## Posts Tagged ‘**statistics**’

## “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.

## “No structure, even an artificial one, enjoys the process of entropy. It is the ultimate fate of everything, and everything resists it.”*…

A 19th-century thought experiment that motivates physicists– and information scientists– still…

The universe bets on disorder. Imagine, for example, dropping a thimbleful of red dye into a swimming pool. All of those dye molecules are going to slowly spread throughout the water.

Physicists quantify this tendency to spread by counting the number of possible ways the dye molecules can be arranged. There’s one possible state where the molecules are crowded into the thimble. There’s another where, say, the molecules settle in a tidy clump at the pool’s bottom. But there are uncountable billions of permutations where the molecules spread out in different ways throughout the water. If the universe chooses from all the possible states at random, you can bet that it’s going to end up with one of the vast set of disordered possibilities.

Seen in this way, the inexorable rise in entropy, or disorder, as quantified by the second law of thermodynamics, takes on an almost mathematical certainty. So of course physicists are constantly trying to break it.

One almost did. A thought experiment devised by the Scottish physicist James Clerk Maxwell in 1867 stumped scientists for 115 years. And even after a solution was found, physicists have continued to use “Maxwell’s demon” to push the laws of the universe to their limits…

A thorny thought experiment has been turned into a real experiment—one that physicists use to probe the physics of information: “How Maxwell’s Demon Continues to Startle Scientists,” from Jonathan O’Callaghan (@Astro_Jonny)

* Philip K. Dick

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**As we reconsider the random,** we might send carefully-calculated birthday greetings to Félix Édouard Justin Émile Borel; he was born on this date in 1871. A mathematician (and politician, who served as French Minister of the Navy), he is remembered for his foundational work in measure theory and probability. He published a number of research papers on game theory and was the first to define games of strategy.

But Borel may be best remembered for a thought experiment he introduced in one of his books, proposing that a monkey hitting keys at random on a typewriter keyboard will – with absolute certainty – eventually type every book in France’s Bibliothèque Nationale de France. This is now popularly known as the infinite monkey theorem.

## “An imbalance between rich and poor is the oldest and most fatal ailment of all republics”*…

… so, how we measure it matters…

In 2015, Greece, Thailand, Israel, and the UK were equally unequal. That is, all four countries had the same Gini coefficient, a common measure of income inequality.

The number suggests that the spread of incomes in the four nations was the same. However, a close look at the poorest and wealthiest in those societies reveals a very different picture. The ratio between income held by the richest 10% and the poorest 10% ranged significantly, from 13.8 in Greece to 4.2 in the UK.

The fact is, just because the Gini coefficient is so well known doesn’t mean it’s a particularly useful measurement. Its appeal comes from its simplicity—a number between 0 and 1 that can encapsulate a complex distribution in a single figure—as well as its popularity. It is also regularly published and updated by powerful international organizations like the OECD, the World Bank, and the International Monetary Fund.

However, it has a number of serious limitations. So many, in fact, that the World Inequality Database, one of the world’s leading sources of income inequality data, steers clear. And it’s not alone. While some economists defend the Gini coefficient’s continued use, most agree that as a way to understand income inequality, it’s insufficient on its own…

A primer on the dominant measure of economic inequality, and on some alternatives/supplements to it: “Gini coefficient: An introduction.”

* Plutarch

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**As we aim to understand,** we might note that today is the Summer Solstice, the day on which the earth’s north pole is maximally tilted toward sun, and there are more hours of daylight than on any other day of the year (in the Northern Hemisphere; in the Southern, it is the Winter Solstice, the shortest day). The June solstice is the only day of the year when all locations inside the Arctic Circle experience a continuous period of daylight for 24 hours. And perhaps more immediately, it is the “official” start of Summer.

(The 21st is the traditional date; in the event, the solstice falls on the 20th, 21st, or 22nd– this year, on the 20th… still, the traditional date is the one folks tend to mark.)

Not coincidentally, today is also National Daylight Appreciation Day.

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