Blog Profile / The Endeavour

Filed Under:Academics
Posts on Regator:1617
Posts / Week:4.5
Archived Since:April 26, 2011

Blog Post Archive

Painless project management

This evening, after I got off a phone call discussing a project with a colleague, I thought “Huh, I guess you could call that project management.” I worked as a project manager earlier in my career, but what I’m doing now feels completely different and much more pleasant. Strip away the bureaucracy and politics, and […]

New animation feature for exponential sums

There’s now an “Animate” link on the exponential sum pages that lets you watch the curves being drawn. Sometimes these are surprising. The plot of the partial sums might bounce all over in the process of filling in a very symmetric plot. Here’s an example of that.  

Quantifying normal approximation accuracy

Probability is full of theorems that say that probability density approximates another as some parameter becomes large. All the dashed lines in the diagram below indicate a relationship like this.   You can find details of what everything in the diagram means here. How can you quantify these approximations? One way is to use Kullback-Leibler […]

Ordinary Potential Polynomials

I’ve run into potential polynomials a lot lately. Most sources I’ve seen are either unclear on how they are defined or use a notation that doesn’t sit well in my brain. Also, they’ll either give an implicit or explicit definition but not both. Both formulations are important. The implicit formulation suggests how potential polynomials arise […]

Bell polynomials: partial, ordinary, and exponential

Bell polynomials come up naturally in a variety of contexts: combinatorics, analysis, statistics, etc. Unfortunately, the variations on Bell polynomials are confusingly named. Analogy with differential equations There are Bell polynomials of one variable and Bell polynomials of several variables. Show More Summary

Ways to connect

If you visit this blog once in a while, here are a few ways to hear from me more regularly. Subscription You can subscribe to the blog via RSS or email. I often use SVG images because they look great on a variety of devices, but most email clients won’t display that format. If you […]

Apple design, squircles, and curvature

A “squircle” is a sort of compromise between a square and circle, but one that differs from a square with rounded corners. It’s a shape you’ll see, for example, in some of Apple’s designs. A straight line has zero curvature, and a circle with radius r has curvature 1/r. So in a rounded square the […]

Liouville-Green approximation

Suppose you have a differential equation of the form If the function f(x) is constant, the differential equation is easy to solve in closed form. But when it is not constant, it is very unlikely that closed form solutions exist. But there may be useful closed form approximate solutions. There is no linear term in […]

Iterating between theory and code

Yesterday I said on Twitter “Time to see whether practice agrees with theory, moving from LaTeX to Python. Wish me luck.” I got a lot of responses to that because it describes the experience of a lot of people. Someone asked if I’d blog about this. The content is confidential, but I’ll talk about the […]

Eloquent mathematical writing

Sir Michael Atiyah recommends Hermann Weyl’s book The Classical Groups for its clarity and beautiful prose. From my interview with Atiyah: Hermann Weyl is my great model. He used to write beautiful literature. Reading it was a joy because he put a lot of thought into it. Hermann Weyl wrote a book called The Classical Groups, […]

Focus on the most important terms

Consider the following Taylor series for sin(?/7) and the following two functions based on the series, one takes only the first term def short_series(x): return 0.14285714x and a second that takes five terms (two of which have zero coefficients). def long_series(x): return 0.1425714x - 4.85908649e-04x3 + 4.9582515e-07x5 Which is more accurate? Let’s make a couple […]

Scaling and Monte Carlo integration methods

Here’s an apparent paradox. You’ll hear that Monte Carlo methods are independent of dimension, and that they scale poorly with dimension. How can both statements be true? The most obvious way to compute multiple integrals is to use product methods, analogous to the way you learn to compute multiple integrals by hand. Unfortunately the amount […]

Integrating polynomials over a sphere or ball

Spheres and balls are examples of common words that take on a technical meaning in math, as I wrote about here. Recall the the unit sphere in n dimensions is the set of points with distance 1 from the origin. The unit ball is the set of points of distance less than or equal to 1 from the […]

Bounding the 3rd moment by the 4th moment

For a random variable X, the kth moment of X is the expected value of Xk. For any random variable X with 0 mean, or negative mean, there’s an inequality that bounds the 3rd moment, m3 in terms of the 4th moment, m4: The following example shows that this bound is the best possible. Define u […]

Putting a brace under something in LaTeX

Here’s a useful LaTeX command that I learned about recently: \underbrace. It does what it sounds like it does. It puts a brace under its argument. I used this a few days ago in the post on the new prime record when I wanted to show that the record prime is written in hexadecimal as […]

Emacs features that use regular expressions

The syntax of regular expressions in Emacs is a little disappointing, but the ways you can use regular expressions in Emacs is impressive. I’ve written before about the syntax of Emacs regular expressions. It’s a pretty conservative subset of the features you may be used to from other environments as summarized in the diagram below. But […]

Easter eggs and yellow pigs

An Easter egg is a hidden feature, a kind of joke. The term was first used in video games but the idea is broader and older than that. For example, Alfred Hitchcock made a brief appearance in all his movies. And I recently heard that there’s a pineapple or reference to a pineapple in every […]

Big derivatives

Suppose you have a function of n variables f. The kth derivative of f is a kth order tensor [1] with nk components. Not all those tensor components are unique. Assuming our function f is smooth, the order in which partial derivatives are taken doesn’t matter. It only matters which variables you differentiate with respect […]

Are coffee and wine good for you or bad for you?

One study will say that coffee is good for you and then another will say it’s bad for you. Ditto with wine and many other things. So which is it: are these things good for you or bad for you? Probably neither. That is, these things that are endlessly studied with contradictory conclusions must not […]

Distribution of matches between two shuffled decks

Take two desks of cards and shuffle them. They can be standard 52-card decks, though the number of cards in the decks doesn’t matter as long as they’re the same and the decks are fairly large. Now count the number of times the two desks match, i.e. how many times the same card is in […]

Copyright © 2015 Regator, LLC