From reddit.

1:15 video

Hat tip to Forgetomori, forwarded by Robert Breedlove.

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.

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

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

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.

*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.

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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

From the Australian News.Com.Au:

President Mahmoud Ahmadinejad has sought to justify his confidence the US will not attack Iran, saying the proof comes from his mathematical skills as an engineer and faith in God, the press reported today.

Mr Ahmadinejad told academics in a speech that elements inside Iran were pressing for compromise in the nuclear standoff with the West over fears the US could launch a military strike.

“In some discussions I told them ‘I am an engineer and I am examining the issue. They do not dare wage war against us and I base this on a double proof’,” he said in the speech yesterday, reported by the reformist Etemad Melli and Kargozaran newspapers.

“I tell them: ‘I am an engineer and I am a master in calculation and tabulation.

“I draw up tables. For hours, I write out different hypotheses. I reject, I reason. I reason with planning and I make a conclusion. They cannot make problems for Iran.”’

Meanwhile, he is also boasting of having reached a milestone in his quest for an Iranian nuclear weapon.

AFP:

President Mahmoud Ahmadinejad said Sunday Iran had put into operation over 3,000 uranium-enriching centrifuges at a nuclear plant, reaching a key goal of its atomic drive, state broadcasting reported.

“They (world powers) thought that by issuing any resolution Iran would back down,” Ahmadinejad told Islamist students, referring to the two sanctions resolutions imposed against Tehran by the UN Security Council.

“But after each resolution the Iranian nation took another step along the path of nuclear development,” he said.

“Now it has put into operation more than 3,000 centrifuges and every week we install a new series.”

The installation of 3,000 centrifuges has always been earmarked by Iran as the key medium-term goal of its nuclear programme which it had originally hoped to reach by March.

Legend has it that vampires feed on human blood and once bitten a person turns into a vampire and starts feasting on the blood of others.

Efthimiou’s debunking logic: On Jan 1, 1600, the human population was 536,870,911. If the first vampire came into existence that day and bit one person a month, there would have been two vampires by Feb. 1, 1600. A month later there would have been four, and so on. In just two-and-a-half years the original human population would all have become vampires with nobody left to feed on.

If mortality rates were taken into consideration, the population would disappear much faster. Even an unrealistically high reproduction rate couldn’t counteract this effect.

“In the long run, humans cannot survive under these conditions, even if our population were doubling each month,” Efthimiou said.

Every simply connected closed three-manifold is homeomorphic to the three-sphere, where a three-sphere is simply a generalization of the usual sphere to one dimension higher.

Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.

After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.

Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.

As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.