math

‘Learning many things does not teach understanding.’ –Heraclitus

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Another power law […] is Benford’s Law, which states that the distribution of digits in a lot of data are not even, but power law distributed. For example, in base 10, the number one should, all things being equal, appear 10% of the time. But in many data sources one appears around 30% of the time. This fact is actually used to help detect fraud in, for example, tax returns.

{ Oscillatory Thoughts | Continue reading | NY Times }

M.C. Busy Le Disco fooled around in Fresno

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On August 31, 2012, Japanese mathematician Shinichi Mochizuki posted four papers on the Internet.

The titles were inscrutable. The volume was daunting: 512 pages in total. The claim was audacious: he said he had proved the ABC Conjecture, a famed, beguilingly simple number theory problem that had stumped mathematicians for decades. […]

The problem, as many mathematicians were discovering when they flocked to Mochizuki’s website, was that the proof was impossible to read. The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”

This is not just gibberish to the average layman. It was gibberish to the math community as well.

{ Caroline Chen/Project Wordsworth | Continue reading }

art { Cy Twombly, Coronation of Sesostris, 2000 }

They swear to God that it’s me sellin the choppas

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“What people do in cities—create wealth, or murder each other—shows a relationship to the size of the city, one that isn’t tied just to one era or nation,” says Lobo. The relationship is captured by an equation in which a given parameter—employment, say—varies exponentially with population. In some cases, the exponent is 1, meaning whatever is being measured increases linearly, at the same rate as population. Household water or electrical use, for example, shows this pattern; as a city grows bigger its residents don’t use their appliances more. […]

If the population of a city doubles over time, or comparing one big city with two cities each half the size, gross domestic product more than doubles. Each individual becomes, on average, 15 percent more productive. Bettencourt describes the effect as “slightly magical,” although he and his colleagues are beginning to understand the synergies that make it possible. Physical proximity promotes collaboration and innovation, which is one reason the new CEO of Yahoo recently reversed the company’s policy of letting almost anyone work from home. […]

Remarkably, this phenomenon applies to cities all over the world, of different sizes, regardless of their particular history, culture or geography. Mumbai is different from Shanghai is different from Houston, obviously, but in relation to their own pasts, and to other cities in India, China or the U.S., they follow these laws.

{ Smithsonian | Continue reading }

art { Alex Roulette }

‘Canada’s four seasons: almost winter, winter, still winter, and road construction.’ –Will Ferrell

Find the pattern for the following series of numbers:

8, 5, 4, 9, 1, 7, 6, 3, 2, 0

{ Solution }

They grab wafers between which are wedged lumps of coal and copper snow. Sucking, they scatter slowly.

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What constitutes a dangerous equation? […] Few would disagree that the obvious winner in this contest would be Einstein’s iconic equation

E=MC2

for it provides a measure of the enormous energy hidden within ordi- nary matter. Its destructive capability was recognized by Leo Szilard, who then instigated the sequence of events that culminated in the construction of atomic bombs.

This is not, however, the direction I wish to pursue. Instead, I am interested in equations that unleash their danger, not when we know about them, but rather when we do not; equations that allow us to understand things clearly, but whose absence leaves us dangerously ignorant.* There are many plausible candidates that seem to fill the bill. But I feel that there is one that surpasses all others in the havoc wrought by ignorance of it over many centuries. It is the equation that provides us with the standard deviation of the sampling distribution of the mean; what might be called De Moivre’s equation:

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For those unfamiliar with De Moivre’s equation let me offer a brief aside to explain it.

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Why have I decided to choose this simple equation as the most dangerous? There are three reasons, related to

(1) the extreme length of time during which ignorance of it has caused confusion,
(2) the wide breadth of areas that have been misled, and
(3) the seriousness of the consequences that such ignorance has
caused.

{ Howard Wainer | PDF }

photo { Adam Broomberg & Oliver Chanarin }

Most of us thought as much

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One of the increasingly famous paradoxes in science is named after the German mathematician Dietrich Braess who noted that adding extra roads to a network can lead to greater congestion. Similarly, removing roads can improve travel times.

Traffic planners have recorded many examples of Braess’ paradox in cities such as Seoul, Stuttgart, New York and London. And in recent years, physicists have begun to study how it might be applied in other areas too, such as power transmission, sporting performance where the removal of one player can sometimes improve a team’s performance and materials science where the network of forces within a material  can be modified in counterintuitive ways, to make it expand under compression, for example.

Today,  Krzysztof Apt at the University of Amsterdam in The Netherlands and a couple of pals reveal an entirely new version of this paradox that occurs in social networks in which people choose products based on the decisions made by their friends. 

They show mathematically that adding extra products can reduce the outcome for everyone and that reducing product choice can lead to better outcomes for all. That’s a formal equivalent to Braess’ paradox for consumers.

{ The Physics arXiv Blog | Continue reading }

‘War doesn’t determine who’s right, just who’s left.’ –Steven Wright

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MATHEMATICS PRIZE: Dorothy Martin of the USA (who predicted the world would end in 1954), Pat Robertson of the USA (who predicted the world would end in 1982), Elizabeth Clare Prophet of the USA (who predicted the world would end in 1990), Lee Jang Rim of KOREA (who predicted the world would end in 1992), Credonia Mwerinde of UGANDA (who predicted the world would end in 1999), and Harold Camping of the USA (who predicted the world would end on September 6, 1994 and later predicted that the world will end on October 21, 2011), for teaching the world to be careful when making mathematical assumptions and calculations.

{ Improbable | Continue reading }

quote { thanks Tim }

video still { Adam Magyar }

‘But where danger is, grows the saving power also.’ –Hölderlin

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Heaven is hotter than hell

Proof:

The temperature of heaven can be rather accurately computed. Our authority is the Bible, Isaiah 30:26 reads,

Moreover, the light of the moon shall be as the light of the sun and the light of the sun shall be sevenfold as the light of seven days.

Thus, heaven receives from the moon as much radiation as the earth does from the sun, and in addition seven times seven (forty nine) times as much as the earth does from the sun, or fifty times in all. The light we receive from the moon is one ten-thousandth of the light we receive from the sun, so we can ignore that. With these data we can compute the temperature of heaven: The radiation falling on heaven will heat it to the point where the heat lost by radiation is just equal to the heat received by radiation. In other words, heaven loses fifty times as much heat as the earth by radiation. Using the Stefan-Boltzmann fourth power law for radiation

(H/E)4 = 50

where E is the absolute temperature of the earth, 300°K (273+27). This gives H the absolute temperature of heaven, as 798° absolute (525°C).

The exact temperature of hell cannot be computed but it must be less than 444.6°C, the temperature at which brimstone or sulfur changes from a liquid to a gas. Revelations 21:8: But the fearful and unbelieving… shall have their part in the lake which burneth with fire and brimstone.” A lake of molten brimstone [sulfur] means that its temperature must be at or below the boiling point, which is 444.6°C. (Above that point, it would be a vapor, not a lake.)

We have then, temperature of heaven, 525°C. Temperature of hell, less than 445°C. Therefore heaven is hotter than hell.

Refutation:

In Applied Optics (1972, 11 A14) there appeared a calculation of the respective temperatures of Heaven and Hell. That of Heaven was computed by substituting the values given in Isaiah 30 26 in the Stefan-Boltzman radiation law. […] This is hard to find fault with. The assessment of the temperature of Hell stands, I suggest, on less firm ground.

{ Applied Optics/Journal of Irreproducible Results | Continue reading }

photo { Harold Diaz }

Numbers are things in time

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How does a number of things become a number in itself, seen as a ‘thing’? In all its simplicity, this is the unsolved question for cognitive science.

{ Per Aage Brandt/SSRN | Continue Reading }

Everything, whether it be more perfect or less perfect, will always be able to persist in existence with the same force wherewith it began to exist; wherefore, in this respect, all things are equal.

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What number is halfway between 1 and 9? Is it 5 — or 3?

Ask adults from the industrialized world what number is halfway between 1 and 9, and most will say 5. But pose the same question to small children, or people living in some traditional societies, and they’re likely to answer 3.

Cognitive scientists theorize that that’s because it’s actually more natural for humans to think logarithmically than linearly: 30 is 1, and 32 is 9, so logarithmically, the number halfway between them is 31, or 3. Neural circuits seem to bear out that theory. For instance, psychological experiments suggest that multiplying the intensity of some sensory stimuli causes a linear increase in perceived intensity.

In a paper that appeared online last week in the Journal of Mathematical Psychology, researchers from MIT’s Research Laboratory of Electronics (RLE) use the techniques of information theory to demonstrate that, given certain assumptions about the natural environment and the way neural systems work, representing information logarithmically rather than linearly reduces the risk of error.

{ MIT | Continue reading }

photo { Rupp Worsham }

At four, she said. Time ever passing.

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{ When Tarzan leaps from a swinging rope, when should he let go to jump furthest? The answer isn’t as simple as you might think. }

The entropy formula for the Ricci flow and its geometric applications

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Grigori Perelman is one of the greatest mathematicians of our time, a Russian genius who solved the Poincaré Conjecture, which plagued the brightest minds for a century. At the height of his fame, he refused a million-dollar award for his work. Then he disappeared. Our writer hunts him down on the streets of St. Petersburg. […]

Word was that someone had solved an unsolvable math problem. The Poincaré conjecture concerns three-dimensional spheres, and it has broad implications for spatial relations and quantum physics, even helping to explain the shape of the universe. For nearly 100 years the conjecture had confused the sharpest minds in math, many of whom claimed to have proven it, only to have their work discarded upon scrutiny. The problem had broken spirits, wasted lives. By the time Perelman defeated the conjecture, after many years of concentrated exertion, the Poincaré had affected him so profoundly that he appeared broken too.

Perelman, now 46, had a certain flair. When he completed his proof, over a number of months in 2002 and 2003, he did not publish his findings in a peer-reviewed journal, as protocol would suggest. Nor did he vet his conclusions with the mathematicians he knew in Russia, Europe and the U.S. He simply posted his solution online in three parts—the first was named “The Entropy Formula for the Ricci Flow and Its Geometric Applications”—and then e-mailed an abstract to several former associates, many of whom he had not contacted in nearly a decade.

{ Brett Forrest | Continue reading }