Solution of the Grazing Goat Problem: A Conflict between Beauty and Pragmatism

What is the ideal solution of a problem in mathematics? It depends on your nerd ideology. Pure mathematicians worship the beauty of a mathematics result. Closed form solutions are particularly beautiful. Engineers and applied mathematicians, on the other hand, focus on the result independent of its beauty. If a solution exists and can be calculated, that's enough. The job is done. An example is solution of the grazing goat problem. A recent closed form solution in the form of a ratio of two contour integrals has been found for the grazing goat problem and its beauty has been admired by pure mathematicians. For the engineer and applied mathematician, numerical solution of the grazing goat problem comes from an easily derived transcendental equation. The transcendental equation, known for some time, was not considered a beautiful enough solution for the pure mathematician so they kept on looking until they found a closed form solution. The numerical evaluation of the transcendental equation is not as beautiful. It is not in closed form. But the accuracy of the solution can straightforwardly be evaluated to within any accuracy desired. To illustrate, we derive and solve the transcendental equation for a generalization of the grazing goat problem.

David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

David J. Benney (1930–2015) was an applied mathematician and fluid dynamicist whose highly original work has shaped our understanding of nonlinear wave and instability processes in fluid flows. This article discusses the new paradigm he pioneered in the study of nonlinear phenomena, which transcends fluid mechanics, and it highlights the common threads of his research contributions, namely, resonant nonlinear wave interactions; the derivation of nonlinear evolution equations, including the celebrated nonlinear Schrödinger equation for modulated wave trains; and the significance of three-dimensional disturbances in shear flow instability and transition.

Michael Selwyn Longuet-Higgins. 8 December 1925—26 February 2016

Michael Longuet-Higgins was a geometer and applied mathematician who made notable contributions to geophysics and physical oceanography, particularly to the theory of oceanic microseism and to the dynamics of finite amplitude, sharp-crested wind-generated surface waves. The latter led to his pioneering studies on breaking waves. On a much larger scale, he showed how ocean waves produce currents around islands in the ocean. He considered wider aspects of the physics of waves, including wave-driven transport of sand along beaches, and the electrical effects of tidal streams. He also contributed to subjects of a geometrical character such as the growth of quasi-crystals, the assembly of protein sheaths in viruses, to chains of circle themes and to a wide variety of other topics. He was an extraordinary applied mathematician, using the simplest forms of mathematics to demonstrate and discover highly complex nonlinear phenomena. In particular, he often thought of problems involving water waves using his unique knowledge of geometry and then tested his theories by experiment. Along with Brooke Benjamin FRS, Sir James Lighthill FRS, Walter Munk FRS, John Miles and Andrei Monin, Michael Longuet-Higgins stands out as one of the towering figures of theoretical fluid dynamics in the twentieth century. His contributions will have a continuing influence on our attempts to understand better the processes that influence the oceans.

Frederick Gerard Friedlander. 25 December 1917 — 20 May 2001

Gerard Friedlander was the son of Austrian communist intellectuals, who divorced when he was four. From the age of two he was raised by grandparents in Vienna, while his mother lived in Berlin as a communist organizer. Hitler came to power in 1933; Friedlander was sent to England, aged 16, in 1934; two years later, he won a scholarship to Trinity College, Cambridge. By 1940 he was a fully fledged applied mathematician who came to embrace both the European and British traditions of that subject. His work was marked by profound originality, by the importance of its applications and by the mathematical rigour of his treatment. The applications of his work changed over the years. The first papers (written between 1939 and 1941, but published only in 1946 for security reasons) were a contribution to Civil Defence: they presented entirely new and explicit results on the shielding effect of a wall from a distant bomb blast. The late papers were contributions to the general, more abstract theory of partial differential equations, but, characteristically, with concrete examples that illuminated obscure aspects of the general theory. Between these two, the middle years brought a flowering of results about the wave equation (including results for a curved space-time), of importance to both physicists and mathematicians.

Mathematics that has intrigued me

I trace the main steps of the first fifty-five years of my career as an applied mathematician, pausing from time to time to describe problems that arose in asymptotics and numerical analysis and had far-reaching effects on this career. Lecture delivered at Asymptotics and Applied Analysis, Conference in Honor of Frank W. J. Olver's 75th Birthday, January 10–14, 2000, San Diego State University, San Diego, California. Editors' Note: Frank W. J. Olver died on April 23, 2013. The following text was typed by his son, Peter J. Olver, from handwritten notes found among his papers. At times the writing is unpolished, including incomplete sentences, but the editors have decided to leave it essentially the way it was written. However, for clarity, some abbreviations have been written out in full. A couple of handwritten words could not be deciphered, and a guess for what was intended is enclosed in brackets: […]. Endnotes have been made into footnotes within the body of the article. References were mostly not included in the handwritten text, but rather listed in order at the end. Citations to references have been included at the appropriate point in the text.