Sunday, March 21, 2010

More Bach

Today is Bach's 325th birthday, if you take the Julian calender date. What are you doing to celebrate? I'd recommend listening to the St. Matthew Passion or a cantata. Maybe play some yourself, if you're so inclined.

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Saturday, March 20, 2010

Landau: Mechanics, 3rd Ed., p. 32

WARNING!!! Serious mathematical question ahead.

On the page mentioned above in the book mentioned above, Landau makes the following claim, in the context of talking about motion in a central field; that is, the motion of two particles, where the only forces between the two particles are directed on the straight line joining the two particles.

Such cases are exceptional, however, and when the form of $U(r)$ is arbitrary the angle $\Delta\phi$ is not a rational fraction of $2\pi$. In general, therefore, the path of a particle executing a finite motion is not closed. It passes through the minimum and maximum distances an infinity of times, and after infinite time it covers the entire annulus between the two bounding circles. The path shown in Fig. 9 is an example.

A note to the reader: Fig. 9 looks very much like the graph on the front of this book.

Now, in classical mechanics, we have no tunneling or other means of teleportation. This implies that the particle must continuously traverse whatever path it is on. We should also note that a particle is like a mathematical point - it has no extension.

Now, mathematically, Landau is saying that there is a continuous bijection from the half-infinite line $[0,\infty)$ to the annulus $\overline{B}(0,r_{\text{max}})\setminus B(0,r_{\text{min}})$. Is this possible? The annulus obviously requires two continuous variables to locate a point in its interior. However, given the path of the particle as a function of time, we need only specify one variable (time) in order to locate its position.

So help me out, you topologists. Is there a homeomorphism between these two sets? I could believe that the particle's path is dense in the annulus, but I'm not sure about traversing every point.

In Christ.

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Wednesday, March 17, 2010

Some Hard Numbers

According to the wiki article on Texas, its area is 268,820 square miles. Now, 1 square mile is exactly 27,878,400 square feet. Ergo, Texas has approximately 7.5 x 10^12 square feet. Now, according to the wiki article on World Population, the current population of the world is estimated by the US Census Bureau to be 6,808,900,000. Suppose we were to fit the entire world population into the state of Texas. How much room would each person have? Well, you'd take your 7.5 x 10^12 and divide by 6,808,900,000. According to my calculator, that gives about 1100 square feet / person. That's a small home with two bedrooms, a living room, a kitchen, and one bathroom. That's not too bad. If you do the same calculation, except that you use the area of the entire United States, you end up with approximately 15500 square feet / person. That's an extremely large mansion.

Have you ever noticed that the people who argue that the world is over-populated tend to be liberals/socialists? And have you also noticed that liberals and socialists tend to be concentrated in large cities? And have you ever noticed that in large cities, the amount of living area the average person has is considerably smaller than in the country? It makes you wonder, doesn't it, as to whether liberals are able to look beyond the confines of their cities!

To take another angle, let us consider wheat. According to the wiki on Norman Borlaug,

Biologist Paul R. Ehrlich wrote in his 1968 bestseller The Population Bomb, "The battle to feed all of humanity is over ... In the 1970s and 1980s hundreds of millions of people will starve to death in spite of any crash programs embarked upon now." Ehrlich said, "I have yet to meet anyone familiar with the situation who thinks India will be self-sufficient in food by 1971," and "India couldn't possibly feed two hundred million more people by 1980." - Ehrlich, Paul: The Population Bomb, 1968.

A little bit later, the wiki has the following: "By 1974, India was self-sufficient in the production of all cereals." So, ok, maybe India took a little time to get there, but they did. Ehrlich's basic idea was flat-out wrong. Norman Borlaug has been credited with saving the lives of over a billion people from starvation.

When are we going to get it into our heads that Malthus and Ehrlich are wrong? It seems, in the face of such hard numbers as I've given above, that the over-population people are incorrect. So why would supposedly rational people ignore numbers such as these? There might be any number of reasons. Possibly one reason might be that science and hard data is not as important to such people as control over other people. We've seen this first-hand with the climategate scandal, which the extreme environmentalists have ignored to the best of their ability. Their support is waning, however.

I would issue a call to liberals and socialists everywhere to re-examine your philosophy and how well it matches up with your rhetoric. Do not invoke science and statistics when you intend to twist them for your own political agenda.

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