Math I Love

Two notes, which are unrelated except that I wish everyone had heard of them:

1. Sofya Kovalevikaya

Sofya Kovalevikaya, the first ever woman to obtain a Ph.D in Mathematics.

On being committed and getting things done:

“Sofia Kovalevskaya (née Korvin-Krukovskaya), was born in Moscow. …

Despite her obvious talent for mathematics, she could not complete her education in Russia. At that time, women there were not allowed to attend the universities. In order to study abroad, she needed written permission from her father (or husband). Accordingly, she contracted a “fictitious marriage” with Vladimir Kovalevsky, then a young paleontology student who would later become famous for his collaborations with Charles Darwin. They emigrated from Russia in 1867.

In 1869, Kovalevskaya began attending the University of Heidelberg, Germany, which allowed her to audit classes as long as the professors involved gave their approval. … After two years of mathematical studies at Heidelberg under such teachers as Helmholtz, Kirchoff and Bunsen, she moved to Berlin, where she had to take private lessons from Karl Weierstrass, as the university would not even allow her to audit classes. In 1874 she presented three papers—on partial differential equations, on the dynamics of Saturns rings and on elliptic integrals —to the University of Göttingen as her doctoral dissertation. With the support of Weierstrass, this earned her a doctorate in mathematics summa cum laude, bypassing the usual required lectures and examinations.

She thereby became the first woman in Europe to hold that degree. Her paper on partial differential equations contains what is now commonly known as the Cauchy-Kovalevski theorem, which gives conditions for the existence of solutions to a certain class of those equations.”

(seleted excerpts from wikipedia: http://en.wikipedia.org/wiki/Sofia_Kovalevskaya)

2. Braess’s paradox

Maybe this is considered popular math, but I fucking love Braess’s paradox. Basically, it states that adding a shortcut route to a traffic network can theoretically increase traffic times, counterintuitively, instead of decrease them. Crazy!

“The paradox is stated as follows: “For each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times.”

The reason for this is that in a Nash equilibrium, drivers have no incentive to change their routes. If the system is not in a Nash equilibrium, selfish drivers must be able to improve their travels time by changing the routes they take. In the case of Braess’s paradox, drivers will continue to switch until they reach Nash equilibrium, despite the reduction in overall performance.

If the latency functions are linear then adding an edge can never make total travel time at equilibrium worse by a factor of more than 4/3.”

http://en.wikipedia.org/wiki/Braess_paradox

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