Posts Tagged ‘Markov’
“Why, sometimes I’ve believed as many as six impossible things before breakfast”*…
Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…
Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.
In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.
Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…
Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.
* The Red Queen, in Lewis Carroll’s Through the Looking Glass
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As we get real, we might spare a thought for two great mathematicians…
Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
Andrey (Andrei) Andreyevich Markov died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…
Maryam Mirzakhani did not enjoy mathematics to begin with. She dreamed of being an author or politician, but as a top student at her all-girls school in Tehran she was still disappointed when her first-year maths exam went poorly. Her teacher believed her – wrongly – to have no particular affinity with the subject.
Soon that would all change. “My first memory of mathematics is probably the time [my brother] told me about the problem of adding numbers from 1 to 100,” she recalled later. This was the story of Carl Gauss, the 18th-century genius whose schoolteacher set him this problem as a timewasting exercise – only for his precocious pupil to calculate the answer in a matter of seconds.
The obvious solution is simple but slow: 1+2+3+4. Gauss’s solution is quicker to execute, and far more cunning. It goes like this: divide the numbers into two groups: from 1 to 50, and from 51 to 100. Then, add them together in pairs, starting with the lowest (1) and the highest (100), and working inwards (2+99, 3+98, and so on). There are 50 pairs; the sum of each pair is 101; the answer is 5050. “That was the first time I enjoyed a beautiful solution,” Mirzakhani told the Clay Mathematics Institute in 2008.
Since then, her appreciation for beautiful solutions has taken her a long way from Farzanegan middle school. At 17 she won her first gold medal at the International Mathematics Olympiad. At 27 she earned a doctorate from Harvard University. The Blumenthal Award and Satter Prize followed, and in 2014 she became the first woman to be awarded the Fields Medal, the highest honour a mathematician can obtain.
Before this particular brand of wonder became perceptible to Mirzakhani, she experienced feelings many of us can relate to: to the indifferent, her subject can seem “cold”, even “pointless”. Yet those who persist will be rewarded with glimpses of conceptual glory, as if gifted upon them by a capricious god: “The beauty of mathematics,” she warned, “only shows itself to more patient followers.”
This concept of “beauty” found in maths has been referred to over centuries by many others; though, like beauty itself, it is notoriously difficult to define…
For an experienced mathematician, the greatest equations are beautiful as well as useful. Can the rest of us see what they see? “What makes maths beautiful?”
[From The New Humanist, via the ever-illuminating 3 Quarks Daily]
Maryam Mirzakhani died last Friday, a victim of breast cancer; she was 40. As Peter Sarnak (a mathematician at Princeton University and the Institute for Advanced Study) said, her passing is “a big loss and shock to the mathematical community worldwide.” See also here.
* Bertrand Russell, A History of Western Philosophy
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As we accede to awe, we might spare a thought for Andrey (Andrei) Andreyevich Markov; he died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological and social sciences.
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