the equator. Distances between geodesics on a spherical earth and a WGS84 ellipsoidal earth are also analyzed, using the WGS84 ellipsoid as the oracle. Bounds on the 2D distance between a straight line and a great circle path, in an orthographically projected plane rather than on the surface of the earth, have been formally verified in the PVS theorem prover, meaning that they are mathematically correct in the absence of floating point errors.

Neeley, Paula↗