madhadron

Mathematical miscellany

Last updated: May 13, 2020

This collection is pure fun…well, if you’re trained as physicist or mathematician. Otherwise it’s probably going to be pretty inscrutable. These are little treasures and odd and ends, often dead ends, that I have accumulated over the years. Many were useful at the time I did the work, but now they’re just keepsakes from earlier fields I have worked in.

Physics

The apothecary’s balance
I spend too much time thinking instead of sleeping, but I run across such gems. For example, I recently lost sleep when I realized that my naive idea of how an apothecary’s balance worked was wrong.
Conservation of energy
Last night it occurred to me that, although I knew Landau’s derivation of conservation of energy in Lagrangian mechanics, I had never seen it done from Newton’s third law. A few minutes of scribbling produced a proof; a few more produced ththis exposition that might be of use to some poor physics student somewhere.
Material derivative
Section 2 of Landau and Lifshitz’s Fluid Mechanics starts by writing down equations of motion for a fluid particle in coordinates attached to that particle, then transforms to coordinates fixed in space. The argument is physically clear, but mathematically opaque. This was bothering me, so I cleaned it up.

Statistics & Computation

Asymptotically heavier tailed distributed
One of Casella’s papers contains a decision theory paradox in interval estimation. I became suspicious of interval estimation a while back, triggered by Savage’s work, but Casella’s paper uses a cute analytical tool that I haven’t seen elsewhere, and which they pass over without comment.
Calculating the golden ratio
My wife was watching the Khan academy video on the golden ratio, in which they used the quadratic equation to calculate the value of the constant. Most of us have access to a fast square root function today to make that calculation easy, and calculating a constant like the golden ratio only has to happen once so having a very efficient algorithm to calculate it is unnecessary. But I picked up the taste for designing little numerical algorithms from Acton’s superb book Real Computing Made Real, and while she finished watching the video happily produced a couple.
Calculating pagination bounds for display
How do you calculate the limits for a pagination widget like “« 4 5 6 7 »”?
Determining affine transforms from three points
One of my labmates needed to visit a grid of points on a microscopy stage, something the software provided with the microscope wouldn’t handle. I wrote a program to generate grids of points for her. The heart of it is this calculation that three distinct points in the plane determine an affine transformation.

Pure math

Drying categories
Category theory is rife with commutative diagrams. I asked myself how to make them look like simple things such as commutative (a\\cdot b = b\\cdot a) or associative (a \\cdot (b \\cdot c) = (a \\cdot b) \\cdot c) laws; about which John Baez commented, “Alas, 1-dimensional representations of commutative diagrams are sort of like dried roses…” So here is my recipe for drying commutative diagrams.
The unit circle deformed
Other measures than the usual Euclidean definition of the unit circle are possible. And it turns out that if you massage them, you can get an estimator for the number of non-zero components in a vector.

Biology

Lockless-Ranganathan formalism
Are all monomers of a polypeptide chain equally important? Are some merely incidental—does something needs to be there, but it doesn’t matter what? Lockless and Ranganathan developed a way of answering this in the context of thermodynamic mutant cycles, and found that only sparse pairs of amino acids appear to be important.
A clean calculation Luria-Delbruck mathematics
All biology students are made to read Luria and Delbruck’s paper “Mutations of bacteria from virus sensitivity to virus resistance,” which begins with a series of ad hoc probability calculations. When the paper was written in 1943, they represented the best tools available in probability. Today they are quaint to the point of being incomprehensible. Here is the whole calculation from the paper in modern form.
Intrinsic and extrinsic noise in gene expression
The question of noise in gene expression has been more and more important recently, both as an interesting question in its own right, and as a necessary piece of information for constructing random processes to describe other aspects of biology. There is a problem with measuring noise, however: how much of it is actually noise associated with the components of a pathway operating in a thermal bath, and how much is cell to cell variation, the local density of a necessary enzymes, and other things that are extraneous to the pathway?