Solution Of Mathematical Problems Research Articles

The Problem Bureau And Some of Its Problems

The first reference to the Problem Bureau in The Mathematical Gazette was in July 1928: “The number of applications to individual Officers of the Association for solutions of mathematical problems is now so considerable as to give reason for the belief that many members would appreciate the formation of something akin to a Problem Section to which all such enquiries may be sent.

Speculations on Stonehenge

Proposals that Stonehenge was con- structed as an astronomical observatory [1, 2, 3] with the purpose of predicting eclipses [4, 5] imply that the builders of Stonehenge, even Stonehenge I, were possessed of a degree of intellectual sophistication that seems inconsistent with the usual picture of the population of S. England in the and millennium B.C. Either the people were not just primitive farmers or these proposals must be substantially in error. It is not the purpose of this paper to attempt a decision between these alternatives but to explain the nature of the astronomical arguments in more detail than has been done heretofore. My hope is that with a moderate effort it will be possible for the reader to rework the calculations on which the astronomical suggestions have been based; for only when archaeologists and astronomers have understood each other's arguments can we expect to resolve this intriguing dilemma. At several places in the discussion I shall quote the solution of mathematical problems, thereby permitting the whole discussion to be confined to simple trigonometry. These few places will be marked by an asterisk. The reader wishing to cover these gaps will need to consult a text on spherical astronomy, e.g. [6].

Simple Mathematical Solutions of Problems Associated with Heat Sterilization of Milk

SUMMARYThree simple equations have been developed along with tabulated values that allow calculation of sterilizing time, sterilizing value, or heat penetration factor for a process when only two of the three are known. A short trial‐and‐error calculation can yield the retort temperature if other data are known. A mathematical derivation associated with the problem is included as an appendix.

A Comparison of “Best” Polynomial Approximations with Truncated Chebyshev Series Expansions

Introduction. In the numerical solution of mathematical problems it is common to represent a function of a real variable by the leading terms of its infinite Chebyshev series expansion. The purpose of this paper is to compare the accuracy of such a polynomial approximation with that of the polynomial approximation of the same degree (the being the one whose greatest error is as small as possible). We shall refer to these approximations as the CS and BA respectively; they are defined and discussed in [1]. By definition the BA is more accurate than the CS; on the other hand, the CS is easier to compute. It is practical, therefore, to enquire whether in any given situation the increase in accuracy is sufficient to justify the extra labour of computing the BA. This paper aims to provide some guidance on this question. (We may note that the arguments for and against expressing a polynomial approximation in terms of powers of the independent variable rather than in terms of Chebyshev polynomials are irrelevant here; the BA and the CS can, of course, both be written in either form.) Our procedure is to calculate the maximum absolute errors of the BA and the CS of degree n (which errors we shall denote by En and Sn re? spectively, following Bivlin [2]) for an arbitrary polynomial of degree n + r + 1, with r = 0, 1, 2 and 3. In each case we compute R(r), the maxi? mum value of Sn/En over all polynomials of degree n + r + 1. From these results we conjecture the value of R(r) for arbitrary r, and support this conjecture by numerical experiment in the case r ? 5. The method of attack is based on a device used by Bernstein [4] for a closely related prob? lem. Some related work on the comparison of the CS with the BA has been carried out by Rivlin [2]. He presents some interesting evidence which, as he says, seems to support the practice of truncating the Chebyshev series as an approximation to the best approximation. Using some theorems proved by Blum and Curtis [3], he gives bounds for Sn/En which are most useful in the case of rapidly convergent Chebyshev series. Here, however, we shall consider the problem under less restrictive conditions.

The application of the Ferranti Mercury computer to linguistic problems

In a book published in 1958 1 (Booth et al. , 1958) and hereafter called MRLP Booth, Brandwood, and Cleave discussed at some length the various ways in which electronic computers could be applied to resolving linguistic problems, and gave an account of the results obtained at the Birkbeck College Computational Laboratory. Since that time, the University of London has acquired a Ferranti Mercury computer which, though primarily designed for the solution of mathematical problems (and making use of floating point arithmetic), has a large immediate-access store and a great speed which make it very suitable for linguistic work. This acquisition has made it possible to put into practice on a full scale many of the ideas which the book contained. The present paper elaborates upon some of these ideas with particular reference to the way in which they are affected by the design of the Mercury. The paper is divided into several sections and the first devoted to a description of the Mercury insofar as it affects the design of linguistic programmes. The basic ideas behind the dictionary searching methods described in Section V appeared in MRLP. The minor discrepancies between some of the mathematical results in this section and the corresponding ones in MRLP are due to the assumption in the latter that this size of dictionary is large. It will become apparent that this assumption cannot always be made in dealing with the Mercury.

Networks for digital-to-analogue shaft-position transducers

DIGITAL computers have been used extensively in the solution of mathematical problems requiring both speed and accuracy in their computation. More recently, digital computers are being employed to control analogue equipment. The control of machine tools is one example of this application.1

New Computer

The successful operation of a new electronic digital computer, designed and constructed by staff members of the physics division of the Argonne National Laboratory, was announced at the end of January. The $250,000 computer, based on an instrument recently constructed at the Institute for Advanced Study at Princeton, New Jersey, has been named AVIDAC, which is reported to stand for “Argonne's Version of the Institute's Digital Automatic Computer”. AVIDAC is expected to facilitate the solution of mathematical problems of Argonne scientists engaged in reactor engineering and theoretical physics research. In the past, the lack of such a computing device has meant that many complex problems had to be solved by time-consuming methods or by the use of existing machines in other parts of the country. An additional computer is now being constructed at Argonne for use at the Oak Ridge National Laboratory. Another member of the same family of computers is the Los Alamos Scientific Laboratory's MANIAC, a device that was designed and constructed by members of the Los Alamos scientific staff.

The Theory of Noise in Continuous Media

It is shown that the mathematical solution of problems involving the propagation of noise is materially aided by the introduction of the space correlation function ψ(x1, x2, τ), defined as the average, over t, of p(x1t)p(x2t−τ), p being the acoustic pressure in the noise field. The differential equations satisfied by ψ are derived. Its relation to Ψ(α, β, γ) is discussed, α, β, γ being the propagation vector of a sinusoidal wave, and Ψ/2ρc2 being the density of potential energy in the α, β, γ space. The theory of uniform noise fields, both isotropic and anisotropic, is developed in detail. The anisotropy caused by a reflecting plane is discussed. The radiation of noise by a vibrating plane is discussed, neglecting the reaction of the radiation on the motion of the surface. In particular, it is shown that to this approximation the sea surface cannot radiate subsonic energy into the high atmosphere because the velocity of the surface gravity waves is less than the velocity of sound in air. The theory of noise, as here developed, is analogous to the theory of turbulence, and the relation between noise and turbulence is briefly discussed.

On the motion of floating bodies. I

We shall restrict ourselves here to floating bodies without any means of propelling themselves. The body may, of course, be a ship lying dead in the water, but there is no real limitation to practical shapes of any particular sort except that we shall suppose the body to be hydrostatically stable. This will restrict the extent of this survey in an important way: we are able to slough off all effects associated with an average velocity of the body. Since mathematical solution of problems almost inevitably proceeds by way of linearization of the boundary conditions, this means that we may avoid introducing a linearization parameter whose smallness expresses the fact that the body doesn't disturb the water much during its forward motion, for example, slenderness or thinness. If we do introduce such a geometrical assumption, it will be an additional approximation, not one forced upon us by the physical situation. Fortunately, Newman's (1970) article treats, among other things, the recent advances in the theory of motion of slender ships under way. More can be found in a paper by Ogilvie (1964) . We shall assume from the beginning that motions are small and take this into account in formulating equations and boundary conditions. Further­ more, we shaH assume the fluid inviscid, and without surface tension. It is not difficult to write down equations and boundary conditions for a less restricted problem. However, since most results are for the case of small motions and since the perturbation expansions associated with the deriva­ tion of the linearized problem from the more exact one do not present any special points of interest, it seems more efficient to start with the simpler problem. Even so, some account will be given of recent attempts to consider nonlinear problems.

A photo-electric integraph

A machine for the purpose of facilitating mathematical solution of problems requiring the evaluation of an integral in which the integrand involves a variable parameter is described in this article. It involves the use of an optical system in which the transmission of light is limited in a definite manner by apertures having the shape of the area under curves representing mathematical functions. The accuracy of the machine itself is from 2 to 5 per cent. Examples of the use of the device in the solution of the superposition theorem integral and a form of integral equation are given. Its application to other problems including the Fourier transform, correlation analysis, and periodogram analysis is discussed, and its limitations are considered.