I wonder sometimes if our modern structured and red taped upper education model could ever cope with such exceptional individuals at all. I cannot stop thinking they will never get the unbridled space they need to express their ideas and concot together without an heavy amount of supervision and channeling toward 'agreed' topics and behaviours, I think education and the application of the law should be very flexible when exceptional individuals are at stake...
That is already the case. Many Hollywood stars walk penalty free from repeated drug incidents, while poorer people spend many years in prison for one time minor offenses, for one example amongst many.
You might argue that these people are not the “exceptional” you have in mind, but many are likely to not argue with your definition either.
They get off crimes mostly because they can hire good lawyers. A lot of successful entrepreneurs didn't finish college and instead ran out to do their own thing (gates, jobs, zuckerberg).
Steinhaus, Ulam and many other mathematicians in the Lviv school were Jewish, and had to flee or go into hiding to survive the German occupation. Interesting that there is no mention of this in the article.
I'm not sure how that relates to the OP since the article is on a Polish website. "Remains hostile" seems to suggest it was hostile before WWII. Remember, Lviv today is largely Ukrainian (as was Bandera). Most Poles, Jew or gentile, were either expelled after or killed during the war.
It's mentioned briefly through a single paragraph but not much beyond that.
"He lived through the war under the name Grzegorz Krochmalny and hid in an estate near Lviv and at a manor neat Nowy Sącz. He gave private lessons in exchange for firewood, oil and milk. In his spare time he played chess and worked on designing a solar clock. During the war he resorted to solving serious mathematical equations by post but continuing his research from before the war was out of the question."
For the Banach match box problem, that is, flip a fair coin x + y times and find the probability of getting heads exactly x times. So, it's a binomial probability problem.
A Banach space is a complete, normed linear space. There is a nice chapter on Banach space in
Walter Rudin,
Real and Complex Analysis.
There are some nice applications of the Hahn-Banach theorem in
David G. Luenberger,
Optimization by Vector Space Methods.
In
Patrick Billingsley,
Convergence of Probability Measures.
is a nice presentation of Ulam's result in measure theory
Le Cam called tightness: Roughly, IIRC, for any probability measure P and any a > 0 no matter how small, there exists a sphere S of finite radius so that P(S) > 1 - a. Intuitively the probability mass can't just keep avoiding all spheres; eventually some sphere, if large enough, must cover nearly all the mass. There are some cute technical details.
Once in a paper I used Ulam's result to show that a goofy distribution-free statistical hypothesis test was not trivial. And I've seen other applications.
The hypothesis test was to improve on our work in artificial intelligence for zero day monitoring for problems in server farms and networks. So, Ulam's tightness has played a role in at least one piece of work intended to be practical!
IIRC, Ulam was long head of Los Alamos. Once I heard his lecture on the role in evolution of having two sexes.
There was a Time-Life book on math with a few pages on Ulam. IIRC, Ulam did by hand or mechanical calculator some of the early Monte-Carlo evaluations of critical mass.
There is an intriguing reference here: "...the concept of a spatial x-ray locating device came to [Steinhaus] during a winter stroll spent observing snowflakes falling on his fur coat." Elsewhere, I have found the comment "Steinhaus designed and instrument for localization of strange bodies in the body of a sick person by means of X-rays, based on a simple and elegant geometrical conception (1938) [1]." Does anyone know what this is? A form of tomography, perhaps, or an application of stereology? - which is distinct from tomography, according to this article [2], and is apparently somehow related to his Longimeter, recently discussed here [3].
”The first setup augmenting imaging data registered to an object was described in 1938 by the Austrian mathematician Steinhaus. He described the geometric layout to reveal a bullet inside a patient with a pointer that is visually overlaid on the invisible bullet. This overlay was aligned by construction from any point of view and its registration works without any computation. However, the registration procedure is cumbersome and it has to be repeated for each patient. The setup involves two cathodes that emit X-rays projecting the bullet on a fluoroscopic screen (see Fig. 2). On the other side of the X-ray screen, two spheres are placed symmetrically to the X-ray cathodes. A third sphere is fixed on the crossing of the lines between the two spheres and the two projections of the bullet on the screen. The third sphere represents the bullet. Replacing the screen with a semi-transparent mirror and watching the object through the mirror, the third sphere is overlaid exactly on top of the bullet from any point of view. This is possible because the third sphere is at the location to which the bullet is mirrored. Therefore, the setup yields stereoscopic depth impression. The overlay is restricted to a single point and the system has to be manually calibrated for each augmentation with the support of an X-ray image with two X-ray sources.”
Steinhaus was Polish, the confusion probably comes from the fact that he was born in then occupied by Austro-Hungary part of Poland (city of Jaslo, now within borders of Poland).
I've noticed more than a few Poles saddled with non-Polish roots in various history books. Banach himself was at one point incorrectly identified as a Russian mathematician in an old edition of the Encyclopedia Britannica. Sometimes the cause of error may be the historian's ignorance, sometimes nationalistic appropriation by others. Many Poles also often worked abroad during the 19th and 20th centuries because of foreign oppression by the Germans, Russians, and the Austrians (the last to a lesser degree than the first two).
