Category: Math

Quantum leaps

A literal quantum leap is a discrete change, typically extremely small [1]. A metaphorical quantum leap is a sudden, large change. I can’t think of a good metaphor for a small but discrete change. I was reaching for such a metaphor recently and my first thought was “quantum leap,” though that would imply something much […]

Easter and exponential sums

For the last couple years, the exponential sum of the day for Easter Sunday has been a cross. This was not planned, since the image each day is determined by the numbers that make up the date, as explained here. This was the exponential sum for last Easter last year, April 1, 2018: and this […]

Groups in categories

The first time I saw a reference to a “group in a category” I misread it as something in the category of groups. But that’s not what it means. Due to an unfortunately choice of terminology, “in” is more subtle than just membership in a class. This is related to another potentially misleading term, algebraic […]

What is an isogeny?

The previous post said that isogenies between elliptic curves are the basis for a quantum-resistant encryption method, but we didn’t say what an isogeny is. It’s difficult to look up what an isogeny is. You’ll find several definitions, and they seem incomplete or incompatible. If you go to Wikipedia, you’ll read “an isogeny is a […]

Isogeny-based encryption

If and when large quantum computers become practical, all currently widely deployed method for public key cryptography will break. Even the most optimistic proponents of quantum computing believe such computers are years away, maybe decades. But it also takes years, maybe decades, to develop, test, and deploy new encryption methods, and so researchers are working […]

Random projection

Last night after dinner, the conversation turned to high-dimensional geometry. (I realize how odd that sentence sounds; I was with some unusual company.) Someone brought up the fact that two randomly chosen vectors in a high-dimensional space are very likely to be nearly orthogonal. This is a surprising but well known fact. Next the conversation […]

Squircle perimeter and the isoparametric problem

If you have a fixed length of rope and you want to enclose the most area inside the rope, make it into a circle. This is the solution to the so-called isoparametric problem. Dido’s problem is similar. If one side of your bounded area is given by a straight line, make your rope into a […]

Taking the derivative of a muscle car

I’ve been getting a lot of spam lately saying my web site does not rank well on “certain keywords.” This is of course true: no web site ranks well for every keyword. I was joking about this on Twitter, saying that my site does not rank well for women’s shoes, muscle cars, or snails because […]

Ratio of Lebesgue norm ball volumes

As dimension increases, the ratio of volume between a unit ball and a unit cube goes to zero. Said another way, if you have a high-dimensional ball inside a high-dimensional box, nearly all the volume is in the corners. This is a surprising result when you first see it, but it’s well known among people […]

Higher dimensional squircles

The previous post looked at what exponent makes the area of a squircle midway between the area of a square and circle of the same radius. We could ask the analogous question in three dimensions, or in any dimension. (Is a shape between a cube and a sphere a cuere?) In more conventional mathematical terminology, […]

History of the term “Squircle”

Architect Peter Panholzer coined the term “squircle” in the summer of 1966 while working for Gerald Robinson. Robinson had seen a Scientific American article on the superellipse shape popularized by Piet Hein and suggested Panholzer use the shape in a project. Piet Hein used the term superellipse for a compromise between an ellipse and a […]

Putting topological data analysis in context

I got a review copy of The Mathematics of Data recently. Five of the six chapters are relatively conventional, a mixture of topics in numerical linear algebra, optimization, and probability. The final chapter, written by Robert Ghrist, is entitled Homological Algebra and Data. Those who grew up with Sesame Street may recall the song “Which […]

Elementary solutions to differential equations

Differential equations rarely have closed-form solutions. Some do, and these are emphasized in textbooks. For this post we want to look specifically at homogeneous second order linear equations: y ” + a(x) y‘ + b(x) y = 0. If the coefficient functions a and b are constant, then the solution can be written down in terms […]

Finite rings

It occurred to me recently that I rarely hear about finite rings. I did a Google Ngram search to make sure this isn’t just my experience. Source Why are finite groups and finite fields common while finite rings are not? Finite groups have relatively weak algebraic structure, and demonstrate a lot of variety. Finite fields […]

Monads and generalized elements

Paolo Perrone gives a nice, succinct motivation for monads in the introduction to his article on probability and monads. … a monad is like a consistent way of extending spaces to include generalized elements of a specific kind. He develops this idea briefly, and links to his dissertation where he gives a longer exposition (pages […]

Mixing error-correcting codes and cryptography

Secret codes and error-correcting codes have nothing to do with each other. Except when they do! Error-correcting codes Error correcting code make digital communication possible. Without some way to detect and correct errors, the corruption of a single bit could wreak havoc. A simple example of an error-detection code is check sums. A more sophisticated […]

US Army applying new areas of math

Many times on this blog I’ve argued that the difference between pure and applied math is motivation. As my graduate advisor used to say, “Applied mathematics is not a subject classification. It’s an attitude.” Traditionally there was general agreement regarding what is pure math and what is applied. Number theory and topology, for example, are […]

Riffing on mistakes

I mentioned on Twitter yesterday that one way to relieve the boredom of grading math papers is to explore mistakes. If a statement is wrong, what would it take to make it right? Is it approximately correct? Is there some different context where it is correct? Several people said they’d like to see examples, so […]

A genius can admit finding things difficult

Karen Uhlenbeck has just received the Abel Prize. Many say that the Fields Medal is the analog of the Nobel Prize for mathematics, but others say that the Abel Prize is a better analog. The Abel prize is a recognition of achievement over a career whereas the Fields Medal is only awarded for work done […]

Thermocouple polynomials and other sundries

I was looking up something on the NIST (National Institute of Standards and Technology) web site the other day and ran across thermocouple polynomials. I wondered what that could be, assuming “thermocouple” was a metaphor for some algebraic property. No, it refers to physical thermocouples. The polynomials are functions for computing voltage as a function […]