Comments on: Maybe you don't need to

By: Life lessons from functional programming — The Endeavour

Life lessons from functional programming — The Endeavour — Tue, 13 Nov 2012 13:01:54 +0000

[...] Maybe you don’t need to Just-in-case vs just-in-time Henri Poincaré’s work schedule [...]

By: Larry

Larry — Fri, 02 Nov 2012 02:06:33 +0000

One example is finding the center of mass of a semicircle (radius = 1). That is, the distance, call it x, from the diameter to the COM. This appears to require calculus.
However: If you rotate the semi-circle about its diameter, it forms a sphere. Pappus’ Theorem then applies:
(area) (distance COM travels) = (volume of the solid swept out)
(pi/2)(2pi*x) = (4pi/3)
From which x = 4/(3pi)

By: Josh Braun

Josh Braun — Thu, 01 Nov 2012 21:48:37 +0000

@Joshua Zucker: I knew someone had to have thought o it before. Glad to put a name to the phenomenon.

By: Joshua Zucker

Joshua Zucker — Thu, 01 Nov 2012 19:10:39 +0000

Josh Braun’s response is equivalent to Euclidean Algorithm; another way of saying what he said is that it lets you find common factors without prime factorizing.

For me the most miraculous items in this category seem to be with expected value, where first of all linearity lets you dispense with all kinds of information about covariances, and second of all you can sum a whole bunch of infinite series that come up in expected value calculations by solving a simple linear equation instead.

By: Rick Wicklin

Rick Wicklin — Thu, 01 Nov 2012 17:38:52 +0000

Solving a differential equation by guessing a solution and then showing that it works. If the problem has uniqueness of solutions, you are done.

By: Franklin Chen

Franklin Chen — Thu, 01 Nov 2012 17:08:59 +0000

There are a lot of examples I can think of. They share in common the fact that one can reason about a system or problem rather than within it. In chess, for example, there are blocked positions in which it is obvious to a human that neither side can break through, but where a computer just using brute force search would never “understand” this structural fact. In mathematical logic, results such as Goedel’s theorem (or equivalently, in computer science the undecidability of the halting problem) result from reasoning about a formal system rather than just following rules within it. More generally, all of mathematics is about being able to prove something without going through all possibilities: something as simple as showing that there are infinitely many primes is fundamental to the whole enterprise (something that I have found many non-mathematicians cannot grasp, because they think in terms of having to show every case, which can’t be done of course, and therefore they think it’s somehow an open question, therefore, whether there could be a final prime).

By: Anton

Anton — Thu, 01 Nov 2012 15:29:11 +0000

If you want to know the integral of a product of two Gaussians, all you need is one evaluation of a third Gaussian.

By: Iain Murray

Iain Murray — Thu, 01 Nov 2012 15:15:09 +0000

A favourite from this blog: find inv(A)*b without inverting matrix A.

Rejection sampling: draw a sample from a distribution defined by a probability density function, without finding its inverse cumulative distribution.

Homomorphic encryption: compute with encrypted data without first decrypting it: http://www.americanscientist.org/issues/pub/2012/5/alice-and-bob-in-cipherspace/1

Maximum likelihood estimation of exponential family models by stochastic approximation: no need to approximate the likelihood being optimized, just go straight for its derivatives (easier to estimate).

A nice class of things…I look forward to seeing more!

By: Will Fitzgerald

Will Fitzgerald — Thu, 01 Nov 2012 15:04:15 +0000

Some people are really good at knowing, as if via proprioception, where the cardinal directions are. This is a challenge for me. One afternoon I was walking with a friend, and asked him which direction was north — I was striving to come up with what I knew about the surroundings we were in to get hints on directions. Look, he said, there’s the sun.

By: fogus

fogus — Thu, 01 Nov 2012 14:50:17 +0000

9 has more interesting properties that fall into this category: http://blog.fogus.me/2010/10/30/09/

By: Josh Braun

Josh Braun — Thu, 01 Nov 2012 14:49:56 +0000

When I was a kid, I discovered a trick for determining whether fractions are reducible. If…

(denominator-numerator)/denominator

…is irreducible, then so is the original fraction, and if it is reducible, so is the original fraction. Occasionally, it’s easier to eyeball whether the expression above is reducible or not than dealing with what you’re initially given.

By: Dave Tate

Dave Tate — Thu, 01 Nov 2012 13:51:54 +0000

Your example of a probability argument rang a bell. One of my graduate professors told the story of stumping colleagues in other branches of math with a challenge to prove a somewhat hairy-looking equation involving a complicated sum on one side and an annoying integral on the other. In the context of a probability class, the proof was by inspection: the left hand side was the probability that the number of events prior to time T in a poisson process with rate L is at least N, and the right-hand side was the probability that an Erlang(N,L) random variable takes a value less than or equal to T.

By: Patrick Honner

Patrick Honner — Thu, 01 Nov 2012 13:04:00 +0000

I wrote about a similar experience I had computing an integral: in this case, applying integration by parts made the problem look more complicated, but actually made it easier to solve. I suspect this sort of thing happens a lot in computing integrals (and other cook-bookery type math) where no one algorithm works.

By: Evpok

Evpok — Thu, 01 Nov 2012 12:41:15 +0000

The most obvious is in my opinion the summation of the first n terms of an arithmetic sequence

E.g. S = 1 + 2 + … + 2713
2×S = (1 + 2713) + (2 + 2712) + … + (2713+1) = 2713×(2713+1)
So S = 2713×(2713+1)/2 = 3681541
So there is no need to waste one’s time with 2712 additions.