@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.

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dmm, to math
@dmm@mathstodon.xyz avatar

This is pretty amazing:

In 1801 the great German mathematician Carl Friedrich Gauss made the following conjecture [2,3]:

The ring of integers for a quadratic imaginary field
K = Q(√-d) is principal only for a finite number of d,
where d ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} .

Said another way, Gauss conjectured that the nine numbers in the sequence {1, 2, 3, 7, 11, 19, 43, 67, 163} are the only numbers who's negative square root can be adjoined to the integers to produce a ring with unique factorization [4].

So how did exactly did Gauss come up with the amazing conjecture that there were nine numbers (and just these nine numbers) that could be adjoined to ℤ to produce a unique factorization ring? The answer involves Gauss' work on the determinants of binary quadratic forms. There is quite a bit of commentary on this around the net; see [5] for example.

Gauss was unable to prove this conjecture; that would have to wait until 1952 when amateur mathematician Kurt Heegner proved the conjecture (up to a few minor flaws) [6]. Today these numbers are known as Heegner numbers [7].

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

References

[1] "Carl Friedrich Gauss", https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

[2] "Disquisitiones Arithmeticae", https://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae

[3] "A conjecture which implies the theorem of Gauss-Heegner-Stark-Baker", https://www.jstor.org/stable/43678652

[4] "Unique Factorization", https://mathworld.wolfram.com/UniqueFactorization.html

[5] "How did Gauss conjecture there were nine Heegner numbers?", https://math.stackexchange.com/questions/3138747/how-did-gauss-conjecture-there-were-nine-heegner-numbers

(1/2)

dmm, to math
@dmm@mathstodon.xyz avatar

Finite fields is one of my favorite topics.

These notes are really nice: https://www.maths.tcd.ie/pub/Maths/Courseware/373-2000/FiniteFields.pdf.

dmm,
@dmm@mathstodon.xyz avatar

@johncarlosbaez I struggle with that one; I changed to "are" 🙂

Thanks! --dmm

dmm, to random
@dmm@mathstodon.xyz avatar

This code works when \nrows ≤ 7 but blows up when \nrows > 7. Why?

LaTeX appears to have some hidden limitation somewhere (or something)...

The LaTeX source is here: https://www.overleaf.com/read/tgkjgjnjhfjm.

Pascal's triangle

dmm,
@dmm@mathstodon.xyz avatar

Another possiblity: lualatex

Pascal's triangle rendered with lualatex, nrows = 14

dmm,
@dmm@mathstodon.xyz avatar

BTW, the reason I started all of this is because diagonal 4 row 14 of Pascal's triangle has the number 364, which is related to "The 12 Days of Christmas".

The hint is "tetrahedral numbers" 🙂

See the image for some of the figurative numbers found in Pascal's triangle.

dmm,
@dmm@mathstodon.xyz avatar

@johncarlosbaez BTW, I stumbled on this when was looking for some interesting things to add to my "Christmastime is Christmas Math Time!" document (https://davidmeyer.github.io/qc/christmastime_is_christmas_math_time.pdf).

I'm thinking I'll add this (12 Days of Christmas) this year 🙂

christianp, to random
@christianp@mathstodon.xyz avatar

Would it be feasible to automatically gather all mathematical terms defined on wikipedia?
I looked at wikidata, but it doesn't look like there's a property to say that an item is a definition of a term, so I think the best I could do would be to wade through things in the Wikiproject Mathematics.

If I was looking at the text of articles, I suppose looking for the phrase "is called" in the page description would get most definitions.

dmm,
@dmm@mathstodon.xyz avatar

@christianp I finally had to make this: https://davidmeyer.github.io/qc/algebraic_structures.pdf...

mcnees, (edited ) to random
@mcnees@mastodon.social avatar

Astronomer and mathematician Pierre-Louis Moreau de Maupertuis, exact date of birth unknown, was born around this time in 1698 and baptized .

He was among the first to articulate the Principle of Least Action, one of the most beautiful ideas in physics. (1/n)

Image: Wellcome Trust

dmm,
@dmm@mathstodon.xyz avatar

@mcnees

I wrote a bit about the Principle of Least Action here: https://davidmeyer.github.io/qc/pola.pdf. As always, questions/comments/corrections/* greatly appreciated.

Fascinating stuff...

dmm, to math
@dmm@mathstodon.xyz avatar

f you color the sub-triangles in Pascal's triangle [1] you see that additional structure is revealed: Pascal's triangle is actually a Sierpinski gasket [2,3].

Scale invariance turns up in many perhaps unexpected places...

References

[1] "Pascal's triangle", https://en.wikipedia.org/wiki/Pascal's_triangle

[2] "The Sierpinski gasket", https://www.math.stonybrook.edu/.../Sierpinski_gasket.html

[3] "Order from Chaos: The Sierpinksi Gasket", http://www.nmsr.org/digdudle.htm

dmm, to random
@dmm@mathstodon.xyz avatar

If you've never looked into the Antikythera Mechanism [1] (or even if you have) there is a nice overview article that contains quite a bit of interesting background information on the Mechanism here: https://greekreporter.com/2023/08/26/ancient-greece-antikythera-mechanism.

How the ancient Greeks conceived of, designed and built the Mechanism is really mind boggling and worth a look.

A few of my notes on how the Mechanism works (and in particular, how Derek J. de Solla Price’s [2] proposed Metonic cycle gear train works, as well as Michael Wright’s [3] scheme for turning the Mechanism's Metonic pointer, which appears to be the generally accepted architecture) are here: https://davidmeyer.github.io/astronomy/prices_metonic_gear_train.pdf. The LaTeX source is here: https://www.overleaf.com/read/ndpvkytkhmbv.

As is so frequently the case (or so it seems), these notes were a work in progress that has never quite been finished...In any event, questions/comments/corrections/* are greatly appreciated (as always).

References

[1] "An Ancient Greek Astronomical Calculation Machine Reveals New Secrets", https://www.scientificamerican.com/article/an-ancient-greek-astronomical-calculation-machine-reveals-new-secrets

[2] "Derek J. de Solla Price", https://en.wikipedia.org/wiki/Derek_J._de_Solla_Price

[3] "Michael T. Wright", https://en.wikipedia.org/wiki/Michael_T._Wright

How the ancient Greeks conceived of, designed and built the Mechanism is really mind boggling and worth a look.
How the ancient Greeks conceived of, designed and built the Mechanism is really mind boggling and worth a look.

dmm, to math
@dmm@mathstodon.xyz avatar

Here's some math/LaTeX/TikZ fun for today: The Spiral of Theodorus

What is the Spiral of Theodorus?

The Spiral of Theodorus [1] is the fantastic object shown below. It is composed of right triangles placed edge-to-edge, where the side opposite the (inside) acute angle has length 1 and where the length of the hypotenuse of the n^{th} triangle is √(n+1). Pretty cool.

The spiral is named after Theodorus of Cyrene, the ancient Greek mathematician who lived during the 5th century BC and who apparently discovered the spiral [2].

So can you draw the Spiral of Theodorus in TikZ? But of course! I don't know if I love the approach I used to draw the figure below (I've tried a few different approaches), but this is where I stopped...

A few of my notes are here: https://davidmeyer.github.io/qc/spiral_of_theodorus.pdf. The LaTeX source is here: https://www.overleaf.com/read/scsptwzqxzdt.

References

[1] "Spiral of Theodorus", https://en.wikipedia.org/wiki/Spiral_of_Theodorus

dmm, to science
@dmm@mathstodon.xyz avatar

I love this picture (from [1]). It shows just how much of a latecomer adaptive immunity really is. Adaptive immunity is only a couple of hundred million years old; plants, invertebrates and fungi (and really most other organisms) survive just fine without adaptive immune systems.

The use of combinatorial diversity to make all kinds of randomly generated antigen receptors from our couple of hundred antibody genes is really spectacular.

BTW, these Mayo Clinic videos are really informative.

References

[1] "2015 B Cell Biology and T Follicular Helper Cells – The Fundamentals", https://www.youtube.com/watch?v=tuITLpoWdXU

dmm, to machinelearning
@dmm@mathstodon.xyz avatar

I've been working on my (old now) notes on Fisher information to make them more in line with how I write these days.

The Fisher information turns out to be important in both statistics and machine learning, as well as many other fields.

"Essentially the Fisher Information tells us how much information a random variable X contains about the parameter vector θ, where X is distributed according to a probability distribution parameterized by θ. Intuitively, the Fisher information captures the variability of the gradient of the score function ∇θ log pθ (x). In a family of distributions for which the score function has high variability we expect estimation of the parameter θ to be easier; essentially (and perhaps counter-intuitively) events with lower probability contain more information."

A few of my notes are here: https://davidmeyer.github.io/ml/fisher.pdf.

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

dmm, to random
@dmm@mathstodon.xyz avatar

Happy Pi Approximation Day!

Pi Approximation Day is so named because

22/7 ≈ 3.14285714286

which is a popular approximation of π.

dmm, to math
@dmm@mathstodon.xyz avatar

Here's a crazy thing. This beautiful formula for π was devised by John Wallis in 1655. It says that π/2 is equal to the infinite product of even numbers squared divided by the product of their adjacent odd numbers, and is sometimes known as the Wallis product.

A few of my notes are here: https://davidmeyer.github.io/qc/basel.pdf. The LaTeX source is here: https://www.overleaf.com/read/xsnvysdcfcvd.

See https://en.wikipedia.org/wiki/Wallis_product for more.

dmm, to random
@dmm@mathstodon.xyz avatar

62 years ago John Lewis was released from Parchman Farm Penitentiary after being arrested in Jackson, MS for using a "white only" restroom during the Freedom Rides of 1961.

References

[1] "Rep. John Lewis on the time he was sent to prison for using a 'white' restroom", https://www.vox.com/2014/7/7/5877957/rep-john-lewis-memories-of-a-mississippi-prison-during-the-freedom

christianp, to random
@christianp@mathstodon.xyz avatar

mathstodon.xyz is going to go down in about 20 minutes. It might be unavailable for around an hour, depending on how well this database work goes.

dmm,
@dmm@mathstodon.xyz avatar

@christianp Thank you!

lauren, to random
@lauren@mastodon.laurenweinstein.org avatar

Bluesky's (former Twitter CEO) Dorsey and Twitter's CEO Musk pushing anti-vaccine crackpot Kennedy for president - https://thehill.com/homenews/campaign/4034584-ex-twitter-leader-jack-dorsey-endorses-rfk-jr-for-president/

dmm,
@dmm@mathstodon.xyz avatar

@lauren I've already said this elsewhere (so sorry for duplicates), but this is worth listening to, especially if you happen to be listening to RFK, Jr.

https://www.youtube.com/watch?v=tGoJeLyMG5I

dmm, to math
@dmm@mathstodon.xyz avatar

This is kind of cool: the Cistercian numerals are a forgotten number system developed by the Cistercian monastic order in the early thirteenth century [1,2].

Interestingly, Cistercian numerals are much more compact than Arabic or Roman numerals; with a single character you could write any integer from 1 to 9999.

References

[1] "Cistercian numerals", https://en.wikipedia.org/wiki/Cistercian_numerals

[2] "The Forgotten Number System", https://www.youtube.com/watch?v=9p55Qgt7Ciw

dmm,
@dmm@mathstodon.xyz avatar

@bencurthoys I never heard of that number system. Thanks!

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

In math there's an ordered hierarchy of concepts: stuff, structure and properties. This is because categories have 3 things: objects, morphisms between objects, and equations between morphisms. But we see it in the usual form of math definitions, like:

"A group is a set (that's the stuff) with some operations (that's the structure) obeying some equations (those are the properties)."

To build a group we start with an object in the category of sets - that's the stuff. Then we give it some operations, i.e. some morphisms - that's the structure. Then we impose some equations - those are the properties.

These ideas let us talk about how much a forgetful functor forgets. There are 3 main options. Some functors forget nothing at all. Some forget properties but not structure or stuff. Some forget properties and structure, but not stuff. And some forget properties, structure and stuff. We can get more fancy than this, but this is a good place to start.

Let me say all this more technically, since the point of this junk is that we can make it all precise and prove theorems with it.

(1/2)

dmm, (edited )
@dmm@mathstodon.xyz avatar

@johncarlosbaez This is, I guess, only peripherally related but I've been reading this (sort of model theoretic) paper titled "On Automorphism Criteria for Comparing Amounts of Mathematical Structure": https://arxiv.org/abs/2204.11973.

And while the authors point out the limitations of using automorphisms to measure structure, it could still be informative (again even though the authors conclude that automorphisms alone do not encode all of the relevant facts about the structure of a mathematical object).

But here's a question: Why didn't we ever try to compare network complexity by comparing the sizes of the respective automorphism groups (or maybe someone did, but if so I don't know about it)? After all, complexity is structure that forms in networks to provide robustness (unless of course it's gratuitous complexity), and if we could build a mathematical structure that represents a network (this would be the trick) then in theory we could use the size of the automorphism groups to get some sense of relative network complexity. Maybe anyway.

Beyond all of that, the idea that we could be more quantitative about Occam's Razor is an interesting philosophical endevor.

In any event, I found this paper interesting. Maybe you will too.

lauren, to random
@lauren@mastodon.laurenweinstein.org avatar

With great reluctance, I'm forced to the conclusion that affirmative action as implemented over these many years in the U.S. has become increasingly stretched and distorted, like a ball of Silly Putty that began as a perfect sphere and over time became unrecognizable tatters with pieces breaking off at random. A valiant effort, however.

dmm,
@dmm@mathstodon.xyz avatar

@lauren

Justice Jackson summarized what many see as the problem with SCOTUS' ruling in herdissent in the UNC case (independent of how deformed the silly putty might be):

“With let-them-eat-cake obliviousness, today, the majority pulls the ripcord and announces "colorblindness for all' by legal fiat. But deeming race irrelevant in law does not make it so in life.”

dmm, (edited ) to math
@dmm@mathstodon.xyz avatar

How to calculate (and display) Pascal's Triangle in TikZ?

%
% Draw Pascal's Triangle with binomial coefficients.
% The triangular numbers are shown in blue.
%
\newcommand \nrows {8} % number of rows
%
% Draw the picture
%
\begin{figure}[H]
\centering % center everything
\begin{tikzpicture} [framed,scale=0.90] % frame & scale
\foreach \n in {0,...,\nrows} { % iterate over rows
\foreach \k in {0,...,\n} { % iterate over columns
\node at (\k-\n/2,-\n) { % put coefficient here
\ifnum \k = 2 {\color{blue}
$\mathbf{\binom{\n}{\k}}$}
\else {\color{black}
$\mathbf{\binom{\n}{\k}}$}
\fi}; % end \ifnum
}. % end \foreach \k ...
} % end \foreach \n ...
\end{tikzpicture}
\caption{Pascal's Triangle with Binomial Coefficients}
\label{fig:pascals_triangle_binomial_coefficients}
\end{figure}

Happy 400th BD Blaise Pascal!

dmm,
@dmm@mathstodon.xyz avatar
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