The well-known treatment of the path integral for the Coulomb potential, by means of the Kustaanheimo–Stiefel transformation and a time transformation, is made more transparent by reducing the problem to the equality of two measures on the space of paths. For this equality two proofs are given: an elementary computational one and a short one recurring to general features of stochastic processes. It is shown that the time transformation is a special case of the well-known time change for a continuous local martingale, by which the process is changed to a Brownian motion and which is determined by its quadratic variation.
The march of Alexander from the Granicus to Issus is given by Arrian in less than a dozen pages scattered among various sieges that are more fully described; Plutarch, Diodorus and Quintus Curtius do less, and no more than a page or two apiece has come down to us on the whole of these movements.Although his first meeting with Asia was probably the most important experience in Alexander's adult life, and though the Anatolian campaigns lasted a year and a half, or even a little more, out of the short total of eleven years that were left him, the poverty of the sources has imposed its brevity on modern historians also. Professor Tarn—who is as much a bedside book to modern devotees as the Iliad was to their hero—describes the marches and countermarches of Asia Minor in little more than three pages; and there is a great gap left us from classical times between Xanthus and Phaselis in Lycia. It would be absurd to think of filling it. But after sailing down the coast, I believed that some evidence might be gathered by comparing the written scraps left us with the nature of the places recorded, provided this were done before the road-building policy of modern Turkey succeeds in changing the pace of living in these mountains. Hitherto their ruins have scarcely been altered except by a natural decay; and the methods of travel being as slow as ever they were before, except along a very few roads, the flavour of their past is preserved.In this essay the geography is attempted, with the problems and such answers to them as my rather intermittent journeys seemed able to provide. Someone better equipped than I am may find the outline useful and venture more profitably, before too much time goes by; for the interest is not one of geography merely. By visualising the routes which were chosen, the motives and processes by which that choice was made become clearer; and behind these motives and processes is the most dynamic being that the world has perhaps ever known.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.