dmm

dmm, 20 days 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, 22 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, 25 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, 28 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

johncarlosbaez, 1 month ago (edited 1 month 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, 1 month 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

johncarlosbaez, 2 months ago (edited 2 months 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.

dmm, 2 months ago to math

"So here is the crux of my argument. If you believe in an external reality independent of humans, then you must also believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure. In other words, we all live in a gigantic mathematical object — one that is more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names like Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced theories. Everything in our world is purely mathematical — including you." -- Max Tegmark, "The Mathematical Universe", https://arxiv.org/abs/0704.0646.

Something to think about...

See also "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf.

#math #maths #physics #erf #muh

dmm, 2 months ago to math

Here's an interesting series:[S=\sum\limits_{n=1}^{\infty} {\left (\frac{a}{b}\right)}^{n}
]Does it converge, and if so, to what?

A few of my notes on all of this are here:
https://davidmeyer.github.io/qc/infinite_sum_a_over_b.pdf, and as always, questions/comments/corrections/* greatly appreciated.

#convergentseries #math #maths

dmm, 2 months ago to math

Born #onthisday in 1835, Josef Stefan was an ethnic Carinthian Slovene physicist, mathematician, and poet of the Austrian Empire [1].

During his lifetime Stefan published nearly 80 scientific articles, most appearing in the Bulletins of the Vienna Academy of Sciences.

Stefan is perhaps best known for his study of blackbody radiation [2] and for discovering what we now call Stefan's law, a physical power law which states that the total radiation from a blackbody is proportional to the fourth power of its (thermodynamic) temperature. Stefan's law was later extended to grey bodies by one of Stefan's students, Ludwig Boltzmann [3], and is now known as the Stefan–Boltzmann law [4].

I wrote a bit about blackbody radiation and the famous Stefan–Boltzmann law here: https://davidmeyer.github.io/qc/oscillators.pdf, but it looks like I got distracted (again) and never finished. The LaTeX source is here: https://www.overleaf.com/read/xjmyvksvtztb. In any event, as always questions/comments/corrections/* greatly appreciated.

References

[1] "Josef Stefan", https://en.wikipedia.org/wiki/Josef_Stefan

[2] "Josef Stefan’s – Black Bodies and Thermodynamic Temperature", http://scihi.org/josef-stefans-thermodynamics/

[3] "Ludwig Boltzmann", https://mathshistory.st-andrews.ac.uk/Biographies/Boltzmann/

[4] "Stefan–Boltzmann law", https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law

#josefstefan #stefanboltzmannlaw #blackbodyradiation #math #maths #physics

dmm, 2 months ago to random

@christianp Is there a bug in 4.2.7 in which the Notifications bell turns on with a whole bunch of notifications, many of which are old, and can't be cleared?

In any event I've been observing that. Please let me know if there is additional information I can provide.

--dmm

johncarlosbaez, 3 months ago (edited 3 months ago) to random

There's a bunch of bigshots on YouTube who pontificate about string theory, the mysteries of quantum mechanics, and other profound issues in physics. But you can't really learn much physics from most of them. It's just chat.

Angela Collier here is so much better! So much more humble - and so much more fun if you really care about physics. I actually learned something: how to estimate the distance of a pulsar!

When pulses of radio waves from a pulsar move through space, they get smeared out as they go, and you can use this to guess how far away the pulsar is. Why? Because waves of lower frequency move a bit slower. Why? Because they interact more with the ionized gas in the Milky Way.

But how much slower, and why? That's what she explains - and actually this part, how radio waves interact with ionized gas, is what will stick with me.

This is the first episode of a series she calls Coffee and The Problem:

"We have coffee and I solve a problem, and the idea is that it's like a cozy weekend morning and you pull out your notebook and you solve the problem right along with me. I will give you time to pause and solve it yourself if you want and compare your answer with mine if you want. That's the game! That's the fun."

This time she's solving a problem about estimating the distance of pulsar. The problem just hands you a formula. But she's good. She doesn't just use the formula, she shows how to derive it from more fundamental principles! And also, at the end, she raises the question I was worrying about all along: how reliable is this method in practice? So she's not blindly solving a problem: she's thinking about physics.

https://www.youtube.com/watch?v=iox8Z-NGGS8