Abstract: Cubic splines are employed, experimentally, to approximate to the solution of a simple two-point boundary value problem for a linear ordinary differential equation. Checked by comparison with the analytical solution, the results are encouraging. Comparison is also made with the application of Hermite interpolation, which gives results of the same order of accuracy—but, generally, involves the solution of many more linear equations for the same number of nodes.

Abstract: A generalized logistic equation is proposed for the mathematical representation of batch culture kinetic data. Properties of the equation are discussed. A computer program is used to fit the generalized equation to both artificial and actual batch culture data. The equation is shown to be capable of fitting data exhibiting lag, exponential, deceleration, stationary, and death phases, as well as diauxic growth. The fitted equation is useful for differentiation, interpolation, and other manipulations of the data, and it is a convenient means of data storage.

Abstract: One simple method of data compression relies on the approximation of the source output by polynomial segments or "interpolators." The parameters of each polynomial are transmitted in place of the original data. This paper presents a method of theoretically analyzing such techniques. Straight line interpolation is considered specifically although the ideas can he readily generalized. It is shown that the class of compression methods considered may or may not perform well depending on the data and, thus, that in some cases more complex techniques, including possibly adaptive methods, might he used, depending on the knowledge of the data statistical model.

Abstract: The object is to provide a simple scheme for calculating the dependence of the $d$-band structure on the position of the atoms in the crystal. The Korringa-Kohn-Rostoker method in the narrow-band approximation has been used to calculate the nonhybridizing $d$ bands for different crystal structures and for various values of volume/atom, in terms of the position and width of the $d$-scattering resonance. These bands have been fitted by near-neighbor linear combination of atomic orbitals (LCAO) parameters, as in the interpolation schemes of Hodges, Ehrenreich, and Lang, and of Mueller. It is shown that the dependence of the LCAO parameters on structure and volume can be described to a good approximation by simple structure-independent functions. The accuracy of the description for the $d$ bands of the fcc and bcc phases of Fe is about 0.01 Ry. The results are discussed in relation to recent developments in transition-metal band theory.

Abstract: In many problems of potential scattering, and particularly in the inverse‐scattering problem, two equations prove to be of essential interest: the Gel'fand‐Levitan equation and the Regge‐Newton equation. These and their generalizations apply, respectively, to energy‐independent and to λ‐independent potentials. In this paper, a method is devised for obtaining equations applying to more general cases. The requirement is the existence of analytic properties of the wavefunctions corresponding to special classes of potentials, enabling one to construct interpolation formulas of the Lagrange form. From these formulas, it is possible to derive integral equations which may then be generalizable to much larger classes of potentials. This method is fully developed in the case of potentials depending linearly on λ. Interpolation formulas and analytic properties of the wavefunctions in the λ plane are exhibited. Integral equations are given and proved to apply to very large classes of potentials. Existence, uniqueness, and analytic properties of their solutions are thoroughly studied. An example is given for which all the wavefunctions are calculated exactly. Application of the method to the inverse scattering problem in the presence of a spin‐orbit potential will be the object of a forthcoming paper.

Abstract: 650 levels belonging to the configurations (3d + 4s)n of all second spectra of the iron group were calculated, and 472 observed levels were fitted to them. In addition to the usually used approximation, a complete set of two-body and three-body effective-interaction parameters between 3d electrons was also introduced. Using only two-body effective interactions, we obtained a mean error of ±192 cm−1; while the addition of three-body effective interactions reduced the mean error to ±87 cm−1.

Abstract: The concept of reducing the required transmission rate for a given system through prediction, interpolation, or other such techniques loosely labeled as "data compression" is now well known. The problems in analyzing such systems by theoretical means are formidable in even the simplest situations due to the inherent nonlinear nature of the operations performed. This paper discusses these difficulties, presents some approximate and exact solutions, and suggests areas where further work is needed.

Abstract: : Techniques of interpolating from irregularly spaced known function values in two dimensions are discussed. First, a function defined as a weighted average of the given function values, with the weighting based on the inverse distance from the corresponding points, is explored. To shorten computation and improve the quality of interpolation, additional terms based on distances, directions and slopes are added. An interpolation function everywhere continuously differentiable and fitting the data exactly is finally presented. A provision for barriers to interpolation is described for applications in geographical analysis. The interpolation function has been incorporated into a computer mapping program in FORTRAN called SYMAP (Versions IV and V). (Author)

Abstract: From (l.~) it follows that the norm 11.111 on a Banach fu町tion space is semicontinuous, i. e円。 ζfn ↑f, f-fdX implies ilfii=yp1ijpt il.The space (工 11 . 1¥)is called Jでarrangcment ùruaバωzt, if 0ζfε X implies 9 ε . r-and I\\f\\\\ 二\\\\9 ¥ for each fu町tion g , equimeasurable with f・ Let V , L∞ be the Lebes只ue spaces over (0 , 1), and let R (V; L∞) be the set of al boundecl linear operators from each of the spaces .L1, I~∞ into itself. By ¥¥Tlli (i=l , or ニ∞) we denote the norm of an operator Tε R(V 、 L∞) on the corresponding spaces. For each a>O , J\" is the functionεiven by j;, (刈 =f(([.r) , if ([.r l. We write also

Abstract: In part I of this paper the SLATER-sums of two charged particles were expanded in TAYLOR-series with respect to the distance between the particles. Using these expansions we calculate the binary SLATER-sums for small values of r(r ≪ λ) λ-thermal wavelength). In the case of r ≫ λ the binary SLATER-sums can be approximated by the classical BOLTZMANN-factor. In the intermediate region r ≈ λ we get the binary SLATER-sums by interpolation. For high temperatures we obtain KELBG's result. For special cases the binary SLATER-sums and the binary distribution functions are presented graphically and discussed.

Abstract: A numerical path control system including a time shared digital computer generating for each of a plurality of axes regularly recurring numerical excursion commands. A two axis system is shown to be capable of deriving from a small number of data words successive straight line path or circular arc path command signals at a rate of 50 times a second and of converting these into intermediate or secondary command signals which are applied to the servos of the machine axes at a rate of 500 times a second.

Abstract: The theory of generalized multistep methods using an off-grid point is extended to the special second-order equation y″ = f(x, y). New high-order methods for solving this equation, based on quasi-Hermite interpolating polynomials, are shown to exist, as well as new explicit generalized methods for a first-order equation. Some results in the theory of quasi-Hermite interpolation are given, and results of computations of an unperturbed orbit trajectory are presented.

Abstract: Successive linear or circular arc path segments are represented by successive ''''blocks'''' of data. A corrective change is made in the data block representing a given path segment if the preceding path segment did not terminate exactly at its designated end point.