2025-04-01 05:20:52
Differential equations, especially nonlinear differential equations, rarely have a closed-form solution, but sometimes it happens. As I wrote about a year ago
It is unusual for a nonlinear PDE to have a closed-form solution, but it is not unheard of. There are numerous examples of nonlinear PDEs, equations with important physical applications, that have closed-form solutions.
This post will present some closed-form solutions of the minimal surface equation
(1 + |ux|²) uyy − 2 uxuyuxy + (1 + |uy|²) uxx = 0
One trivial class of closed-form solutions are planes.
u(x, y) = ax + by + c.
There are three non-trivial classes of solutions as far as I know. Jean Baptiste Marie Meusnier discovered two of these in 1776, namely the helicoild
u(x, y) = tan−1(y/x)
and the catenoid
u(x, y) = cosh−1(a (x² + y²)½) / a
Heinrich Scherk discovered another closed form solution in 1830:
u(x, y) = log( cos(ay) / cos(ax) ) / a
Here’s a plot.
The surface formed by the graph of the solution is known as Scherk’s surface. You could image that if the edges of this surface were made of wire and the wire was dipped in soapy wanter, it would form a bubble like Sherk’s surface.
Note that the closed-form solutions satisfy the minimal surface PDE itself, but do not satisfy any given boundary conditions, unless the boundary values you’d like to specify happen to be exactly the values this function has.
The post Closed-form minimal surface solutions first appeared on John D. Cook.2025-03-30 20:46:48
The “fundamental solution” to a PDE solves the equation with the right-hand side set to δ. Intuitively, you can think of the delta function as striking something with a hammer in order to see how it rings.
A novice might be OK with the explanation above.
A sophomore might rightly object that this doesn’t make sense. This delta “function” isn’t even a function. How can you set one side of a differential equation to something that isn’t even a function?
An expert would understand that calling δ a function is just a convenient figure of speech for a rigorous construction using distribution theory. You can find a high-level introduction here.
As with many of the bell curve memes, the horizontal axis is really experience rather than intelligence. “Whatever you say” could be an intelligent response to someone talking about things they understand but you don’t. And objecting that something doesn’t make sense (as stated) is an intelligent response when you’re exposed to a metaphor that you didn’t realize was a metaphor. A mature response is to appreciate the value of rigor and the value of metaphor.
The reason a fundamental solution is called “fundamental” is that once you have the fundamental solution, you can find more solutions by convolving the right-hand side with it.
So if L is a linear differential operator and F is a fundamental solution, i.e.
L F = δ
then the convolution f = F * h is a solution to
L f = h.
The fundamental solution to Poisson’s equation
∇² f = h
depends on dimension.
For dimension d > 2 the solution is proportional to rd−2 where r is the radial distance to the origin.
For dimension d = 2 the solution is proportional to log r.
This is an example of the phenomenon alluded to in the article titled A Zeroth Power Is Often a Logarithm Yearning to Be Free by Sanjoy Mahajan. If we naively stuck d = 2 into the fundamental solution rd−2 for higher dimensions we’d get r0 = 1, which doesn’t work. But we’d read Mahajan’s article, we might guess that log r and then verify that it works.
I give a couple more examples of “logarithms yearning to be free” in this post.
The post Fundamental solution first appeared on John D. Cook.
2025-03-30 19:53:09
The following quote stuck with me when I read it years ago. Looking back I appreciate it even more.
Now, when solving differential equations, or indeed solving any problem, it is permissible to use any methods at all, no matter how dubious, provided that once the solution has been found it can be proved to satisfy all the conditions of the problem.
You could make a bell curve meme out of this. A novice would say “Sure, if it works it works.” An expert would agree. But someone in between who has recently been introduced to rigorous mathematics would object. They might say, for example, “You can’t just treat dy/dx like a fraction!” even though they did a few weeks ago.
Mathematics is discovered inductively but taught deductively. This creates the false impression that math advances deductively. It does not. That is a rationalist fantasy. To use Iain McGilchrist’s metaphor from “The Master and His Emissary,” the intuitive master solves problems, then assigns his analytical emissary to check the work.
I’ve looked for the source of the quote above several times without success. I was convinced it was a footnote in Boyce and DiPrima, but could never find it there. I recently ran across the line when I was looking for something else. The quote come in fact it comes from Applied Functional Analysis by D. H. Griffel, 1985.
The post No matter how dubious first appeared on John D. Cook.2025-03-30 05:45:25
What to do first after scoping out and starting a data science project?
I’ve started an unsupervised learning project based on textual data. The first thing I like to do is actually look at the data. Is it noisy? What are the features—complex feature engineering needed? How heterogeneous? What generalization and overfitting challenges?
Analysis can take many forms: actually looking at the numbers, using visualization tools, Excel spreadsheet, Jupyter notebooks with Matplotlib, computing various statistics on the whole dataset or portions of it.
Some may believe this is not important. Just throw a barrage of classification or regression methods at the data, treat the data as a black box. Of course testing on a suite of ML methods is not a bad thing. But I can’t imagine not using every avenue available, including looking at the data. I’m certainly not alone in this view (see for example here, here and here).
I spent a few hours developing a simple custom data viewer for my problem that colored different parts of the textual data to give insight as to what was going on. I used ChatGPT to develop parts of this tool; some of it was incorrect and needed fixing, but having at least a draft of the code definitely saved time. Seeing the actual data in person was insightful and generated ideas for solving the problem.
While inspecting the data can help identify issues, it also risks biasing the modeling process by imposing assumptions that a flexible model might otherwise uncover on its own. One must also beware of data leakage. That being said—in general I think understanding as much as you can about the data is not a bad thing.
The post Looking at Your Data first appeared on John D. Cook.2025-03-27 22:11:23
An ellipse has equation
and a hyperbola has equation
Similarly the superellipse has equation
and the superhyperbola
When p = 2, the absolute value signs are unnecessary and the superellipse and superhyperbola reduce to the ellipse and hyperbola respectively.
Increasing p makes the superellipse more like a rectangle. But unlike a rectangle with rounded corners, the change in curvature is continuous.
Increasing p makes the superhyperbola more blunt at the vertices.
The superellipse is a fairly well known variation on an ellipse. Even if you’re not familiar the term, you’ve probably seen the shape. I give a couple examples here. The superhyperbola is the obvious analog of a superellipse, but the term is far less common. I’d never hear the term until yesterday.
It’s not clear why the superellipse would be common and the superhyperbola obscure, but here’s some speculation. First of all, the superellipse had an advocate, Piet Hein. If the superhyperbola has an advocate, he’s not a very effective advocate.
The name is also off-putting: juxtaposing super and hyper sounds silly. The etymology makes sense, even if it sounds funny. Piet Hein used the prefix super– to refer to increasing the exponent from the usual value of 2. Its unfortunate that hyperbola begins with a root that is similar to super.
2025-03-27 00:51:46
The glass disk game is played on a grid. You have translucent colored glass disks you can either place on an edge or a vertex.
There are two kinds of disks that can be placed on an edge: blue or yellow. A vertex with a blue and yellow disk looks green.
There are two kinds of disks that can be placed on a vertex: red or white. A vertex with a red and white disk looks pink.
For this post we only need red disks. We may use white and pink in a future post.
The following rules tell you how you are allowed to add disks.
If you have this configuration
then you may add a green disk in the middle.
Mnemonic: Think of copying the blue and yellow dots on the end and placing them both in the middle.
If you have this configuration
you may add a blue disk between the two blue disks.
If you have this configuration
you may add a yellow disk between the two yellow disks.
The rules above are all the rules of the game. You do not need to know any mathematics to play the game.
But the game does have a mathematical interpretation. The grid is a commutative diagram. A red disk means the horizontal diagram is exact at that vertex, i.e. the image of the function coming in from the left is the kernel of the function going out to the right.
A blue disk means a function is surjective (onto) and a yellow disk means a function is injective (one-to-one).
The diagram need not represent sets and functions. It could represent objects in a category along with morphisms. In that case blue disks represent epimorphisms and yellow disks monomorphisms.
Rule 1 is known as the five lemma. Rule 2 is called the four lemma.
The glass disk game is a pair of theorems in algebraic topology, or more generally homological algebra.
The post The glass disk game first appeared on John D. Cook.
