Set Theory Proof

Here's a proof that I have always found fascinating.

Theorem: Given any set A, the power set of A has strictly larger cardinality than A.

For two sets to have the same cardinality, there must exist a bijection between the two sets. So, let us assume that there exists such a bijection between A and P(A), denoted f:A->P(A).

For each x in A, either x is in f(x) or x is not in f(x). Consider the subset S of A defined as
S = {x in A | x is not in f(x)}.

Since we have assumed that f:A->P(A) is a bijection and S is in P(A), then there exists some t in A such that f(t) = S.

Either t is in S or t is not in S. Suppose t is in S. Then t satisfies the condition that t is not in f(t) which is S. Therefore t is not in S. Suppose t is not in S. Then t does not satisfy the condition, and t is in f(t)=S. Therefore t is in S.

So in either case, we arrive at the contradiction that t is in S and simultaneously t is not in S. The source of the contradiction is the assumption that there exists a bijection f:A->P(A). Therefore, there exists no such bijection and the cardinality of P(A) is not the same as the cardinality of A. Since the cardinality of P(A) cannot be smaller than the cardinality of A, the conclusion is that P(A) has strictly larger cardinality than A. (Q.E.D.)

Finite Simple Group (of Order Two)

The lyrics of this song can be found here. More information on the group (from Northwestern University) can be found at http://www.kleinfour.com/.

[Thanks to TMB for sending me this video.]

Fields Medals 2006

Awarded once every 4 years by the International Mathematical Union, the Fields Medal recognizes outstanding contributions to mathematics.

This year's winners...

Andrei Okounkov “for his contributions briding probability; representation theory and algebraic geometry.”

Grigori Perelman “for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.” (Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow. Perelman is quite the recluse and actually turned down the Fields Medal.)

Terence Tao “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.” (At 31, Tao is one of the youngest mathematicians to be awarded the Fields Medal.)

Wendelin Werner “for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.”

Busy year for Pluto

The International Astronomical Union defined the term "planet" for the first time. This definition excluded Pluto from planethood, so Pluto was reclassified under the new category of "dwarf planet" along with and Ceres and Eris.

At Berkeley

On a bulletin board in the mathematics department at UC Berkeley, somebody had left a flier offering a $300 reward for a solution to the equation x^5 + ax^4 + bx^3 + cx^2 + dx = e in terms of a, b, c, d and e. If you studied college level math, you know this problem is unsolvable. The insolubility of the quintic is proved as part of Galois theory in the second semester of undergraduate abstract algebra. I am not sure if this was a hoax or exactly what the purpose of this flier might have been, but this being Berkeley several people had scribbled sarcastic offers to square the circle (another problem that has been proven to be unsolvable) or to prove that pi is a natural number.

Nix and Hydra

Two additional moons of Pluto (in addition to Charon) were imaged by astronomers working with the Hubble Space Telescope on 5/15/2005. The International Astronomical Union officially named the moons Nix and Hydra. These moons orbit Pluto at approximately two and three times the distance of Charon: Nix at 48,700 kilometres and Hydra at 64,800 kilometres from the barycenter of the system.

Geeky Math Humor

Q: What's the value of a contour integral around Western Europe?
A: Zero, because all the Poles are in Eastern Europe.

Itzykson & Zuber is back in print

Dover rocks!

http://www.amazon.com/gp/product/0486445682/

Ridiculous

On today's Law & Order Criminal Intent, I learned that elementary age kids without any training in advanced mathematics or supervision by a knowledgeable mathematician nevertheless have a reasonable chance of solving the Riemann Hypothesis just because they are geniuses.

A couple of weeks ago, I learned that in theoretical physics "theories" are formulated without any appeal to mathematics, and then mathematics is used to prove those theories.

Ridiculous

Having a bad day? No wonder. According to one scientist, today, January 23, is the gloomiest day of 2006. Cliff Arnall, a health psychologist at the University of Cardiff in Wales, has devised a formula that combines personal and seasonal factors to calculate the year's emotional low point.

Caught this story in today's paper. The same story made the rounds last year. It's utterly ridiculous. According to this "scientist," the variables are weather (W), debt (D), monthly salary (S), time since Christmas (T), time since failure to quit a bad habit (Q), low motivational levels (M) and the need to take action (N). Sounds reasonable enough. But now let's look at his "formula."

Gloominess = [W + (D-S)xTxQ] / [M x N]

So we take time (since Christmas) times time (since failure) times money. So we have a quantity that is measured in $ x days squared. Then we add this to a weather variable, which is measured in . . . ?

Then we take this already meaningless concoction and further divide it by "low motivational levels" and "the need to take action." I'm unclear on how either of these can be measured well enough to allow any mathematical comparisons to be made between January 23 and January 20 or January 26 ... or July or November or any other random date I pull out of a hat. Furthermore, why do we divide by them; why not multiply?

I am appalled that some idiot can get this "research" published. What a freakin' joke.

Ridiculous

I'm watching an episode of Law & Order: Criminal Intent, and in just a five minute interval, I've learned the following:

• "After three tries, most password programs give you a hint." Funny, most password programs I've ever seen after three tries LOCK YOU OUT!
• Senior math majors take multivariable calculus. (I took it first semester of sophomore year, and I was far from the most advanced student in my cohort.)
• Seniors who have not completed multivariable calculus are given research grants by the Department of Defense.
• Random number generator programs increase your odds of winning the lottery.

2005 Nobel Prize in Physics

Roy J. Glauber
"For his contribution to the quantum theory of optical coherence."

John L. Hall
Theodor W. Hänsch
"For their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique."

http://nobelprize.org/physics/laureates/2005/

MIT Open CourseWare

MIT has a website, Open Course Ware, where professors present a lot of the teaching materials from their various courses. When the site first started, the math and physics offerings were pretty thing. However, over the last year, many of the courses have been substantially enhanced, to the point where they are now quite useful resources, including extensive lecture notes and many solutions to homework assignments. I am also providing the link to the MIT math and physics departments, since those sites still have the home pages of professors with resources for courses being offered in the current academic years.

Physics OCW

Physics Department Website

Mathematics OCW

Mathematics Department Website

Dr. Serge Lang, R.I.P.

Serge Lang died today. Rather than create a lengthy post that would still not do this great mathematician justice, I'll just link to his wikipedia entry.

Methods of Theoretical Physics

The famous mathematical physics textbook by Morse & Feshbach has finally been republished. I've long wanted to get my hands on a copy, but not at this price ($329 for the set, up from the original price of $250).

Not sure why the price is going up or what is he thinking, but I don't think his company will never be successful at this price. It's a shame.

Thirteen things that do not make sense

New Scientist has an article on thirteen things that do not make sense in science. Historically, such paradoxes have provided the impetus for new discoveries. The most notable examples are the Michelson-Morley null experiment (leading to special relativity), the unexpected blackbody radiation spectrum (leading to the quantum hypothesis), and the anomalous precession of the perihelion of Mercury (leading to general relativity).

1. The placebo effect
2. The cosmological horizon problem
3. Ultra-energetic cosmic rays
4. Belfast homeopathy results
5. Dark matter
6. Viking's discovery methane on Mars
7. Tetraneutrons
8. The Pioneer anomaly
9. Dark energy
10. The Kuiper cliff
11. The "Wow" signal
12. Not-so-constant constants
13. Cold fusion

[Thanks to DVD for the link]

Superstrings <> Cosmic Strings

I have seen this article twice today, so I thought I'd make a quick post about it.

Could two lookalike galaxies, barely a whisker apart in the night sky, herald a revolution in our understanding of fundamental physics? Some physicists believe that the two galaxies are the same - its image has been split into two, they maintain, by a "cosmic string." If this interpretation is correct, then CSL-1 - the name of the curious double galaxy - is the first concrete evidence for "superstring theory": the best candidate for a "theory of everything", which attempts to encapsulate all the phenomena of nature in one neat set of equations.

Not sure why the author of this article switched from "cosmic strings" to "superstrings" without a word of explanation, but these two concepts are completely unrelated. Superstrings are the extension of Quantum Field Theory of point particles to one-dimensional "strings," a theory which leads to highly speculative predictions about a 10 dimensional universe and may succeed in unifying quantum field theory with classical General Relativity. Cosmic strings are astronomical objects described by the rules of classical general relativity. The two things have absolutely nothing to do with each other.

http://www.astronomycafe.net/qadir/ask/a11828.html

Astronomers find star-less (dark matter?) galaxy

Astronomers say they have discovered an object that appears to be an invisible galaxy made almost entirely of dark matter. The team, led by Cardiff University, claimed it is the first to be detected. It was found 50 million light years away using radio telescopes in Cheshire and Puerto Rico.

[Thanks to GB for the link.]

No Evidence of Life on Mars

News reports on February 16, 2005, that NASA scientists from Ames Research Center have found strong evidence that life may exist on Mars are incorrect. NASA does not have any observational data from any current Mars missions that supports this claim. The work by the scientists mentioned in the reports cannot be used to directly infer anything about life on Mars, but may help formulate the strategy for how to search for martian life. Their research concerns extreme environments on Earth as analogs of possible environments on Mars. No research paper has been submitted by them to any scientific journal asserting martian life.

Life on Mars? Never mind!

A pair of NASA scientists told a group of space officials at a private meeting Sunday that they have found strong evidence that life may exist today on Mars, hidden away in caves and sustained by pockets of water. The scientists, Carol Stoker and Larry Lemke of NASA's Ames Research Center in Silicon Valley, told the group that they have submitted their findings to the journal Nature for publication in May, and their paper currently is being peer reviewed. What Stoker and Lemke have found, according to several attendees of the private meeting, is not direct proof of life on Mars, but methane signatures and other signs of possible biological activity remarkably similar to those recently discovered in caves here on Earth.