How to Write a Review for a Scientific or Mathematical Paper

Reviewing a paper or grant proposal is a difficult and thankless task. Yet tens of billions of dollars in grants are dispersed on the basis of peer review. And careers are made and broken through publications that are judged through peer review. The goal of this report is to provide an education resource to improve the quality and efficiency of reviews.

M. L. Urquhart

To assess the mathematical work of the late M. L. Urquhart is, paradoxically, both easy and extremely difficult. It is easy, in that he never published a mathematical paper in any of the journals. Thus one does not have to read a large volume of published work. On the other hand, Urquhart was far from being mathematically inactive, but he communicated his ideas verbally to his associates. At this point of time it is well nigh impossible to recall all the ideas that he discussed so freely during his lifetime. Conversations have long since been forgotten and ideas are now only vaguely remembered. Consequently any objective assessment of Urquhart's mathematical work is very difficult.

Jacques Hadamard, 1865-1963

Jacques Hadamard died on 17 October 1963 at the age of 98. He published his first mathematical paper of importance in 1888, and continued working until he was over 90 covering an immense range of mathematical subjects including educational, philosophical and psychological aspects of mathematics. To classical analysts his name has been well known as the author of the Hadamard gap theorem, the Hadamard three circle theorem , the Hadamard factorization theorem for integral functions and other results on Taylor series published before 1900, but perhaps it was his proof of the prime number theorem in 1896 more than anything else which made Hardy describe him in 1944 as the ‘living legend’ in mathematics. To some pure mathematicians he may be better known by his theorem on the modulus of a determinant which plays such an essential part in the Fredholm theory of integral equations, or because he invented the name for functional analysis. His work on the theory of propagation of waves and partial differential equations was no less significant, but it is less easy to pinpoint particular results. It is at the base of the modern theory of both subjects; it includes much work on the Cauchy problem and on the technique of the finite parts of integrals which, although superseded by the theory of distributions, proved to be very useful. He was no physicist, but he helped to lay the foundations of the modern theory of shock waves. No one person could do justice to such an enormous range of mathematical activity which was matched by wide interests outside mathematics, and what I have to say owes much to the assistance of others. In particular Mile Jacqueline Hadamard has supplied me with much information and copies of articles and speeches about her father, but, owing to the fact that all their belongings were stolen by the Germans during the war, and also, because as she admitted, ‘he had so little order and method, keeping everything about others and so little about himself’, the information, and in particular the list of published work, is probably incomplete, and the latter is certainly difficult to check. There are over 300 items, many of them published long ago in periodicals not easily traced from the brief descriptions given in the list in the Selecta [284].* I have drawn largely from speeches made at his Scientific Jubilee [284a] in 1936, from articles by Lévy, Mandelbrojt, Fréchet, Julia and Kahane and from Hadamard’s own accounts of events and influences extracted from various articles, books and speeches. I am also particularly grateful to Professor Fréchet, Professor Leray, Professor Kahane, Professor Temple, Professor G. N. Ward, Mr A. E. Ingham, Professor Truesdell, Dr F. G. Friedlander, Dr L. S. Bosanquet and Sir Edward Collingwood for their help. * Numbers in square brackets refer to the numbered items in the bibliography described at the end of the memoir, pp. 96 to 99.

TEST OF A QUANTITATIVE MOUNTAIN BUILDING THEORY BY APPALACHIAN STRUCTURAL DIMENSIONS

A mathematical paper by Ross Gunn (Journal of the Franklin Institute, volume 224, pages 19–53, 1937) gives a mountain building theory which makes possible the calculation of certain structural dimensions in terms of thickness and strength of the geologic elements involved. The present paper applies various formulae from Gunn’s paper to the calculation of width and amplitude of folds, to basin widths and depths, and to mountain heights of the Appalachian Mountain system. The numerical values resulting are in surprisingly good agreement with previously published values measured or estimated from the known geology of the area. A brief discussion is also given of the theoretical interpretation, by Gunn’s theory, of the large isostatic gravity anomaly here as due to possible crustal stress.

Obituary notices of fellows deceased

Among the losses which the Royal Society has recently sustained none has evoked deeper regret than the death of Sir Alfred Bray Kempe, who for twenty-one years, as its Treasurer and one of its Vice-Presidents, took a leading share in the management of its affairs and in the promotion of its prosperity. Some grateful record of his career could not find a more appropriate place than in the pages of the ‘Proceedings’ of the Society with which he was so long and so closely associated. The third son of Prebendary John Edward Kempe, Rector of St. James’s, Piccadilly, he was born on July 6, 1849. From St. Paul’s School, as Camden Exhibitioner, he passed to Trinity College, Cambridge, where, in 1872, he took his degree with special distinction in Mathematics. In the same year he published his first mathematical paper, the title of which—“A general method of solving equations of the n th degree by mechanical means”—showed the bent of his mind in scientific enquiry. For some years he continued to publish mathematical essays, but having chosen the Law as his profession, and become a Barrister of the Inner Temple and Western Circuit, he was soon immersed in legal business. To the last, however, he never wholly relinquished his mathematical studies. He used to say of himself that his favourite recreations were Mathematics and Music. He was hardly ever without some problem at which, in such leisure as he could find, he steadily worked. But he refused, as he said, to “empty his note-books into the ‘Proceedings’ of the Royal Society.” He would not be induced to publish his studies until he had really got to the bottom of his enquiry.

Obituary notices of fellows deceased

Among the losses which the Royal Society has recently sustained none has evoked deeper regret than the death of Sir Alfred Bray Kempe, who for twenty-one years, as its Treasurer and one of its Vice-Presidents, took a leading share in the management of its affairs and in the promotion of its prosperity. Some grateful record of his career could not find a more appropriate place than in the pages of the ‘Proceedings’ of the Society with which he was so long and so closely associated. The third son of Prebendary John Edward Kempe, Rector of St. James’s, Piccadilly, he was born on July 6, 1849. From St. Paul’s School, as Camden Exhibitioner, he passed to Trinity College, Cambridge, where, in 1872, he took his degree with special distinction in Mathematics. In the same year he published his first mathematical paper, the title of which—“A general method of solving equations of the n th. degree by mechanical means”—showed the bent of his mind in scientific enquiry. For some years he continued to publish mathematical essays, but having chosen the Law as his profession, and become a Barrister of the Inner Temple and Western Circuit, he was soon immersed in legal business. To the last, however, he never wholly relinquished his mathematical studies. He used to say of himself that his favourite recreations were Mathematics and Music. He was hardly ever without some problem at which, in such leisure as he could find, he steadily worked. But he refused, as he said, to “empty his note-books into the ‘Proceedings’ of the Royal Society.” He would not be induced to publish his studies until he had really got to the bottom of his enquiry.