First was a great talk by Jordan Ellenberg on the importance of uncertainty and contradiction in math and why mathematics is actually part of the humanities at the Santa Fe Institute https://www.youtube.com/watch?v=L1pXQNaS9Oo (2/11) #math#statistics
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.
i just took out my ear lobe studs for the first time and i NEED to show off my klein bottles. Also I'm accepting suggestions for fun #math#earrings !! #MathCoffeeSelfie
Approaching the annual "running out of beekeeping equipment" date. 😬
(Bees follow the rule of exponential growth and decay: you either have WAY TOO MANY BEES or NOT NEARLY ENOUGH BEES -- in spring they grow by the power of two or more, but they also tend to drop in half in winter). #beekeeping
I genuinely think that a fanatical emphasis on real world uses has done more to harm mathematical literacy than any other academic policy.
Kids love weird shit, and teachers should 100% take advantage of that and roll into class bein' like "Ok nerds, here's the math that proves you're a donut"
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.
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.
🐍 aprxc — A #Python#CLI tool to approximate the number of distinct values in a file/iterable using the Chakraborty/Vinodchandran/Meel’s (‘coin flip’) #algorithm¹.
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].
College Precalculus – Full Course with Python Code by Ed Pratowski and freeCodeCamp focus on the foundation of calculus with Python implementation. This 12 hours course covers the following topics:
✅ Core trigonometry
✅ Matrix operation
✅ Working with complex numbers
✅ Probability
watching #QI. An infinity of mathematicians go into a bar and place an order. "I'll have a pint, a half pint, a quarter pint, an eighth, an ounce.." The barman says "I'll stop you there," and pours out two pints. "The problem with you mathematicians is you don't know your limits" #math#joke
Can someone help me with this #Geometry volume problem?
I'm trying to calculate the volume of water in my storage pond. It is a section of a cone with these dimensions: the bottom is roughly circular and 40 feet diameter; the top is roughly circular and 65 feet diameter. It is 16 feet deep when full.
I probably could've done this in high school geometry class, but that was…er...a very long time ago. #math#FediHelp
First authorb paper out in the wild. It's challenging as an independent researcher, but it can be done. This has been a long time coming. Maybe more in the future https://zenodo.org/records/11214976 #paper#math#matheducation#proofs
Should show up in a couple other locations as well hopefully (pending reviews)
The Ultimate IB Math Coursework Handbook with Examples (snokido.uk)