Abstract: This paper presents the results of an investigation to develop new techniques to enhance the moire method of mechanical interferometry. The program demonstrated the practicality of developing the use of orthogonal bicolored moire fringes with the unique capability to selectively display either set of fringes while obscuring the other set. An interpolation method for the determination of fractional fringe orders is also discussed.

Abstract: On the basis of a 10-yr record of rawinsonde observations in the Tropics, experiments were run to illustrate the manner in which climatology may be used to minimize the root-mean-square errors of interpolation from data of mixed quality that are irregularly located in time and space. The procedure, based on the theory of optimum interpolation, determines the relative weights of the data used in the interpolation on the basis of their error characteristics, their location, and the scale and variability of the meteorological fields that they sample.

Abstract: A combined linear and quadratic interpolation method for Brillouin-zone integration is described. Comparisons of the combined method with previously introduced linear and quadratic approaches indicate that the combined scheme will require the least computer time in most cases. Calculations of the densities of states of phonons in copper and electrons in nickel are given as examples. Problems associated with critical points and the intersection of dispersion curves are also investigated numerically to a limited extent.

Abstract: We construct an optimal interpolation formula for a particular class of analytic functions, optimization being over a set of interpolation methods which are not necessarily linear. Optimal nodes and the norm of the error are found for the optimal interpolation formula.

Abstract: [5]; they form a complete orthonormal set in the Hilbert function space known as the Paley-Wiener functions, and wn(m) = 5nm (Kronecker's Symbol) for all integers n and m. This means that the cardinal series is not only an orthogonal expansion for the Paley-Wiener functions, but it also provides a process for interpolation at the integers since the series reduces formally to am when x is an integer m. In the present note we shall consider further sets of this type and in particular a set involving Bessel functions. Cardinal series interpolation has important applications in information theory, where it was introduced by C. E. Shannon [11]. It is, for example, important for the electrical engineer to know that a certain type of transmitted signal, a function of time, lies in a subspace (the Paley-Wiener functions) of L(-oo, oo), and that this subspace possesses an orthogonal basis with respect to which the \" coordinates \" of the signal are actually values taken by the signal at certain instants of time. It was with this application in mind that H. P. Kramer introduced a generalisation of the cardinal series in a lemma which we adopt as the starting point for the present discussion. LEMMA 1 (Kramer [7]). Let (a, b) be a finite interval of U (the real numbers). Let K(x, t) e L(a, b)for each xeU and suppose that the sequence of real numbers {xn} (where n runs over some indexing set of integers) is such that {K(xn, t)} forms a complete orthogonal set (COS) in L(a, b). If

Abstract: This paper describes two algorithms which find the visible portions of surfaces in a picture of a cluster of three-dimensional quadric patches. A quadric patch is a portion of quadric surface defined by a quadratic equation and by zero, one, or several quadratic inequalities. The picture is cut by parallel planes called scan planes. The visibility problem is solved in one scan plane at a time by making a "good guess" as to what is visible according to the visible portions found in the previous scan plane.

Abstract: A method is described for using a digital computer to construct contour maps automatically. Contour lines produced by this method have correct relations to given discrete data points regardless of the spatial distribution of these points. The computer‐generated maps are comparable to those drawn manually. The region to be contoured is divided into quadrilaterals whose vertices include the data points. After supplying values at each of the remaining vertices by using a surface‐fitting technique, bicubic functions are constructed on each quadrilateral to form a smooth surface through the data points. Points on a contour line are obtained from these surfaces by solving the resulting cubic equations. The bicubic functions may be used for other calculations consistent with the contour maps, such as interpolation of equally spaced values, calculation of cross‐sections, and volume calculations.

Abstract: Let L be a first-order finitary predicate language with equality. For each pair of infinite cardinals K and X with fc^/lwe let LKX be the logic extending L which allows the conjunction ( A ) and disjunction ( v ) of fewer than K formulas and the simultaneous universal or existential quantification of fewer than k variables. We set L^x — \JKLKk. The standard syntactical and semantical concepts are defined as usual (see [1], [2]). If 0 is a sentence we write 211= 0 to mean that 0 is true on the model 21. 31 = Kk 23 means that 2Ï and 93 have the same true sentences of LKA. 21,93, and 21, are always used for models for L, and we follow the convention that their universes are A, B9 At respectively. The cardinality of a set X is denoted by \X\. If L' is some other language, then LKk is the corresponding infinitary logic built on L'. For ease in stating many of our results we assume, except in the last section, that L has only countably many nonlogical symbols. A detailed presentation of these and related results is in preparation for publication elsewhere.

Abstract: The well-known error formula for Lagrange interpolation is used to derive an expression for a truncation error bound in terms of the sampling rate and Nyquist frequency for regular samples and central interpolation. The proof is restricted to pulse-type functions possessing a Fourier transform. The formula finds application to the estimation of convergence rate in iterative interpolation, thus providing a criterion for the choice of sampling rate to achieve a specified truncation error level in a given number of steps. The formula can also be used as a guide when the samples are not regular but fairly evenly distributed.

Abstract: The differential method of van't Hoff is a powerful method for kinetic data analysis if accurate rates of reaction can be found. Differentiation methods reported in the literature are inadequate, particularly when applied to data which is unequally spaced or which is subject to random error. A need is therefore expressed for a general numerical method of differentiation which can be readily applied in kinetic analysis. This paper suggests the use of the cubic spline, an interpolation method which overcomes the difficulties of other methods and gives exceptionally accurate derivatives. The theory of the cubic spline, its properties, and uses are discussed. Its application to experimental data is demonstrated and its performance is quantitatively compared with better known but less accurate methods.