URL : | http://www.johndcook.com/blog/ | |
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Filed Under: | Academics | |

Posts on Regator: | 1359 | |

Posts / Week: | 4.8 | |

Archived Since: | April 26, 2011 |

We’re about to see a lot of new, powerful, inexpensive medical devices come out. And to my surprise, I’ve contributed to a few of them. Growing compute power and shrinking sensors open up possibilities we’re only beginning to explore. Even when the things we want to observe elude direct measurement, we may be able to infer them from […]

For an article to be published, it has to be published somewhere. Each journal has a responsibility to select articles relevant to its readership. Articles that make new connections might be unpublishable because they don’t fit into a category. For example, I’ve seen papers rejected by theoretical journals for being too applied, and the same papers […]

One of my lightbulb moments in college was when my professor, Jim Vick, explained the Lagrange multiplier theorem. The way I’d seen it stated in a calculus text gave me no feel for why it should be true, but his explanation made sense immediately. Suppose f(x) is a function of several variables, i.e. x is a vector, and g(x) = c […]

Suppose you did a pilot study with 10 subjects and found a treatment was effective in 7 out of the 10 subjects. With no more information than this, what would you estimate the probability to be that the treatment is effective in the next subject? Easy: 0.7. Now what would you estimate the probability to be […]

I’m starting a new Twitter account for logic and formal methods: @FormalFact. Expect to see tweets about constructive logic, type theory, formal proofs, proof assistants, etc. The image for the account is a bowtie, a pun on formality. It’s also the symbol for natural join in relational algebra.

The number 3435 has the following curious property: 3435 = 33 + 44 + 33 + 55. It is called a Münchausen number, an allusion to fictional Baron Münchausen. When each digit is raised to its own power and summed, you get the original number back. The only other Münchausen number is 1. At least in […]

Beta reduction is essentially function application. If you have a function described by what it does to x and apply it to an argument t, you rewrite the xs as ts. The formal definition of ?-reduction is more complicated than this in order to account for free versus bound variables, but this informal description is sufficient […]

I was catching up on Engines of our Ingenuity episodes this evening when the following line jumped out at me: If I flip a coin a million times, I’m virtually certain to get 50 percent heads and 50 percent tails. Depending on how you understand that line, it’s either imprecise or false. The more times you […]

Experience with the normal distribution makes people think all distributions have (useful) sufficient statistics [1]. If you have data from a normal distribution, then the sufficient statistics are the sample mean and sample variance. Show More Summary

The other day I found myself saying that I preferred org-mode files to Jupyter notebooks because with org-mode, what you see is what you get. Then I realized I was using “what you see is what you get” (WYSISYG) in exactly the opposite of the usual sense. Jupyter notebooks are WYSIWYG in the same sense […]

Most programmers are at one extreme or another when it comes to floating point arithmetic. Some are blissfully ignorant that anything can go wrong, while others believe that danger lurks around every corner when using floating point....Show More Summary

Here’s an interesting quote omparing writing proofs and writing programs: Building proofs and programs are very similar activities, but there is one important difference: when looking for a proof it is often enough to find one, however complex it is. On the other hand, not all programs satisfying a specification are alike: even if the […]

Statistics can be useful, even if it’s idealizations fall apart on close inspection. For example, take English letter frequencies. These frequencies are fairly well known. E is the most common letter, followed by T, then A, etc. The string of letters “ETAOIN SHRDLU” comes from the days of Linotype when letters were arranged in that order, […]

The applied math featured here tends to be fairly sophisticated, but there’s a lot you can do with the basics as we’ll see in the following interview with Trevor Dawson of Borrego Solar, a company specializing in grid-connected solar PV systems. JC: Can you say a little about yourself? TD: I’m Trevor Dawson, I’m 25, born in the […]

When people ask me what calculus is, my usual answer is “the mathematics of change,” studying things that change continually. Algebra is essentially static, studying things frozen in time and space. Calculus studies things that move, shapes that bend, etc. Algebra deals with things that are exact and consequently can be fragile. Calculus deals with […]

As systems get larger and more complex, we need new tools to test whether these systems are correctly specified and implemented. These tools may not be new per se, but they may be applied with new urgency. Dimensional analysis is a well-established method of error detection. Simply checking that you’re not doing something like adding […]

This evening I ran across an unexpected reference to spherical trigonometry: Thomas Hales’ lecture on lessons learned from the formal proof of the Kepler conjecture. He mentions at one point a lemma that was awkward to prove in its original form, but that became trivial when he looked at its spherical dual. The sides of […]

The set of primitive recursive (PR) functions is the smallest set of functions of several integer arguments satisfying five axioms: Constant functions are PR. The function that picks the ith element of a list of n arguments is PR. The successor function S(n) = n+1 is PR. PR functions are closed under composition. PR functions are closed under primitive […]

There’s no notion of continuity in linear algebra per se. It’s not part of the definition of a vector space. But a finite dimensional vector space over the reals is isomorphic to a Euclidean space of the same dimension, and so we usually think of such spaces as Euclidean. (We’ll only going to consider real vector spaces […]

Let S be the area of triangle T in three-dimensional space. Let A, B, and C be area of the projections of T to the xy, yz, and xz planes respectively. Then S2 = A2 + B2 + C2. There’s an elegant proof of this theorem here using differential forms. Below I’ll sketch a less elegant but more elementary proof. You could prove the identity above by using the fact that the […]

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