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Most read

This page lists the top ten most read articles for this journal based on the number of full text views and downloads recorded on Cambridge Core over the last 30 days. This list is updated on a daily basis.

Contents
  • Bessel-integral functions
  • Pierre Humbert
  • Published online by Cambridge University Press: 20 January 2009, pp. 276-285
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    In a very remarkable work on the operational Calculus, Dr Balth. van der Pol has introduced a new function, playing with respect to Bessel function of order zero the same part as the cosine- or sine-integral with respect to the ordinary cosine or sine. He showed that this function—which he called Bessel-integral junction—can be used to express the differential coefficient of any Bessel function with respect to its index. But he did not investigate the further properties of his new function. I propose to give here some of them, which appear to be interesting, and to introduce and study the functions connected, in the same way, with Bessel functions of any order.

  • A Matrix Form of Taylor's Theorem
  • H. W. Turnbull
  • Published online by Cambridge University Press: 20 January 2009, pp. 33-54
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    The following pages continue a line of enquiry begun in a work On Differentiating a Matrix, (Proceedings of the Edinburgh Mathematical Society (2) 1 (1927), 111-128), which arose out of the Cayley operator , where xij is the ijth element of a square matrix [xij] of order n, and all n2 elements are taken as independent variables. The present work follows up the implications of Theorem III in the original, which stated that

    where s (Xr) is the sum of the principal diagonal elements in the matrix Xr. This is now written ΩsXr = rXr – 1 and Ωs is taken as a fundamental operator analogous to ordinary differentiation, but applicable to matrices of any finite order n.

  • The Definition of Lie Derivative
  • T. J. Willmore
  • Published online by Cambridge University Press: 20 January 2009, pp. 27-29
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    The differential operation known as Lie derivation was introduced by W. Slebodzinski in 1931, and since then it has been used by numerous investigators in applications in pure and applied mathematics and also in physics. A recent monograph by Kentaro Yano (2) devoted to the theory and application of Lie derivatives gives some idea of the wide range of its uses. However, in this monograph, as indeed in other treatments of the subject, the Lie derivative of a tensor field is defined by means of a formula involving partial derivatives of the given tensor field. It is then proved that the Lie derivative is a differential invariant, i.e. it is independent of a transformation from one allowable coordinate system to another. Sometimes some geometrical motivation is given in explanation of the formula, but this is seldom very satisfying.

  • Mean Value Theorems for Vector Valued Functions
  • Robert M. McLeod
  • Published online by Cambridge University Press: 20 January 2009, pp. 197-209
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    The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we have

    for some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to be

    where cka, b, 0 k, and .

  • A note on skew-symmetric determinants
  • Walter Ledermann
  • Published online by Cambridge University Press: 20 January 2009, pp. 335-338
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    A short proof, based on the Schur complement, is given of the classical result that the determinant of a skew-symmetric matrix of even order is the square of a polynomial in its coefficients.

  • Applications of Mathematics to Medical Problems
  • A. G. M'Kendrick
  • Published online by Cambridge University Press: 20 January 2009, pp. 98-130
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    In the majority of the processes with which one is concerned in the study of the medical sciences, one has to deal with assemblages of individuals, be they living or be they dead, which become affected according to some characteristic. They may meet and exchange ideas, the meeting may result in the transference of some infectious disease, and so forth. The life of each individual consists of a train of such incidents, one following the other. From another point of view each member of the human community consists of an assemblage of cells. These cells react and interact amongst each other, and each individual lives a life which may be again considered as a succession of events, one following the other. If one thinks of these individuals, be they human beings or be they cells, as moving in all sorts of dimensions, reversibly or irreversibly, continuously or discontinuously, by unit stages or per saltum, then the method of their movement becomes a study in kinetics, and can be approached by the methods ordinarily adopted in the study of such systems.

  • Bounds for modified Bessel functions of the first and second kinds
  • Árpád Baricz
  • Published online by Cambridge University Press: 05 August 2010, pp. 575-599
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    Some new inequalities for quotients of modified Bessel functions of the first and second kinds are deduced. Moreover, some developments on bounds for modified Bessel functions of the first and second kinds, higher-order monotonicity properties of these functions and applications to a special function that arises in finite elasticity, are summarized. The key tool in our proofs is a frequently used criterion for the monotonicity of the quotient of two Maclaurin series.

  • Notes on Everett's Interpolation Formula
  • G. J. Lidstone
  • Published online by Cambridge University Press: 20 January 2009, pp. 21-26
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    Writing of his now well-known Interpolation Formula, Professor Everett said, “The only novelty in the formula is the simplicity of its form.… The best known formulae for interpolation by central differences are difficult to carry in the memory on account of their unsymmetrical aspect, one law being applicable to the odd and another to the even terms. … This disadvantage does not apply to the formula proposed,” viz., that which now goes by Everett's name.

  • On Rational Solutions of x3 + y3 +z3 = R.
  • H. W. Richmond
  • Published online by Cambridge University Press: 20 January 2009, pp. 92-100
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    The earliest proof that every rational number (R) can be expressed as a sum of cubes of three rational numbers (x, y, z), not necessarily positive, was published in 1825 by S. Ryley, a schoolmaster of Leeds: formulae were given for x, y, z in terms of a parameter, such that every value of the parameter led to a system of values of x, y, z satisfying the above relation, and every rational value of the parameter led to a system of rational values of x, y, z. The later solutions referred to by Dickson are found to give the same results as Ryley's formula, as does another method, quoted in a modified form by Landau from a paper by the present writer. Thus it might almost be believed that Ryley's century-old result embodies all that is known with regard to the resolution of a number into three cubes, and that his formula is unique. I propose to examine the rationale of his method and the causes of its success; it will then appear that an infinity of similar formulae exist, and that one of them is at least as simple as his. It is convenient to state Ryley's formula, and the modification made by Landau, in section 2; and to generalize the method in section 3.

  • w. G. Cochran, Sampling Techniques (John Wiley & Sons, 2nd edition, 1963), ix+413 pp., 72s.
  • R. A. Robb
  • Published online by Cambridge University Press: 20 January 2009, pp. 342-343
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