@dmm@mathstodon.xyz avatar

dmm

@dmm@mathstodon.xyz

Retired husband/father/grandfather living in the US. Interests include #science, #math, #evolution, #machinelearning, #physics, #finance, #markets, #climatechange, #biology, #surfing, #music, and our #oceans.

B.Sc. in Biology, M.Sc. in Computer Science.

Former Director, Advanced Network Technology Center at the University of Oregon.

Former Chief Scientist, VP and Fellow at Brocade Communications Systems.

Former Senior Scientist at Sprint.

Former Distinguished Engineer at Cisco Systems.

This profile is from a federated server and may be incomplete. Browse more on the original instance.

dmm, to science
@dmm@mathstodon.xyz avatar

Évariste Galois was a French mathematical prodigy, political activist, and all-around trouble maker who, as a teenager, solved a 350-year-old problem regarding the condition necessary for a polynomial to be solvable by radicals. His work laid a foundation for a major branch of abstract algebra which today we call Galois theory.

in 1832 Galois wrote to a friend accurately predicting that he would die in a duel taking place the next morning (legend has it that the duel was over a woman). In his final letter, Galois describes his work in mathematics and closed with, "Eventually there will be, I hope, some people who will find it profitable to decipher this mess."

Galois was only 20 years old at the time of his death.

References

[1] "Évariste Galois", https://mathshistory.st-andrews.ac.uk/Biographies/Galois

dmm,
@dmm@mathstodon.xyz avatar

@johncarlosbaez "is the whole letter translated into English?"

I'm not an expert on these translations but I just looked and found this: https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

Boolean algebras are a standard way to formalize the concepts of 'and', 'or', 'not', 'true' and 'false' in classical logic. If you aren't careful, the list of axioms is pretty long! Each individual axiom makes perfect sense - but it's not obvious when you've got enough.

Things get simpler if you already know the concept of a 'ring' - roughly, a gadget with +, ×, 0 and 1 obeying a bunch of familiar axioms. Then you can define a 'boolean ring' to be a ring where every element x has x² = x. Boolean algebras turn out to be equivalent to boolean rings!

There's a trick, though. The × in your boolean ring is 'and', but the + in your boolean ring isn't 'or'. It's exclusive or! You can define this in any boolean algebra. In words:

x exclusive or y = (x or y) and not(x and y)

In symbols:

x + y = (x ∨ y) ∧ ¬(x ∧ y)

To go back from boolean rings to boolean algebras, we define

x ∨ y = x + y + xy

An interesting thing about boolean rings is that they automatically obey x + x = 0 and xy = yx. If you like fiddling around with equations - and honestly, who doesn't? - you can have fun trying to prove these from the ring axioms and x² = x. If you give up, you can see the answer here:

https://ncatlab.org/nlab/show/Boolean+algebra#boolean_rings

But why am I talking about this today?

(1/2)

dmm,
@dmm@mathstodon.xyz avatar

@johncarlosbaez Hey, I was just looking at your graphic and it looks like you have "Annihilator for ∨: x ∨ 1 = 1" twice...

dmm, to science
@dmm@mathstodon.xyz avatar

Here's a cool integral involving the golden ratio.

My notes are here: https://davidmeyer.github.io/qc/golden_ratio.pdf. The LaTeX source is here: https://www.overleaf.com/read/mkjdjwtmnzjd.

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

dmm,
@dmm@mathstodon.xyz avatar

@deilann I thought that was what I was doing, but I'll look around to see what I can learn.

dmm,
@dmm@mathstodon.xyz avatar

@deilann Humm, I though you could read it through the URL I provided. Does that mean you can't?

dmm,
@dmm@mathstodon.xyz avatar

@deilann So can you read it now? Thx, --dmm

dmm,
@dmm@mathstodon.xyz avatar

@deilann Humm, I haven't even looked at it...

dmm,
@dmm@mathstodon.xyz avatar

@deilann Hey, attention is all you need 🙂

https://arxiv.org/abs/1706.03762

dmm,
@dmm@mathstodon.xyz avatar

@paulmasson @Marekgluza @deilann I'm sure everyone has had enough of this particular problem, but for completeness....

dmm,
@dmm@mathstodon.xyz avatar

@Marekgluza I saw the identity somewhere and I wanted to understand why it held.

Regarding geometric interpretations, not that I know of (although I did use a trig substitution in my "proof").

dmm,
@dmm@mathstodon.xyz avatar

@deilann @paulmasson @Marekgluza I added this acknowledgement:

@deilann notes that c2 − c − 1 is (or at least should be :-)) immediately identifiable as the golden ratio’s quadratic form and φ’s minimal polynomial which has φ and the negative inverse of φ as roots and so is ”not sure solving it in full is truly necessary once you’ve gotten there”.

dmm,
@dmm@mathstodon.xyz avatar

@deilann @paulmasson @Marekgluza Probably not, but in the future when I've forgotten everything I've learned here I'm sure the extra detail will come in handy 🙂.

All of my notes have this flavor (to remind me of all I have forgotten).

Thank for the comment though; I'll make a note of that.

  • All
  • Subscribed
  • Moderated
  • Favorites
  • •
  • megavids
  • khanakhh
  • mdbf
  • ethstaker
  • magazineikmin
  • GTA5RPClips
  • rosin
  • thenastyranch
  • Youngstown
  • InstantRegret
  • slotface
  • osvaldo12
  • kavyap
  • DreamBathrooms
  • JUstTest
  • Durango
  • everett
  • cisconetworking
  • normalnudes
  • tester
  • ngwrru68w68
  • cubers
  • modclub
  • tacticalgear
  • provamag3
  • Leos
  • anitta
  • lostlight
  • All magazines
    •