dmm

dmm, 2 hours ago to math

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.

A few of my notes on this and related topics are here: https://davidmeyer.github.io/qc/dirac_delta.pdf

As always, questions/comments/corrections/* greatly appreciated.

#laplacetransform #existencetheorem #math #maths #mathematics

dmm, 2 days ago to math

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.

References

[1] "Lucky numbers of Euler", https://en.wikipedia.org/wiki/Lucky_numbers_of_Euler

[2] "Heegner number", https://en.wikipedia.org/wiki/Heegner_number

[3] "Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1)", https://oeis.org/A003173

[4] "Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1", https://oeis.org/A003173

#luckynumbersofeuler #heegnernumber #euler #math #maths #mathematics

dmm, 2 days ago to math

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].

A few of my notes on this and related topics are here: https://davidmeyer.github.io/qc/galois_theory.pdf. As always, questions/comments/corrections/* greatly appreciated.

References

[1] "Heegner Number", https://mathworld.wolfram.com/HeegnerNumber.html

[2] "Splitting Field", https://mathworld.wolfram.com/SplittingField.html

[3] "Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).", https://oeis.org/A003173

[4] "Ramanujan Constant", https://mathworld.wolfram.com/RamanujanConstant.html

#galois #gauss #heegnernumber #ramanujan #math #maths #mathematics

dmm, 5 days ago to random

Category theory friends: Is there a standard way to describe a functor?

I was using a two-case function to describe functor, where one case is what the functor does to objects and the other case is what the functor does to morphisms (see the image). However, I haven't been able to find a standard form in any of the literature I've been reading...

Thx, --dmm

#categorytheory #functors

dmm, 8 days ago to random

On May 17, 1902, Valerios Stais discovered the Antikythera Mechanism in a wooden box in the Antikythera shipwreck on the Greek island of Antikythera. The Mechanism is the oldest known mechanical computer and can accurately calculate various astronomical quantities.

As Tony Freeth says, "It is a work of stunning genius" [1].

A few of my notes on the Mechanism are here: https://davidmeyer.github.io/astronomy/prices_metonic_gear_train.pdf. The LaTeX source is here: https://www.overleaf.com/read/ndpvkytkhmbv.

As always, questions/comments/corrections/* greatly appreciated.

#antikytheramechanism

References

"The Antikythera Mechanism: A Shocking Discovery from Ancient Greece", https://www.youtube.com/watch?v=xWVA6TeUKYU

dmm, 10 days ago to random

Later that very same morning...

#dow40K

johncarlosbaez, 11 days ago (edited 11 days ago) to random

There's a dot product and cross product of vectors in 3 dimensions. But there's also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There's nothing really like this in other dimensions.

We can get the dot and cross product in 3 dimensions by taking the imaginary quaternions and defining

v⋅w= -½(vw + wv), v×w = ½(vw - wv)

We can get the dot and cross product in 7 dimensions using the same formulas, but starting with the imaginary octonions.

The following stuff is pretty well-known: the group of linear transformations of ℝ³ preserving the dot and cross product is called the 3d rotation group, SO(3). We say SO(3) has an 'irreducible representation' on ℝ³ because there's no linear subspace of ℝ³ that's mapped to itself by every transformation in SO(3).

Much to my surprise, it seems that SO(3) also has an irreducible representation on ℝ⁷ where every transformation preserves the dot product and cross product in 7 dimensions!

It's not news that SO(3) has an irreducible representation on ℝ⁷. In physics we call ℝ³ the spin-1 representation of SO(3), or at least a real form thereof, while ℝ⁷ is called the spin-3 representation. It's also not news that the spin-3 representation of SO(3) on ℝ⁷ preserves the dot product. But I didn't know it also preserves the cross product on ℝ⁷, which is a much more exotic thing!

In fact I still don't know it for sure. But @pschwahn asked me a question that led me to guess it's true:

https://mathstodon.xyz/@pschwahn/112435119959135052

and I think I almost see a proof, which I outlined after a long conversation on other things.

The octonions keep surprising me.

https://en.wikipedia.org/wiki/Seven-dimensional_cross_product

dmm, 18 days ago to random

Epic #onthisday in blues/music history:

Robert Johnson was born on this day in 1911 in Hazlehurst, Mississippi. His landmark recordings in 1936 and 1937 display a combination of singing, guitar skills, and songwriting talent that has influenced later generations of musicians. Although his recording career spanned only seven months, he is recognized as a master of the blues, particularly the Delta blues style, and as one of the most influential musicians of the 20th century.

If you are not familiar with Johnson's music, there is a nice playlist here: https://www.youtube.com/playlist?list=PLYPx-lRv1uyB8Rrw1GNrluTznSqO5-FBN

The Wikipedia also has a nice piece on Johnson: https://en.wikipedia.org/wiki/Robert_Johnson.

And of course, there's this: https://www.youtube.com/watch?v=ycNtYoxNuW8.

[Image credit: https://en.wikipedia.org/wiki/Robert_Johnson#/media/File:Robert_Johnson.png]

#robertjohnson #blues #deltablues

dmm, 29 days ago to math

Just started writing up a few of my notes on introductory Category Theory. Not much here yet (it took me awhile to get Figure 1 to look right, and it's still not perfect).

In any event, the pdf, such as it is, is here: https://davidmeyer.github.io/qc/category_theory.pdf. The LaTeX source is here: https://www.overleaf.com/read/wnptmrwwfjgv#a36a79. As always, questions/comments/corrections/* greatly appreciated.

#categorytheory #math #maths