Showing papers in "Mathematics of Computation in 1960"

Calculation of transient motion of submerged cables

Abstract: The system of nonlinear partial differential equations governing the transient motion of a cable immersed in a fluid is solved by finite difference methods. This problem may be considered a generalization of the classical vibrating string problem in the following respects: a) the motion is two dimensional, b) large displacements are permitted, c) forces due to the weight of the cable, buoyancy, drag and virtual inertia of the medium are included, and d) the properties of the cable need not be uniform. The numerical solution of this system of equations presents a number of interesting mathematical problems related to: a) the nonlinear nature of the equations, b) the determination of a stable numerical procedure, and c) the determination of an effective computational method. The solution of this problem is of practical significance in the calculation of the transient forces acting on mooring and towing lines which are subjected to arbitrarily prescribed motions. 1. Introduction. This problem arose as a result of an urgent requirement by the Navy in connection with a series of nuclear explosion tests which were conducted in the Pacific. In preparation for these tests a number of ships were instrumented and moored at specified locations from the explosion point. These positions had to be maintained intact during the period preceding the explosion. However, the bobbing up and down of the ships due to ocean waves could excite transient forces in the mooring lines sufficient to break them and thus result in the loss of informa- tion from the tests. Several months prior to these tests a request was made to the Applied Mathematics Laboratory to calculate the magnitude of the forces acting on the mooring lines for waves of varying amplitude and frequency. The two factors which made a theoretical solution feasible at this time, whereas it would not have been possible several years ago, 'vere: a) the availability of a high-speed computer and b) the recent progress made in the understanding and development of nu- merical methods for the solution of systems of partial differential equations by finite-difference methods. Although this problem was solved to satisfy a specific request, it is more useful to regard it as the general problem of the two-dimensional motion of a cable or rope immersed in a fluid, and it becomes immediately apparent that its solution i.s applicable to a wide class of engineering problems involving the motion of cables. such as: a) the laying of submarine telegraph cables, b) the towing of a ship or other object in water, or c) the snapping of power lines as a result of transient forces caused by storms. The problem may be stated abstractly as follows: Given the initial conditions (i.e., position and velocity at any time, t0) and boundary

Computation of Fresnel integrals

Abstract: flio = +1.595769140 -0.000001702 -6.808568854 = -0.000576361 = +6.920691902 = -0.016898657 -3.050485660 = -0.075752419 = +0.850663781 = -0.025639041 = -0.150230960 On = +0.034404779 6o &i fe2 63 bt 66 6, 67 6, b, bio bn = -0.000000033 = +4.255387524 = -0.000092810 = -7.780020400 = -0.009520895 = +5.075161298 = -0.138341947 = -1.363729124 = -0.403349276 = +0.702222016 «■ -0.216195929 +0.019547031 = 0 = -0.024933975 = +0.000003936 = +0.005770956 = +0.000689892 = -0.009497136 = +0.011948809 = -0.006748873 = +0.000246420 = +0.002102967 C10 = -0.001217930 di = +0.000233939 do = +0 di = +0 d2 = -0 d3 = +0 dt = +0, db = +0, d. = -0 d, = +0 d8-0 d¡ = +0 dio = -0 du = +0, 199471140 000000023 009351341 000023006 004851466 001903218 017122914 029064067 027928955 016497308 005598515 000838386

Effective cross section values for well-moderated thermal reactor spectra

Abstract: The compilation of effective cross sections (CRRP787) included neutron temperatures up to 76O C. Values are presented up to 128O C. The validity of the lowenergy cut-off of the epithermal neutron spectrum assumed in CRRP-787 was also examined. Calculations are repeated for some of the more important elements for cut-off functions of 4 and 3 kt. (W.D.M.)

The rational approximation of functions which are formally defined by a power series expansion

Abstract: 1. Introduction. The advent of high speed digital computers and the consequent intensification of interest in the study of numerical analysis has caused considerable attention to be paid to the problem of obtaining approximation formulas for functions which occur in the theory of mathematical physics. It is the purpose of this note to describe the theory underlying various methods of obtaining rational approximations to functions which are formally defined by a power series expansion; it is assumed that the power series concerned are quite general in character, and that the functions with which they are associated do not satisfy a particular functional equation which would permit the use of any special method. The theory is then subjected to a detailed analysis in terms of the computational steps involved, and a comparison, with regard to computational efficiency, of the various methods which may be devised for obtaining rational approximations is given. 2. The Pade Table and the E-Array. The approximation to the function defined

A method for calculating solutions of parabolic equations with a free boundary

Abstract: (a) uxx(x, t) =ut(x, t), 0 0 (b) ux(0, t) = -1, t > 0 (1) ~~~~~~(c) x-t) = -UX(X(t), t, t > 0 (d) u (x, t) = O. x _ x (t), t > 0 (e) x(0) = 0.

A method for the numerical evaluation of finite integrals of oscillatory functions

Abstract: However in physical problems the finiteness of the range of integration is often associated with a kind of natural boundary of f(x), such that it is impossible to extend f(x) to values of x beyond the upper limit b while preserving the general character of f(x). Analytically speaking, x = b may be a branch point of f(x). Alternatively, it may be possible to extend the range of integration to infinity as in equation (2), but the infinite integrals may not converge. As an example of the branch point difficulty we can consider the integral

On Liouville’s function

Abstract: is negative or zero for all x > 2, and in fact this is true within the range where this function has been previously calculated. Calculations connected with the present study show that L(x) 0. However, his method does not furnish explicitly an x for which the conjecture fails; and in fact it does not give an upper bound for the first counterexample. In the present paper we shall describe calculations which lead to the result that L(906,180,359) = +1. We have not found a smaller value of ? for which the conjecture fails, but also we have not proved that this is the smallest x greater than 2 for which L(x) is positive.

On the conjecture of Hardy & Littlewood concerning the number of primes of the form ²+

Abstract: taken over all odd primes, w, which do not divide a, and for which (-a/w) is the Legendre symbol. In the trivial cases, a = -k2, since (k2/w) = +1 for every w, we have ha = 0 on the one hand, and on the other there can be at most one prime of the form n2 _ k2 = (n k) (n + k). For any other a, ha > 0, and the conjecture indicates that there are infinitely many primes. But for no a has this been proven. In particular, for a = 1, since (-1 /w) equals +1 or -1 according as w 4m + 1 or 4m1, we have

Numerical integration formulas of degree two

Abstract: where R is a region in n-dimensional, real, euclidean space; x = (xi, X2, ** , ; the ai are constants; and the vP are points in the space. Most previous authors have given formulas for special regions (for a bibliography see [4]). Thacher [71 has given a method for constructing formulas of degree 2 with n + 1 points for general regions and of degree 3 with 2n points for certain symmetric regions; with his method, however, each region must also be treated separately. Our main results are to obtain specific formulas of degree 2 with n + 1 points for a general region satisfying a certain condition of non-degeneracy, and to show that for these regions such formulas cannot be obtained with fewer points. We also give a specific 2n point formula of degree 3 for a general centrally symmetric region. These results are a generalization of those of Georgiev [1, 2, 3] who has obtained similar results (but gives no general formulas) for n = 2, 3 with w(x) -1. Our results are obtained by a different method which was developed without knowledge of Georgiev's work.

Abscissas and weights for Lobatto quadrature of high order

Abstract: In recent years, Gaussian quadrature has become the standard method for numerical integration in many computer installations [1, 2]. In general, Gaussian rules are most economical since an n-point rule is exact for polynomials up to degree 2n 1 and no rule can do better. However, for particular classes of functions and for particular applications, other rules may be more efficient. Thus there are occasions when we prefer a closed rule, i.e. one which includes among its abscissas the two end points of the integration interval. This is the case when the integrand vanishes at the two end points as it does in Longman's method for evaluating integrals of oscillating functions [3]. If we want to check a quadrature over a given interval by doing two additional quadratures, each over half the interval, then a closed rule will save at least two evaluations of the integrand. Let us normalize our integration interval to (-1, 1) and consider the closed symmetric n-point integration rule with n odd, n = 2m + 1,

On the factors of certain Mersenne numbers. II

Abstract: This paper is the second of two papers dealing with the prime factors of Mersenne numbers Mp = 2P — 1 of prime exponent (see Brillhart and Johnson [1]). The 899 new factors given below, which were discovered on the IBM 7090 of the Computing Facility at the University of California at Los Angeles, constitute the remaining factors needed for a complete listing in the literature of all prime factors q < 235 for 103 g p g 257 and q < 2U for 257 < p < 20000 (See Brillhart [2], Karst [4], Kravitz [5], Riesel [6], [7]). In pursuing the present investigation, all factors in the literature were rediscovered, which has thus permitted a thorough checking of the tables cited above. Complete agreement was obtained with the tables of Karst and Kravitz, while errors were found in Riesel [7]. (See [9]. Also see Self ridge [8]). It should be noted that the three small factors omitted in Riesel [7] for p = 1451 and 1459 are given in the table below. Also, it should be mentioned that the factors marked in the following table with an asterisk were discovered earlier by E. Karst. The method of search was essentially the same as in [1]. The complete set of factors produced by the program was tested for correctness and multiplicity by a special program written for that purpose. All factors were found to be correct, and none was found to be multiple, thus adding further weight to the conjecture that no multiple factor of Mp exists for p a prime. The factor limit used by the search program was the first multiple of 23040p = 4« 4>(3-5-7-ll-13) greater than 234 (here is the Euler function). Of the trial divisors > 234 only 18504622999, which is a divisor of Mmm , turned out to be a factor. The running time for each Mp , which is an inverse linear function of p, varied from less than one minute for p R¿ 20000 to approximately fifteen minutes for p = 263. A primary concern in the present investigation was the discovery of all M„, 3300 < p < 5000, which had factors < 234. These Mp, being composite, were of interest in a search for prime Mersenne numbers over the same interval, since they could automatically be excluded from that search. (See Hurwitz [3]). No doubt the following table will be of considerable value to investigators in search of large Mersenne primes, who proceed beyond the present range. The author would like to thank John Selfridge and the other directors of the UCLA Computing Facility for their very generous support of this project.

The formula-controlled logical computer “Stanislaus”

Abstract: The evaluation of a formula of propositional calculus is considerably simplified if this formula is written in the parenthesis-free notation of the Warsaw School, [1]. The Warsaw notation may be formulated in the following way: There are symbols for operations, e.g.: N for negation; K for conjunction; A for disjunction; E for equivalence; C for implication; and symbols p, q, r, 8, t for variables. A variable is a formula. A formula preceded by the symbol N is a formula. Two juxtaposed formulas preceded by any one of the symbols, K, A, E, C are a formula. Evaluation of such a formula is done in the following way: Each of the variable symbols p, q, r,... has a value 0 or 1. The operation symbol acts on the value of the one or two formulas governed by it giving the value of the compound formula. It was remarked in 1950 by H. Angstl, [21, that a mechanical evaluation of a formula, written in Warsaw notation without brackets, can be done in the following easy way: Each of the variable symbols is represented by a box with one output, the negation by a box with one input and one output, and the other operation symbols by a box with two inputs and one output. The meaning of a formula in the Warsaw notation is given by Angstl's rule: The first input of each operation symbol is to be connected with the output of the next following symbol, either variable or operation. The symbol N excepted, the second input of each operation symbol is to be connected with the first remaining free output of a symbol going from left to right. This may be demonstrated by an example, which uses Stanislaus present capacity of eleven symbols: the tautology of transitivity of the implication [(p -+ q) & (q -+ r)] -+ (p -r), written in Warsaw notation

Products of Laguerre polynomials

Abstract: and, in particular, is symmetric in r, s, t. A closed formula has been obtained for Cry by Watson [41. We begin by obtaining the same formula by a very simple argument. In ? wce (lerive a simple recurrence relation suitable for rapidly generating the coefficients as needed when working with a high speed computing machine. This will be seen to be more useful in practice than the formal expression in equation (7). 2. It is known [1t that the Laplace transform of L,(x) is p"'(p 1) while that of Lh(x)Lk(X) is

Tables of values of the modified Mathieu functions

Abstract: When the above infinite hyperbolic series are substituted into equation (2), recurrence relationships inay be derivred for allowable values of the characteristic numbers a.-+, (p = 0 or 1) for the even functions, or b2n+ (s = 1 or 2) for the odd functions for a given value of q. Recurrence equations also give the allowable values of the Fourier coefficients A (2`3 and B 4n+ s The formulas for both the characteristic numbers and the Fourier coefficients associated with each type of solution [1, p. 29, 371] are given below: For y = Ce2^(i, q)