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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.

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

A factorion is an interesting kind of natural number that equals the sum of the factorials of its decimal digits [1].

40585 is the largest known factorion and was discovered in 1964 by Leigh Janes [2].

References

[1] "Factorion", https://en.wikipedia.org/wiki/Factorion

[2] "Mathematician:Leigh Janes", https://proofwiki.org/wiki/Mathematician:Leigh_Janes

dmm,
@dmm@mathstodon.xyz avatar

@christianp Thanks for the comment!

I added a reference which briefly discusses the limitation you mention (https://mathworld.wolfram.com/Factorion.html).

Thanks! --dmm

dmm, to math
@dmm@mathstodon.xyz avatar

Did you know that the roots of a cubic polynomial can be visualized using an equilateral triangle?

In this incredibly cool animation from Freya Holmér (@acegikmo):

🔵 the vertices of the triangle map to the roots
🔴 the incenter is the inflection point
🟢 the incircle boundaries are the local minima/maxima

video/mp4

dmm, to random
@dmm@mathstodon.xyz avatar

Born 208 years ago, Ada Lovelace wrote the world’s first computer program (which calculated Bernoulli numbers) and worked with Charles Babbage on the Analytical engine [1,2,3].

She was also the only legitimate child of poet Lord Byron and Lady Byron [4].

References

[1] "Ada Lovelace’s skills with language, music and needlepoint contributed to her pioneering work in computing", https://theconversation.com/ada-lovelaces-skills-with...

[2] "Charles Babbage", https://en.wikipedia.org/wiki/Charles_Babbage

[3] "ADA BYRON, COUNTESS OF LOVELACE", https://www.sdsc.edu/ScienceWomen/lovelace.html

[4] "Ada Lovelace", https://www.biography.com/scholar/ada-lovelace

dmm, to math
@dmm@mathstodon.xyz avatar

Analytic geometry is a fantastic area of mathematics which is populated by all kinds of crazy objects. Being that it is Christmastime (and therefore Christmas math time) please check out the paradoxical object shown on the left. This object is sometimes known as the Infinite Gift. A related object is known as Gabriel's Wedding Cake [7].

In the Infinite Gift the length of the side of the nth box is 1/√n, so the area of one side of the nth box equals (1/√n)² = 1/n. Since a box has 6 sides the surface area of the nth box is 6·(1/n). Then what you find is that in the limit as n → ∞ that the Infinite Gift has infinite surface area but finite volume!

Here's an interesting aside: In the limit the area of the Infinite Gift equals 6 times the harmonic series (which we know diverges).

The continuous version of this object is known as Gabriel’s Horn (aka Torricelli’s Trumpet) and is shown in the figure on the right [1,2]. Gabriel's Horn is the surface of revolution of the function y = 1/x about the x-axis for x ≥ 1 [1,2]. As we can see in image, in the limit Gabriel’s Horn has volume = π and area = ∞.

These properties lead to an interesting situation known as the Painter’s Paradox [3,4].

This is the Painter's Paradox: Somehow even though you can fill Gabriel’s Horn with paint (its volume is finite), you still won’t have enough paint to cover its inside surface (its area is infinite)!

Merry Christmas everyone!


(1/2)

Gabriel's Horn

dmm, to math
@dmm@mathstodon.xyz avatar

In 1683 the Swiss mathematician Jacob Bernoulli discovered this beautiful expression for the constant e while studying continuous compound interest.

See https://en.wikipedia.org/wiki/Jacob_Bernoulli for more on Bernoulli's work and life.

dmm,
@dmm@mathstodon.xyz avatar

@ColinTheMathmo AFAIU Bernoulli's expression was the first expression for e and the one we use today.

dmm, to machinelearning
@dmm@mathstodon.xyz avatar

This is a nice tutorial if you are interested in how and why overfitting (etc) happens, how/why regularization helps to alleviate overfitting, ...

"The Theory Behind Overfitting, Cross Validation, Regularization, Bagging, and Boosting: Tutorial", Benyamin Ghojogh, Mark Crowley

https://arxiv.org/abs/1905.12787

dmm, to physics
@dmm@mathstodon.xyz avatar

in 1915: Albert Einstein submitted a paper to the journal "Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin" that would fundamentally alter our understanding of the universe [1]. The four page paper contained what became known as the Einstein field equations, which relate the geometry of spacetime to the distribution of matter within it [2].

Einstein's field equations were presented in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor) [3].

[Image credit: https://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/6E3MAXK4/index.meta]

References

[1] "Die Feldgleichungen der Gravitation", https://einsteinpapers.press.princeton.edu/vol6-doc/273

[2] "Einstein field equations", https://en.wikipedia.org/wiki/Einstein_field_equations

3] "Einstein tensor", https://en.wikipedia.org/wiki/Einstein_tensor

dmm,
@dmm@mathstodon.xyz avatar

"Gödel presented his solution to Einstein in the form of a manuscript on the occasion of the latter’s 70th birthday in 1949. Gödel was meant to have his manuscript ready for Paul Arthur Schlipp to include it in a book entitled Albert Einstein: Philosopher-Scientist* (Schlipp, 1949). Gödel, ever the perfectionist, however did not finish his essay until about a month before Einstein’s birthday and even then delayed sending it off for a while (Dawson, 2006). Notorious for his attention to detail, Gödel’s six-page manuscript eventually included 34 footnotes."

https://www.privatdozent.co/p/godels-solution-to-einsteins-field

(2/2)

dmm, to math
@dmm@mathstodon.xyz avatar

Functional Analysis is a very cool topic which sits between infinite dimensional linear algebra and real and complex analysis (a couple of mind-blowing topics in their own right...).

A few of my (very nascent) notes are here: https://davidmeyer.github.io/qc/functional_analysis.pdf. The LaTeX source is here: https://www.overleaf.com/read/fgrrxmkycvry. As always, questions/comments/corrections/* greatly appreciated.

Happy Thanksgiving everyone!

dmm,
@dmm@mathstodon.xyz avatar

@johncarlosbaez Thanks, I will definitely check out Reed and Simon. Thx, --dmm

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

More fall in the . Looking east down 21st from University in Eugene, OR.

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...

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

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