By Stanislaw M. Ulam (auth.), Mark C. Reynolds, Gian-Carlo Rota (eds.)

STANISLAW MARCIN ULAM, or Stan as his neighbors known as him, used to be a kind of nice inventive mathematicians whose pursuits ranged not just over all fields of arithmetic, yet over the actual and organic sciences to boot. Like his friend "Johnny" von Neumann, and in contrast to such a lot of of his friends, Ulam is unclassifiable as a natural or utilized mathematician. He by no means ceased to discover as a lot good looks and pleasure within the functions of arithmetic as in operating in these rarefied areas the place there's a overall un trouble with useful difficulties. In his Adventures of a Mathematician Ulam remembers taking part in on an oriental carpet while he was once 4. The curious styles interested him. while his father smiled, Ulam recalls considering: "He smiles simply because he thinks i'm infantile, yet i do know those are curious styles. i do know anything my father doesn't know." The incident is going to the center of Ulam's genius. He may well see speedy, in flashes of tremendous perception, curious styles that different mathematicians couldn't see. "I am the sort that loves to begin new issues instead of increase or elaborate," he wrote. "I can't declare that i do know a lot of the technical fabric of mathematics.

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**Example text**

Still another example of this kind of parallel and independent stimuli is furnished by the consideration of the classical Riemann spaces in the general theory of relativity. Here, too, the mathematical tools were there for Einstein to use for his formulation of the geometry of space. Einstein's theory of special relativity initiated a revolutionary departure from the restrictions on the scope of geometrical ideas by involving, in addition to the physical "space," the variable or the parameter of time.

The element is a set of points and the distance between two sets can be defined in a simple way as described by the mathematician Hausdorff. Given a point x of set B, look at the nearest point y in set A and then take the maximum of the minima with respect to all choices of x. That will be the distance between sets. We wrote a paper, the first I ever published in this country. We sent it to the Bulletin of the American Mathematical Society in 1932. In it we showed that for two dimensions you get a space that you might imagine just as a triangle, for three dimensions you can imagine it as a sort of tetrahedron, but topologically they are the same as a square or a cube.

The first meaningful applications were made in problems of mathematical physics, where the complexity of the interaction of many elements may be such that no methods of classical mathematical analysis can give even a qualitative picture of their solution. In hydrodynamics (the study of the motion of fluids) one must have recourse (other than in very simple cases) to numerical work which becomes so massive that only the fastest machines can provide useful approximations. In problems of weather prediction or of the circulation of the atmosphere, initial data also lacks the simplicity and symmetry that allow for the use of theoretical methods of analysis.