Category Archive 'Mathematics'
14 Oct 2013

XKCD Meets Ayn Rand

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Hat tip to Karen L. Myers.

07 May 2013

“Jame Austen, Game Theorist”

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The latest wrinkle in the contemporary Jane Austen boom is described at Science Blog:

Austen’s novels are game theory textbooks,” Michael Suk-Young Chwe writes in “Jane Austen, Game Theorist,” which Princeton University Press published April 21. “She’s trying to get readers to use their higher thinking skills and to think strategically.”

At its most basic level, game theory assesses all the choices available to two (or more) people in a given situation and assigns a numerical value to the benefit each person reaps from each choice. Often, the choice that is most valuable to one player comes at the expense of the other; hence, game theory’s best-known phrase — “zero-sum game.” But just as frequently, there is a choice with unexpected benefits for both players.

“In game theory, you make choices by anticipating the payoffs for others,” Chwe explains.

Chwe argues that Austen explores this concept in all six of her novels, albeit with a different vocabulary than the one used by Nash, von Neumann and other game theory greats some 150 years later. In Austen’s romantic fiction, this type of strategic thinking is described as “penetration,” “foresight” or “a good scheme.”

In “Pride and Prejudice,” for instance, Mrs. Bennet, a mother eager to marry off her five daughters, sends her oldest, Jane, on horseback to a neighboring estate, even though she’s aware a storm is on the way. “Mrs. Bennet knows full well that because of the rain, Jane’s hosts will invite her to spend the night, thus maximizing face time with the eligible bachelor there, Charles Bingley, whom Jane eventually marries,” Chwe said.

In “Persuasion,” the unmarried heroine, Anne Elliot, is approached by Sophia Croft, the sister of a man whose marriage proposal Anne spurned eight years earlier — a decision she still bitterly regrets. Mrs. Croft casually asks Anne whether she’s heard that her brother has married. Anne flinches, thinking the reference is to her former beau, Captain Frederick Wentworth, but relaxes upon learning that Mrs. Croft is actually referring to their younger brother, Edward.

“It’s hard to imagine a better way for Mrs. Croft to gauge Anne’s visceral interest in her unmarried brother,” said Chwe, a UCLA associate professor of political science (whose last name is pronounced like “chess” without the “ss”). The rest of the novel involves schemes to give Captain Wentworth so many signals of Anne’s enduring love that he finds the courage to propose to her again.

14 Mar 2013

Applied Math

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click on image

22 Sep 2012

Voting by the Dumb and Ill-Informed

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Beyond a certain point of negative understanding, of course, it is better for someone not to vote.

Joseph C. McMurray discusses the Marquis de Condorcet’s mathematical analysis favoring decision-making by larger numbers of people.

An interesting, if somewhat uncommon, lens through which to view politics is that of mathematics. One of the strongest arguments ever made in favor of democracy, for example, was in 1785 by the political philosopher-mathematician, Nicolas de Condorcet. Because different people possess different pieces of information about an issue, he reasoned, they predict different outcomes from the same policy proposals, and will thus favor different policies, even when they actually share a common goal. Ultimately, however, if the future were perfectly known, some of these predictions would prove more accurate than others. From a present vantage point, then, each voter has some probability of actually favoring an inferior policy. Individually, this probability may be rather high, but collective decisions draw information from large numbers of sources, mistaking mistakes less likely.

To clarify Condorcet’s argument, note that an individual who knows nothing can identify the more effective of two policies with 50% probability; if she knows a lot about an issue, her odds are higher. For the sake of argument, suppose that a citizen correctly identifies the better alternative 51% of the time. On any given issue, then, many will erroneously support the inferior policy, but (assuming that voters form opinions independently, in a statistical sense) a 51% majority will favor whichever policy is actually superior. More formally, the probability of a collective mistake approaches zero as the number of voters grows large.

Condorcet’s mathematical analysis assumes that voters’ opinions are equally reliable, but in reality, expertise varies widely on any issue, which raises the question of who should be voting? One conventional view is that everyone should participate; in fact, this has a mathematical justification, since in Condorcet’s model, collective errors become less likely as the number of voters increases. On the other hand, another common view is that citizens with only limited information should abstain, leaving a decision to those who know the most about the issue. Ultimately, the question must be settled mathematically: assuming that different citizens have different probabilities of correctly identifying good policies, what configuration of voter participation maximizes the probability of making the right collective decision?

It turns out that, when voters differ in expertise, it is not optimal for all to vote, even when each citizen’s private accuracy exceeds 50%. In other words, a citizen with only limited expertise on an issue can best serve the electorate by ignoring her own opinion and abstaining, in deference to those who know more. …

This raises a new question, however, which is who should continue voting: if the least informed citizens all abstain, then a moderately informed citizen now becomes the least informed voter; should she abstain, as well?

Mathematically, it turns out that for any distribution of expertise, there is a threshold above which citizens should continue voting, no matter how large the electorate grows. A citizen right at this threshold is less knowledgeable than other voters, but nevertheless improves the collective electoral decision by bolstering the number of votes. The formula that derives this threshold is of limited practical use, since voter accuracies cannot readily be measured, but simple example distributions demonstrate that voting may well be optimal for a sizeable majority of the electorate.

The dual message that poorly informed votes reduce the quality of electoral decisions, but that moderately informed votes can improve even the decisions made even by more expert peers, may leave an individual feeling conflicted as to whether she should express her tentative opinions, or abstain in deference to those with better expertise. Assuming that her peers vote and abstain optimally, it may be useful to first predict voter turnout, and then participate (or not) accordingly: when half the electorate votes, it should be the better-informed half; when voter turnout is 75%, all but the least-informed quartile should participate. …

If Condorcet’s basic premise is right, an uninformed citizen’s highest contribution may actually be to abstain from voting, trusting her peers to make decisions on her behalf. At the same time, voters with only limited expertise can rest assured that a single, moderately-informed vote can improve upon the decision made by a large number of experts. One might say that this is the true essence of democracy.

His conclusion seems to accord with observed results. Ordinary people are surprisingly well able to correct the follies and delusions which too commonly afflict the experts and elites, but there are also people so clueless that they are always going to vote wrong.

02 Jul 2012

Good Tip

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Hat tip to Jose Guardia.

23 Jun 2012

Gandalf’s Function

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Hat tip to The Meta Picture via Kathleen Wagner.

30 May 2012

The Archimedes Palimpsest

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“How do you read a two-thousand-year-old manuscript that has been erased, cut up, written on and painted over [i.e., a palimpsest]? With a powerful particle accelerator, of course! Ancient books curator William Noel tells the fascinating story behind the Archimedes palimpsest, a Byzantine prayer book containing previously-unknown original writings from ancient Greek mathematician Archimedes and others.”

Archimedes Palimpsest website

Hat tip to Dot Porter.

28 Sep 2010

Trigonometry Illustrated

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From reddit.

22 Jan 2010

Yoshimoto Cube

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Two stellated rhombic dodecahedrons can be folded into a cube. “A very impressive piece of engineering.”

1:15 video

Hat tip to Forgetomori, forwarded by Robert Breedlove.

16 Jan 2010

The Predator Prey Ecology of Vampires and Humans in (Pre-Apocalypse) Sunnyvale, California

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A typical Sunnyvale vampire (Spike) preying on a typical female citizen (Willow).

Brian Dalen Thomas addresses the vexed question of human vampire ecology in the Sunnyvale, California of Joss Wheedon’s Buffy the Vampire Slayer.

Excerpts:

(W)e know from the sign in “Lover’s Walk” that the human population of Sunnydale is 38,500. …

Sunnydale’s human population growth rate is 10% annually, which is certainly at the high end for a budding California community.

A vampire feeds every three days, and encounters about one hundred potential victims in the course of a day, meaning that 1 out of every 300 encounters involves a little refreshment.

An individual vampire sires a victim every other year, or once per 240 feedings.

Buffy and her Slayerettes, busy little beavers that they are, annually stake about 1/3 of the vampires plaguing Sunnydale.

Vampires are flocking to Sunnydale, since the Hellmouth is the underwordly equivalent of Silicon Valley, and the demon labor market is just too good to be true. Thus, we’ll assume a yearly migration rate of about 10%, or the same as for the humans.

A Model

What follows is based on some of the simpler theoretical understandings of predator-prey population dynamics. I’m assuming that human populations are not controlled solely by vampire predation (i.e.- in the absence of vampires, the human population would still eventually be limited by some other factor, like food supply, disease, or access to a well written weekly news magazine. I like The Economist myself, but that’s clearly a digression).

If we let H stand for the size of the human population and V stand for the size of the vampire population, then we can represent the changes in each population over time with a pair of differential equations:

dH/dt = rH (K-H)/K -aHV

dv/dT = baHV + mV – sV

where r is the intrinsic growth rate of the human population, incorporating natural rates of both birth and death as well as immigration

K is the human carrying capacity of the habitat in question

a is a coefficient that relates the number of human-vampire encounters to the number of actual feedings

b is the proportion of feedings in which the vampire sires the victim (i.e.- this is the vampire birth rate)

m is the net rate of vampire migration into Sunnydale

s is the rate at which the Scoobies stake vampires (assumed to be the only important source of vampire deaths).


The following graph shows human population sizes on the horizontal axis and vampire population sizes on the vertical axis. Each line represents a trajectory through time (the tail of each line, scattered around the outer edge of the figure, shows the “initial population size” where we started the model in motion). Any point on a line represents a combination of human and vampire population sizes – a step, if you will, in that beautiful dance between Buffy and the Minions of Evil. Notice that wherever we “start” the trajectories, they all spiral in towards our equilibrium state, indicated in the center by an
asterisk.

Hat tip to Robert M. Breedlove.

31 May 2009

16 Year Old Iraqi Immigrant Scores Math Success in Sweden

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A 16-Year-Old Iraqi immigrant to Sweden working over four months apparently independently produced a formula for simplifying the generation of the Bernoulli Numbers, a sequence of rational numbers significant in number theory first identified in the Swiss mathematician Jacob Bernoulli‘s Ars Conjectandi, published posthumously in 1713.

Mohamed Altoumaimi’s formula was actually already known by mathematicians, but his generation of the same formula independently sufficiently impressed the academic community in Sweden that the young man was immediately offered admission to Upsala University. He has decided to finish secondary school first, however.

AFP

AHN

SoftSailor

31 Jan 2009

The Beatle’s Unduplicatable First Chord

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Jason Brown, Chairman of the Mathematics Department at Dalhousie University, applies math to solving a musical mystery.

Wall Street Journal

It is here, in a cluttered mathematician’s office, under blackboards jammed with equations and functional analysis, that one of Western culture’s greatest mysteries has finally been solved: Why has no one been able to replicate the first chord in The Beatles’ pop hit “A Hard Day’s Night”? …

Mr. Brown realized he could use a discrete Fourier transform, a mathematical technique for breaking up complicated signals into simpler functions and known as DFT. He used digital equipment to show the chord as a series of numbers, tens of thousands per second, and then applied a DFT to convert the chord into dozens of simpler functions, each representing a single sound frequency.

Mr. Brown knew there is no such thing as a pure tone: Each instrument emits one sound for the note played and then sounds that are multiples of that note’s frequency, as the string vibrates back on itself. Of his dozens of frequencies, some were background noise and some–the ones he wanted to ferret out–were the notes the Beatles struck.

The professor started making deductions. The loudest notes were likely Mr. McCartney’s bass. The lowest had to be the original note played, since a string can generate waves along half or a third of its length, but not twice its length. But no matter how he divvied up the notes, something didn’t fit.

It is well-documented that Mr. Harrison played a 12-string guitar for the recording of “A Hard Day’s Night.” For every guitar note played, there had to be another one octave higher, since his guitar strings were pressed down in pairs.

But three frequencies for an F note were left, none of which were an octave apart. Even if Mr. Brown assumed Mr. Lennon played one F note on his six-string guitar, Mr. Brown still had two unexplained frequencies.

After weeks of staring at six-decimal-place amplitude values, Mr. Brown suddenly remembered how, as a child, he used to stick his head inside his parents’ grand piano to see how it worked. He ran to a nearby music shop, and poked his head inside the Yamahas there.

Sure enough, there were three strings under the F key, corresponding to the three sets of harmonics he had seen. Buried under the iconic guitar chord was a piano note.

Other problems have since yielded to Mr. Brown’s mathematics. Fans have always marveled at Mr. Harrison’s guitar solo in “A Hard Day’s Night,” a rapid-fire sequence of 1/16th notes, accompanied on piano, that seemed to require superhuman dexterity.

Mr. Brown noticed that a piano is strung differently in its lower octaves, with two strings, rather than three, under each hammer. He saw only two frequencies for each piano note in the guitar solo, suggesting that the solo had been played one octave lower than the recorded version sounded. It had also been played at half-speed, he concluded, then sped up on tape to make the released version sound as if had been played faster and at a higher octave.

2:33 video

23 Jul 2008

Amazon Tribe Does Not Use Numbers

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Telegraph:

The idea that people have an innate mathematical ability has been questioned by a study of an Amazonian tribe that has no sense of number.

The ability of tribal adults of the Pirahã to conceptualise numbers is no better than that of infants or even some animals and their language, with only 300 speakers, has no word even to express the concept of “one” or any other specific number.

Prof Gibson found that there were no words for ‘one’ or ‘two’ for members of the Pirahã tribe
The team, led by Massachusetts Institute of Technology professor of brain and cognitive sciences Edward Gibson, found that members of the Pirahã tribe in remote northwestern Brazil use language to express relative quantities such as “some” and “more,” but not precise numbers.

It is often assumed that counting is an innate part of human cognition, said Prof Gibson, “but here is a group that does not count. They could learn, but it’s not useful in their culture, so they’ve never picked it up.

Hat tip to MeaninglessHotAir.

16 Nov 2007

“An Exceptionally Simple Theory of Everything”

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The E8 root system, with each root assigned to an elementary particle field

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A. Garrett Lisi, a 39-year-old researcher, equipped with a doctorate in Physics from the University of California at San Diego, and not otherwise affiliated with any university, last month published a paper proposing to link the Standard Model of Particle Physics with Gravity, expressed as an E8* root system of exceptionally simple character in Lie algebra.

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*E8 encapsulates the symmetries of a geometric object that is 57-dimensional and is itself 248-dimensional.

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Abstract: All fields of the standard model and gravity are unified as an E8 principal bundle connection. A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs, and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E8 superconnection are described by the curvature and action over a four dimensional base manifold.

PDF

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Lisi makes for a wonderful news subject, being a perfect California type, a surfing and rock-climbing ultra-bohemian, the sort of person found dancing around the fire at the annual Burning Man Festival. His theory has a wonderful appeal based upon its simplicity (no pun intended) and elegance, but we will have to wait to see whether it is confirmable by testable predictions.

The Telegraph article quotes some scientists who regard Lisi’s theory as “a long shot,” but there is general agreement already on how interesting and elegant it is. However all this comes out, my own (testable) prediction is that A. Garrett Lisi will be receiving some very good offers of academic appointments at major universities.

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Telegraph news story

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A. Garrett Lisi

His CV

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