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, 4 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

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, 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

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, 1 month ago to physics

The accomplishments of the Victorian physicists were (and are) amazing.

Among the great Victorian era scientists, I've been studying the work of James Clerk Maxwell, specifically Maxwell's equations [1] (along with the history of Victorian mathematics and physics [2]). In his short life, Maxwell made important contributions in many areas of physics. Unfortunately Maxwell died at age 48 from abdominal cancer in November of 1879 [3].

Among Maxwell's contributions are Maxwell's equations, which completed the unification of electricity and magnetism, thereby forming the concepts of electromagnetism and the electro-magnetic force. One of the really amazing aspects of Maxwell's equations is their generality. In particular, they apply to all charge and current densities, whether static or time-dependent and together they completely describe the dynamical behavior of the electromagnetic field.

Here's the best I could do with unicode to describe the differential form of Maxwell's equations (there are also integral forms of Maxwell's equations, see below):

(i). ∇·E = ρ/ε0 # Gauss's Law

(ii). ∇·B = 0 # Gauss's law for magnetism

(iii). ∇ × E = ∂B/∂t # Maxwell–Faraday equation (Faraday's law of induction)

(iv). ∇ × B = μ0 (J + ε0 ∂E/∂t)

Ampère's circuit law (with Maxwell's addition)

Maxwell's equations are important not only because they unified electricity and magnetism and completely characterized the electromagnetic field, but also because they paved the way for special relativity and quantum mechanics.

#maxwell #physics #math #maths #victorianphysics #electromagnetism

(1/2)

Propagation of electromagnetic waves...

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

dmm, 1 month ago to random

Squirrels taking it easy in Eugene, Oregon...

[Image credit: Susie Meyer]

#squirrels

dmm, 1 month ago to space

M104 (aka Messier 104, NGC 4594, and the Sombrero Galaxy) is a fantastic spiral galaxy which is famous for its nearly edge-on profile featuring a broad ring of obscuring dust lanes. Seen here in silhouette against an extensive central bulge of stars, the swath of cosmic dust lends a broad brimmed hat-like appearance to the galaxy suggesting its more popular moniker, the Sombrero Galaxy.

This sharp view of the well-known galaxy was made from over 10 hours of Hubble Space Telescope image data, processed to bring out faint details often lost in the overwhelming glare of M104's bright central bulge. The Sombrero galaxy can be seen across the spectrum, and is host to a central supermassive black hole. About 50,000 light-years across and 28 million light-years away, M104 is one of the largest galaxies at the southern edge of the Virgo Galaxy Cluster. Still, the spiky foreground stars in this field of view lie well within our own Milky Way.

APOD: Messier 104 (2022 Apr 23)
Image Credit: NASA, ESA, Hubble Legacy Archive;
Processing & Copyright: Ignacio Diaz Bobillo
https://apod.nasa.gov/apod/ap220423.html

#m104 #sombrerogalaxy. #nasa #astronomy

dmm, 1 month ago to ChatGPT

"No, A → B is not equivalent to - B → - A in logic."

Except that the truth table that ChatGPT [1] generated says the opposite. Also, see the law of contraposition [2].

Claude [3] makes the same mistake.

I've had pretty good luck with the chatbots. This is the first thing that I have asked that all of them seem to get wrong.

Interesting.

References

[1] "ChatGPT", https://chat.openai.com

[2] "Contraposition", https://en.wikipedia.org/wiki/Contraposition

[3] "Claude", https://claude.ai

#chatgpt #claude3 #firstorderpredicatelogic #math #maths #logic

dmm, 1 month ago to math

When I made the figure below I used LaTeX, powerpoint and then LaTeX again. Having learned some TikZ I now think I could draw it using TikZ, but apparently I'm too lazy...

A few of my notes on the subject of this figure (and other stuff) are here: https://davidmeyer.github.io/qc/dual_beam_experiment.pdf. As always, questions/comments/corrections/* greatly appreciated.

#math #maths #physics #quantumphysics #texlatex #tikz

dmm, 1 month ago to physics

RIP Peter Higgs

https://www.bbc.com/news/science-environment-68774195?utm_source=press.coop

#peterhiggs #higgsboson #physics

dmm, 1 month ago to internet

Happy birthday RFC 1!

RFC 1 was published on #onthisday in 1969. Impressive work and insight by Steve and by the IETF community over the last 55 years/9K+ RFCs.

Well done!

#IETF #RFC #internet

johncarlosbaez, 1 month ago (edited 1 month ago) to random

Some wasps are called 'parasitoids' because they lay their eggs in still-living caterpillars. The eggs develop into larvae that eat the caterpillar from the inside.

But turnabout is fair play. Sometimes, other wasps called 'hyperparasitoids' lay their eggs in the larvae of these parasitoids!

The caterpillars also fight back. Their immune system detects the wasp's eggs, and they will do things like surround the eggs in a layer of tissue that chokes them.

But many parasitoid wasps have a trick to stop this. They deploy viruses that infect the caterpillar and affect its behavior in various ways - for example, slowing its immune response to the implanted eggs.

These viruses can become so deeply symbiotic with the wasps that their genetic code becomes part of the wasp's DNA. So every wasp comes born with the ability to produce these viruses. They're called 'polydnaviruses'.

In fact some wasps are symbiotic with two kinds of virus. One kind, on its own, would quickly kill the caterpillar - not good for the wasp. The other kind keeps the first kind under control.

And I'm immensely simplifying things here. There are over 25,000 species of parasitoid wasps, so there's a huge variety of things that happen, which scientists are just starting to understand! I had fun reading this:

• Marcel Dicke, Antonino Cusumano and Erik H. Poelman, Microbial symbionts of parasitoids, Annual Review of Entomology, https://doi.org/10.1146/annurev-ento-011019-024939

Why such diversity? I think it's just that there are so many plants! So insect larvae like caterpillars naturally tend to feed on them... in turn providing a big food source for parasitoids, and so on.