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A numerical method is given for the determination of the impedance of an infinitely long thin-wire helix. The propagation constant of the current for zero tangential electric field is found and used in a variational expression for impedance. Asymptotic values of resistance versus pitch are compared with resistances of infinitely long straight-wire antennas.

In order to solve the waveguide junction problem numerically, we express the fields in the guides by truncated modal expansions and construct an error function which is a measure of the mean-square error in the matching of the boundary conditions at the junction. The minimum of this error leads to a set of linear equations for the modal amplitudes. Offset rectangular waveguides with amplitude and current excitations are studied. A weighting factor which multiplies the error contribution due to the magnetic field is studied, and a criterion for its selection given.

1. In a previous communication I proposed a formula for the wave making resistance of ships, and showed that it expressed certain general qualities of experimental results; further, notwithstanding the limitations of theory and the difficulty of interpretation of experimental data, a good numerical agreement was found in several cases with the published results of tank experiments on models when suitable numerical values were given to the coefficients in the formula. This paper records the results of a more systematic study of the numerical values of some of the coefficients, the data being taken from certain recent series of experiments; for the present the discussion is limited to those types of model whose resistance-speed curves show clearly the humps and hollows which are attributed to interference of wave-systems originating at the bow and stern. It has been remarked that although the mode of disturbance is different, the action of the bows of a ship may be roughly compared to that of a travelling pressure-point, and further, that the stern may he regarded in the same way as a negative pressure-point. This point of view originated in the well-known paper of W. Froude on the effect of the length of parallel middle body, and the theory was developed in a later paper by R. E. Froude; from an inspection of experimental results it was seen that the variations in magnitude and position of the oscillations were in directions which agreed with the above interpretation. On account of the lack of an adequate formula, the available data have not yet been examined numerically in any detail; the present investigation aims at supplying this in some measure. Section 2 is theoretical, with some necessary repetition of previous work; Sections 3 and 4 contain a numerical analysis of some available experimental curves. In Section 5 an attempt is made to estimate the effective horse-power of the "Turbinia," in order to illustrate certain points; while in Section 6 the limitations of the interference theory, in the conventional use of the term, are discussed in connection with the residuary resistance curves of finer-ended models.

A numerical-perturbation method is proposed for the determination of the nonlinear forced response of structural elements. Purely analytical techniques are capable of determining the response of structural elements having simple geometries and simple variations in thickness and properties, but they are not applicable to elements with complicated structure and boundaries. Numerical techniques are effective in determining the linear response of complicated structures, but they are not optimal for determining the nonlinear response of even simple elements when modal interactions take place due to the complicated nature of the response. Therefore, the optimum is a combined numerical and perturbation technique. The present technique is applied to beams with varying cross sections. ~ 4Y large-amplitude deflection of a beam or a plate which is restrained at its ends or along its edges results in some midplane stretching/One must account for this stretching with nonlinear strain-displacement relationships. The nonlinear equations of motion describing this situation were the basis of a number of earlier investigations and are the basis for the present paper as well. The purpose of the present paper is to present a new scheme for determining the response to a harmonic excitation. Emphasis is placed on the case when the frequency of the excitation is near a natural frequency. A convenient way to attack this nonlinear problem involves representing the deflection curve or surface with an expansion in terms of the linear, free-oscillation modes. The deflection is then determined in two steps. First, the damping, the forcing, and the nonlinear terms are deleted and the linear modes (eigenfunctions) and natural frequencies (eigenvalues) are determined. Second, the time-dependent coefficients in the expansion are obtained from a set of coupled, nonlinear, ordinary, second-order differential equations, the linear modes being used to determine the coefficients in these equations. (The procedure is described in detail in Sec. II.) Generally, one cannot obtain the linear modes analytically for structural elements having complicated boundaries and composition, and one cannot easily determine the character of the timedependent coefficients through numerical integration of the set of nonlinear equations. (The results obtained in the present numerical example are typical of the complicated manner in which the steady-state amplitudes of the various modes making up the response can vary with the amplitude and the frequency of the excitation.) Consequently, an optimal procedure involves a numerical method to determine the linear, free-oscillation modes and an analytical method to determine the time-dependent coefficients. The present procedure combines either a finiteelement or a finite-difference method with the method of multiple scales (see, for example, Ref. 1). The following brief review mentions representative examples of the work that was and is

Summary This paper is the first of a series that will examine the effect of earth structure on earthquake displacement, strain and tilt fields at the Earth's surface. Its purpose is to develop the numerical techniques to be applied in the papers that foliow. A general computational procedure for the evaluation of the integral expressions for the surface displacements due to an arbitrary point dislocation source in a layered medium is described. It is shown to be rapid and inexpensive to use, and its accuracy appears to be entirely adequate for practical purposes.

In recent papers, the Prufer transformation has been used to calculate the eigenvalues of second order Sturm–Liouville boundary value problems. In the present paper, a Prufer transformation for fourth order vibration problems is devised using four-dimensional spherical coordinates. Some properties of the resulting angular system are investigated. These results are then used to develop a new computational procedure for finding the eigenvalues of a vibrating beam. This procedure is used to compute the eigenvalues of several singular and nonsingular problems.

The starting point of this paper is a theorem, based on the theory of functions of a complex variable, which essentially gives a representation of the remainder in a quadrature rule as a contour integral. From this theorem it is shown how most of the well-known interpolatory and non-interpolatory quadrature rules may be derived. Some new quadrature rules are also given. The latter part of the paper is devoted to the evaluation of asymptotic estimates of the remainder, by consideration of particular examples. It is shown how this asymptotic analysis frequently gives excellent estimates of the remainder.

Consider the Cauchy transform \[T\varphi (z) = \frac{1}{{\pi i}}\int_\Gamma {\frac{{\varphi (\zeta )d\zeta }}{{\zeta - z}},} \quad z \in \Gamma ,\] in which $\Gamma $ is a simple closed curve with a continuously differentiable parameterization. In order to have direct methods for the numerical solution of equations involving $T\varphi (z)$, it is desirable to have numerical integration methods for evaluating $T\varphi (z)$. In this paper, numerical methods are investigated which are based on replacing $\varphi (z)$ by a uniformly convergent sequence $\varphi _n (z)$; if these approximations are “sufficiently smooth”, then the speed of convergence of $T_n \varphi $ to $T\varphi $ is “essentially the same” as the speed of convergence of $T\varphi _n $ to $\varphi $. Specific methods are analyzed, with $\varphi _n (z)$ defined as a piecewise linear or quadratic interpolating function to $\varphi (z)$ at a given set of nodes on $\Gamma $. Numerical examples conclude the paper.

This paper is primarily concerned with the problem of finding approximants for the solution of initial boundary value problems of parabolic type and obtaining a priori error estimates.Perhaps the most widely used method for the treatment of such problems is that of finite differences. Some difficulties arising in this method are stability of the scheme and the treatment of the boundary. Another approach to this problem is the so-called semidiscrete variational method. As in the finite difference approach, a difficulty in this method is the treatment of boundary values.The methods considered in this paper will be of the least-squares type. Roughly speaking, the method consists in finding that element of a suitable finite-dimensional subspace which best approximates the data of the initial-boundary value problem in a least-squares sense. A principal virtue of this approach is that the elements of the finite-dimensional subspace are not required to satisfy any boundary or initial conditions. A particular cho...

This paper reviews some of the more useful, current and newly developing methods for the solution of electromagnetic fields. It begins with an introduction to numerical methods in general, including specific references to the mathematical tools required for field analysis, e.g., solution of systems of simultaneous linear equations by direct and iterative means, the matrix eigenvalue problem, finite difference differentiation and integration, error estimates, and common types of boundary conditions. This is followed by a description of finite difference solution of boundary and initial value problems. The paper reviews the mathematical principles behind variational methods, from the Hilbert space point of view, for both eigenvalue and deterministic problems. The significance of natural boundary conditions is pointed out. The Rayleigh-Ritz approach for determining the minimizing sequence is explained, followed by a brief description of the finite element method. The paper concludes with an introduction to the techniques and importance of hybrid computation.

The numerical solution of linear integral equations of the types studied by Volterra has formed the subject of a recent memoir by E. T. Whittaker. Numerical methods are needed also for the solution of the linear integral equation studied by Fredholm and Hilbert and the object of this paper is to describe a method which may sometimes be useful. The linear integral equation of the second kind, f(s) = Ф(s) - λ ∫ 1 0 k(s, t) Ф(t)dt , may be solved for the unknown function Ф(t) teither directly or indirectly. In the direct methods of solution the function Ф(t) is expressed by means of infinite series of terms involving repeated integrals. In the so-called method of Neumann only one infinite series is used, while in the more complete method of Fredholm the function Ф(t) is expressed as the ratio of two infinite series which converge for all values of the parameter λ. The method of Neumann has been employed on many occasions to obtain numerical results and is generally more convenient to use than Fredholm’s method on account of the great complexity of the expressions occurring in Fredholm’s series. There are occasions, however, when the Neumann series fails to converge, or converges only slowly, and then some other method such as Fredholm’s must be used. It seems desirable if possible to devise simple approximate methods which possess some of the advantages of Fredholm’s method, because in many cases the evaluation of repeated integrals becomes very tedious and the use of the direct methods of solution becomes impracticable except for a rough approxi­mation.

The present paper is devoted to the study of the direct limits of direct systems of measure (resp. probability) spaces. If I is a right directed preordered set, ( E α ) α ∈ I a family of sets indexed by I, G = ⌣ α∈I E α × {α} the sum of the family ( E α ), M α a σ-algebra in E α for each α ∈ I and M = ⌣ α∈I M α × {{α}} is the sum of the family ( M α ), then it is shown that M is a σ-algebra in G . If E = lim E α is the direct limit of the family ( E α ), if E = lim B ( E α ) the direct limit of the family of power sets ( B ( E α )), if M = lim M α is the direct limit of the family ( M α ), if ( E α , M α ) is a direct system of measurable spaces, then ( E, M) = ( lim E α , lim M α ) is a measurable space. If ( λ α ) α ∈ I is a direct system of measures with values in a complete abelian group, if λ = lim λ α is the direct limit of the family ( λ α ), and if ( E α , M α , λ α ) is a direct system of measure (resp. probability) spaces, then it is shown that the direct limit ( E, M, λ) = ( lim E α , lim M α , lim λ α ) is a measure (resp. probability) space. Further papers will be devoted to the applications of these direct limits in the measure (resp. probability) theory.

Non-linearity in structural Analysis is caused in most cases by two factors ; one is plastification of material and another is finite deformation. These two factors usually make the analysis so complicated that it is difficult to obtain the exact analytical solutions. Therefore, numerical analysis is used to get approximate solutions. In this paper, as a first step for solving square plates subject to lateral pressure, one approximate numerical procedure is used for solving the same kind of problem of strip plates. This procedure is based on linearization of non-linear equations by the concept of small increment and on solving these linearized equations by finite difference method.From the fact that analysis of strip plates can be regarded with good accuracy as the analysis of square plates subject to lateral pressure, provided that their aspect ratios are over about 3, the strip plates have been investigated by several researchers analytically and experimentally. The comparison showed that the computed load-deflection curve by the procedure in this paper coincided with the experimental ones published previously with fairly good accuracy. Another results of this paper is estimation of the effect of end restraints of strip plates. The behaviour of strip plates with linear and non-linear end restraints were investigated.

In the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation S(v) = Y v , S′(v) = Y v ′,…,S (r−1) (v) = Y v (r−1) for all v . The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem S(v) = Y v , S′(v) = Y v ′ for all v by quintic spline functions of the class C‴(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n.

In multivariate analysis many of the noncentral latent root distributions can be expressed in terms of hypergeometric functions vFq of two-argument matrices. This paper is concerned with showing that the function 2F1(a, b; c; R, S) satisfies the partial differential equation ∑ i=1 m R i ∂ 2 F ∂R i 2 + ∑ i=1 m ∑ j=1 j≠1 m R i R i −R j ∂F ∂R i +[c− 1 2 (m−1)] ∑ i=1 m ∂F ∂R i −(a+b+2−m) ∑ i=1 m s i 2 ∂F ∂s i − ∑ i=1 m s i 3 ∂F ∂s i 2 − ∑ i=1 m ∑ j=1 j≠1 m s i 3 s i −s j ∂F ∂s i = ab Tr (S)F where R1, R2,…, Rm and s1, s2,…, sm are the latent roots of the m × m symmetric matrices R and S, respectively. Differential equations for the 1F1, 0F1, 1F0 and 0F0 hypergeometric functions are also obtained. Useful applications of these differential equations will be considered in a later paper.

The approach given in this paper leads to numerical methods for Volterra integral equations which avoid the need for special starting procedures. Formulae for a typical fourth-order method are derived and some numerical examples presented. A convergence theorem is given for the method described. 1. Introduction. In this paper we consider the numerical solution of the equation

veloped for the numerical evaluation of multidimensional integrals. These methods are based on the theory of Diophantine approximation. They are suitable for some problems for which the Monte Carlo method is commonly used and, like the Monte Carlo method, are well fitted for use with an electronic digital computer. We shall show, however, that they are superior to the Monte Carlo method provided that the integrand satisfies certain conditions. We shall also show that they are superior, for integrals in space of several dimensions, to formulas typified by those of Gauss and Simpson; they may be superior even to certain new integration formulas specially constructed for the evaluation of multiple integrals (see for example Hammer [2], who gives a bibliography, and Miller [5], [6], [7]). The method of antithetic variates which is described by Hammersley and others [3], [4] may be used to obtain better estimates than the Monte Carlo method but the author thinks that the method described in the present paper is simpler to apply and gives better results. Various authors have suggested methods which are particular cases of those described in this paper but without the underlying theory. See for example Davis and Rabinowitz [1]. In this section we shall give a short account of the behavior of the error in the Monte Carlo method and the direct-product Gauss-type methods so that we can compare these with the errors of the new methods. We shall not give an account of the method of antithetic variates. Suppose that we wish to estimate the integral We shall denote the vector (x1, x2, * *, Xk) by x. Numerical methods for the evaluation of I involve the calculation of f(x) at a number N of points xi. The most desirable of such methods for use on an electronic computer are those which require the evaluation of f(x) at the smallest number of points xi in order to obtain an estimate with a sufficiently small error. The Monte Carlo method gives as an estimate for I the sum iN - Z f(x)y