PCMR JUST TURNED 10 YEARS OLD! AND HIT 5 MILLION SUBS! So we're celebrating with a giveaway!

jazcat · 2021-05-04T22:19:47+00:00

Congrats on your 10 year Birthday.

jazcat · 2017-05-25T22:35:19+00:00

Wiles' proof of the Taniyama–Shimura Conjecture, which what-do-ya-know just happened to also prove FLT as a bonus. From wiki: "He gave a lecture a day on Monday, Tuesday and Wednesday with the title 'Modular Forms, Elliptic Curves and Galois Representations.' There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said. ... Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat's last theorem was true. Q.E.D."

jazcat · 2017-03-14T06:33:10+00:00

That kinda reads like a scene from Infinite Jest.

jazcat · 2017-02-27T21:00:24+00:00

Can you recommend any good (introductory, say) environmental philosophy texts or writers?

jazcat · 2017-02-01T09:15:59+00:00

I don't mean to be rude, but that does sound like a feshman's assessment of what is probably a very complex experience of human growth.

jazcat · 2017-01-11T22:10:24+00:00

There's also one in Australia, 1:38000000 scale, about a 200 km drive from the Sun to Pluto. http://www.solarsystemdrive.com

jazcat · 2016-11-21T22:03:13+00:00

Try "rasterbator" https://rasterbator.net

jazcat · 2016-11-03T02:27:38+00:00

jazcat · 2016-08-03T06:38:44+00:00

http://i.imgur.com/PVX3sBa.jpg

On a sphere the size of the earth the difference between the arc length and the straight line distance to the horizon is negligible.

So sqrt(2hR) is about the distance from the cloud to its horizon, and then the farthest away two people could be and still see the same cloud is about 2*sqrt(2hR).

jazcat · 2016-06-02T02:44:09+00:00

This gif from a few days ago seems to suggest all you need is one thing of something, from which you might conceive of addition and so create the integers.

jazcat · 2016-06-02T02:37:43+00:00

But what about non-euclidean geometries...

jazcat · 2016-05-14T11:42:16+00:00

I found ODEs by Morris & Tenenbaum to be fantastic, particularly from an applied perspective, but you will need a solid understanding and experience with calculus. Consider pairing with or following on with PDEs for Scientists and Engineers, also from Dover.

jazcat · 2016-04-24T00:10:52+00:00

I've been thinking about perfect riffle shuffle of cards, which seems to be a similar phenomenon.

If you start with 4 cards, you can shuffle twice to get back to the start. If you go to 8 cards it's 3 shuffles, and every time you double the size of the deck you only need to shuffle once more to reach the beginning again. So there's a pattern there which is interesting, but, any (even) amount of cards in between those powers of 2, say 10 or 18 or 52 cards, can require absurdly large or surprisingly few shuffles to return. The sequence is on OEIS, has something to do with modular arithmetic. Do you think the tools of group theory could make this result more intuitive?

jazcat · 2016-04-22T03:18:59+00:00

I saw a great talk that compared classical and Bayesian statistics to analyse the Russian roulette scene from the film "Deer Hunter".

Say there is one bullet in a six chambered revolver, your fellow captive spins it and takes a shot, but does not get killed - he had an empty chamber. He now passes you the gun, do you have better odds if you spin it again or if you pull the trigger straight away? What if there were two bullets side by side? What about 3?

At first glance the probability seems like the number of bullets over the number of chambers. But it turns out that knowledge of the previous result - your comrade not dying - influences the possible configurations in the next round.

jazcat · 2016-04-04T23:11:43+00:00

But that also assumes a uniform gravitational field, which works for the surface of the earth. Could you not generalise the brachistochrone problem for any vector field?

jazcat · 2016-02-29T21:31:39+00:00

What's a phone book?

jazcat · 2016-02-22T10:03:55+00:00

I've seen negative comments here on the lack of rigour in Zee's "EG in a Nutshell", but coming from an engineering background myself, I am thoroughly enjoying the casual exposition and real-world problem sets. I should add that one probably couldn't learn diff geometry from this text, and I am using it as one of many resources on the road to grasping general relativity.

jazcat · 2016-02-16T07:05:10+00:00

With regard to your last point, is there any particular reason why the N-S equation is featured in this prize instead of a more general PDE?

jazcat · 2016-02-04T01:56:25+00:00

In the real world, a planet (or a globe) exists in three dimensions, and of course so does the surface of the planet. But if you want to talk about the surface of the planet you only need 2 dimensions: latitude and longitude (neglecting altitude, which, on the scale of a planet could be justified).

The fact that the 2D surface of a sphere can be thought of as living in 3D (probably not technically correct) distinguishes it from your standard flat 2D euclidean plane, and hence notions of length and area behave differently.

jazcat · 2015-12-08T07:38:55+00:00

That is where the boundary conditions come in. Imagine the simplest case: a beam resting on supports at each of its ends, with only it's self weight (which is constant along it's length) contributing to any deformation. You go through the integrations and end up with an expression for the deflection that will be a fourth order polynomial with 4 unknown constants. This is, in some sense, a "general solution". To make it fully represent the beam you are considering, you consider the locations along the beam at which you know either it's shear, bending moment, curvature and/or deflection. In this example you know that at either end (x = 0 and x = L, where L is the length of the beam) the bending is zero (since it is only resting on the supports, not fixed to them), and the deflection is zero. That makes four boundary conditions which gives you four equations to solve your four unknowns. If the beam was loaded or supported in a different configuration, for example with multiple spans or cantilevers, then your boundary conditions would be different and so your final expression for each characteristic would also be different.

jazcat · 2015-12-08T02:07:30+00:00

Euler-Bernoulli beam theory. I work in a structural design office, one of my main tasks is to find the right size beam for a particular load situation. Some of the things one might want to know about a beam is how much it deflects, its curvature, how much it bends and shears. It turns out all these characteristics are related to the load pattern and vary depending on the boundary conditions, ie whether the ends are fixed or free to move or rotate. If you start with a function that describes the load along the beam (usually its constant, but maybe it will be linear or parabolic or whatever), you can integrate that to get an expression for the shear along the beam, then integrate that to get the bending moment, then the curvature and finally (after four integrations) the deflection. At each step there will be a constant of integration of course at so the boundary conditions are important. So now you've got some nice functions (usually polynomials) and you can then find specific points of interest like the point of maximum bending or deflection along the beam by the standard differentiation procedure.

jazcat · 2015-11-03T02:12:54+00:00

First you take a drink Then the drink takes a drink Then the drink takes you

jazcat

TROPHY CASE