Role of the symmetry energy in atomic mass formulas.
Mass formulas differing in treatment of nuclear symmetry energy with very neutron-rich nuclei being bound in one formula and unbound in other
NEUTRON DISTRIBUTION↗
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Equations for Runge-kutta Formulas Through the Eighth Order
Tables for elementary weights of Runge-Kutta formulas of first eight orders and for relations of explicit formulas through order seven
WEIGHTING /MATH/↗
Analysis and design of integration formulas for a random integrand
Analysis of integration formulas and procedure for designing optimal integration formula
Ho, Y. C.↗
Polynomial-smoothing and derivative-estimating formulas for functions of one or two independent variables
Application of polynomial-smoothing formulas and related derivative-estimating formulas simplifies certain linear least-squares problems. These problems can then be solved by a computer method.
Semtner, A. J.↗
Field dependence of gaseous ion mobility: Test of approximate formulas
The accuracies of three approximate formulas were tested by comparison with special cases for which accurate results could be found. The Wannier free flight theory was found to be superior, and can be extended to yield a formula without further adjustable constants that gives an exact result at low electric fields and good results at medium and high fields. It is applicable for any ion neutral force law and mass ratio.
Hahn, H.↗
Simplified formula for mean cycle-slip time of phase-locked loops with steady-state phase error.
Previous work shows that the mean time from lock to a slipped cycle of a phase-locked loop is given by a certain double integral. Accurate numerical evaluation of this formula for the second-order loop is extremely vexing because the difference between exponentially large quantities is involved. The presented article demonstrates a method in which a much-reduced precision program can be used to obtain the mean first-cycle slip time for a loop of arbitrary degree tracking at a specified SNR and steady-state phase error. It also presents a simple approximate formula that is asymptotically tight at higher loop SNR.
Tausworthe, R. C.↗
A precompiler for the formula manipulation system TRIGMAN.
Discussion of a translator designed to simplify the programming of problems involving the TRIGMAN formula manipulation system. The translator allows for the introduction of a new data type, SERIES, into a FORTRAN program and translates a user's program into legal FORTRAN. The translator is adaptable to other formula manipulation systems presently used in celestial mechanics.
Jefferys, W. H.↗
The ATA-67 Formula for Direct Operating Cost
The ATA formulas for direct operating cost were developed for the purpose of comparing different aircraft, existing or not, on the same route or the same aircraft on different routes. Such characteristics of the airline as crew pay, maintenance procedures, and depreciation schedules are kept constant. In air transportation systems analysis the 1967 ATA formula is usually used with appropriate exceptions or modifications, such as: different maintenance labor rate, total maintenance multiplied by a factor, maintenance burden deleted, different depreciation schedule, or different spares percentages.
Faulkner, H. B.↗
Recursive formulas for the partial fraction expansion of a rational function with multiple poles.
The coefficients in the partial fraction expansion considered are given by Heaviside's formula. The evaluation of the coefficients involves the differential of a quotient of two polynomials. A simplified approach for the evaluation of the coefficients is discussed. Leibniz rule is applied and a recurrence formula is derived. A coefficient can also be determined from a system of simultaneous equations. Practical methods for the performance of the computational operations involved in both approaches are considered.
Chang, F.-C.↗
Gyrosynchrotron radiation formulae
The wave-normal emissivity and the ray emissivity formulas for an electron moving along a helical path in a magnetoactive medium are presented. Simplified formulas for the case of an isotropic plasma are also given. Because of the helical motion of the electron, a difference exists between the radiated power per unit solid angle and the received power per unit solid angle. The relation between these two quantities in a magnetoactive medium is shown. Results are compared with those obtained by others, and the sources of discrepancies are pointed out.
Ko, H. C.↗
Derivation of formulas for root-mean-square errors in location, orientation, and shape in triangulation solution of an elongated object in space
Formulas are derived for the root-mean-square (rms) displacement, slope, and curvature errors in an azimuth-elevation image trace of an elongated object in space, as functions of the number and spacing of the input data points and the rms elevation error in the individual input data points from a single observation station. Also, formulas are derived for the total rms displacement, slope, and curvature error vectors in the triangulation solution of an elongated object in space due to the rms displacement, slope, and curvature errors, respectively, in the azimuth-elevation image traces from different observation stations. The total rms displacement, slope, and curvature error vectors provide useful measure numbers for determining the relative merits of two or more different triangulation procedures applicable to elongated objects in space.
Long, S. A. T.↗
Derivation of transformation formulas between geocentric and geodetic coordinates for nonzero altitudes
Four formulas, for the nonzero altitude transformation from geodetic coordinates (geodetic latitude and altitude) to geocentric coordinates (geocentric latitude and geocentric distance) and vice versa, are derived. The set of four formulas is expressed in each of the three useful forms: series expansion in powers of the earth's flattening; series expansion in powers of the earth's eccentricity; and Fourier series expansion in terms of the geodetic latitude or the geocentric latitude. The error incurred in these series expansions is of the order of one part in 3 x 10 to the 7th power.
Long, S. A. T.↗
A prefiltering version of the Kalman filter with new numerical integration formulas for Riccati equations
A prefiltering version of the Kalman filter is derived for both discrete and continuous measurements. The derivation consists of determining a single discrete measurement that is equivalent to either a time segment of continuous measurements or a set of discrete measurements. This prefiltering version of the Kalman filter easily handles numerical problems associated with rapid transients and ill-conditioned Riccati matrices. Therefore, the derived technique for extrapolating the Riccati matrix from one time to the next constitutes a new set of integration formulas which alleviate ill-conditioning problems associated with continuous Riccati equations. Furthermore, since a time segment of continuous measurements is converted into a single discrete measurement, Potter's square root formulas can be used to update the state estimate and its error covariance matrix. Therefore, if having the state estimate and its error covariance matrix at discrete times is acceptable, the prefilter extends square root filtering with all its advantages, to continuous measurement problems.
Womble, M. E.↗
Timing formulas for dissection algorithms on vector computers
The use of the finite element and finite difference methods often leads to the problem of solving large, sparse, positive definite systems of linear equations. MACSYMA plays a major role in the generation of formulas representing the time required for execution of the dissection algorithms. The use of MACSYMA in the generation of those formulas is described.
Poole, W. G., Jr.↗
The determination of gravity anomalies from geoid heights using the inverse Stokes' formula, Fourier transforms, and least squares collocation
A numerical method for the determination of gravity anomalies from geoid heights is described using the inverse Stokes formula. This discrete form of the inverse Stokes formula applies a numerical integration over the azimuth and an integration over a cubic interpolatory spline function which approximates the step function obtained from the numerical integration. The main disadvantage of the procedure is the lack of a reliable error measure. The method was applied on geoid heights derived from GEOS-3 altimeter measurements in the calibration area of the GEOS-3 satellite.
Rummel, R.↗
Formulas for determining storm movement from the surrounding fields
Formulas are given for determining a conserved property of a storm and its centroid from an scalar field that has a tendency equation. The velocity of the centroid is also given. These formulas depend only on fields that are external to the storm.
Costen, R. C.↗
A note on a corrector formula for the numerical solution of ordinary differential equations
A new corrector formula for predictor-corrector methods for numerical solutions of ordinary differential equations is presented. Two considerations for choosing corrector formulas are given: (1) the coefficient in the error term and (2) its stability properties. The graph of the roots of an equation plotted against its stability region, of different values, is presented along with the tables that correspond to various corrector equations, including Hamming's and Milne and Reynolds'.
Chien, Y.-C.↗
Analysis and testing of numerical formulas for the initial value problem
Three computer programs for evaluating and testing numerical integration formulas used with fixed stepsize programs to solve initial value systems of ordinary differential equations are described. A program written in PASCAL SERIES, takes as input the differential equations and produces a FORTRAN subroutine for the derivatives of the system and for computing the actual solution through recursive power series techniques. Both of these are used by STAN, a FORTRAN program that interactively displays a discrete analog of the Liapunov stability region of any two dimensional subspace of the system. The derivatives may be used by CLMP, a FORTRAN program, to test the fixed stepsize formula against a good numerical result and interactively display the solutions.
Brown, R. L.↗