For a long time I felt like I didn't really understand the Yoneda Lemma. I knew some things that people said about it ('we can understand objects by the maps into them' and 'the Yoneda embedding is full and faithful') but the statement 'Hom(Hom(A, -), F) = F(A)' itself was something I could only use as a symbolic manipulation without understanding.
On the other hand, I did separately know facts like 'In the category of quivers there are objects which look like • and •→•, such that the maps out of them tell you exactly the vertices and edges in your quiver' and 'In the category of simplicial sets there are objects which are just an n-simplex; maps out of them are the n-simplices of the object you are mapping into'.
Somehow I only recently realised that these examples are precisely the Yoneda Lemma. These objects are precisely presheaves of the form Hom(A, -), and the Yoneda Lemma tells you what you get when you map out of them.
In particular I think it would be useful to give the quiver example to students when they learn the Yoneda Lemma.
@OscarCunningham do students learn the two things in the right order, though?
I moved from maths to computing before I got to either category theory or quivers, but I did see a summary of an introductory lecture course on quivers, and it listed category theory as a prerequisite.
@rzeta0 cool! I've heard of Lyx but never had a compelling need to write beautiful equations / math notation so...
I admire your journey as an independent math lover/student you know? I'm always shyly looking at the abiss next of the perimeter of math I don't understand.
Here I tried to prove the Existence Theorem for Laplace Transforms. I don't know what the/a "conventional proof" looks like, but this is what I came up with.
"I can't draw" is almost (but not quite) as common a misconception as "I can't do #maths". Prove yourself wrong for one of these at Olivia's free drop-in #drawing workshop at @NPGLondon on 21st June:
A couple of weeks ago, I posted an #animation of a point on a circle generating a #cycloid.
If you turn the curve "upside down", you get the #BrachistochroneCurve. This curve provides the shortest travel time starting from one cusp to any other point on the curve for a ball rolling under uniform #gravity. It is always faster than the straight-line travel time.
Anyway, the #animation took a bit of thought as it requires a bit of #Mechanics, some #Integration and is made a bit more tricky as the curve is multi-valued and so you need to treat different branches separately. The #AnimatedGif was produce with #WxMaxima.
Here's something I just learned: the lucky numbers of Euler.
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k² − k + n produces a prime number.
Leonhard Euler published the polynomial k² − k + 41 which produces prime numbers for all integer values of k from 1 to 40.
Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).
The Heegner numbers 7, 11, 19, 43, 67, 163, yield prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.
h/t John Carlos Baez
(@johncarlosbaez) for pointing this out.
The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].
Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].
@sjb Ah... Klein-Nishina, the QFT formula related to Compton scattering - I was getting a Compton scattering prac up and running in a 3rd Year Uni teaching lab and found a mistake in relation to calculations using Klein-Nishina for this prac in the classic Melissinos "Experiments in Modern Physics" (if I remember correctly the end result is OK... possibly two mistakes cancelling out... but it was a while back). I emailed the publisher but got no response - oh well.
Early on in my hobby I came to the realization that cryptographic prowess has no viable market price point. More's the pity. Yet I think one day I may change that with my secrecy sauce.