dmm, 1 hour 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, 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, 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, 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

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, 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

dmm, 1 month 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, 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, 1 month ago to math

Born #onthisday 428 years ago, René Descartes was a French mathematician and philosopher. He developed the “cartesian” coordinate system, which is named after him. Among many other things, his work also provided the foundations for discovering calculus a few decades later.

Read more about Descartes' life and times here: https://plato.stanford.edu/entries/descartes/

[Image credit: https://mathigon.org/timeline/descartes]

#descartes #cartesiancoordinates #cogitoergosum #math #maths #philosophy

dmm, 2 months ago to random

I've been reading "Introductory Category Theory Notes", by Daniel Epelbaum and Ashwin Trisal.

Is there a better introductory Category Theory text (that I can find online)? Thx!

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

dmm, 2 months ago to physics

English polymath Isaac Newton, who was a mathematician, physicist, astronomer, alchemist, and theologian, died #onthisday in 1727.

His pioneering book Philosophiæ Naturalis Principia Mathematica (1687) consolidated many previous results and established classical mechanics [1]. He also made seminal contributions to optics (among many other things), and shares credit with Gottfried Wilhelm Leibniz for developing calculus.

Books by Newton at Project Gutenberg: https://www.gutenberg.org/ebooks/author/6288

[Image credits: https://en.wikipedia.org/wiki/Isaac_Newton and https://www.westminster-abbey.org/abbey-commemorations/commemorations/sir-isaac-newton]

References

[1] "Newton’s Philosophiae Naturalis Principia Mathematica", https://plato.stanford.edu/entries/newton-principia/

#newton #calculus #physics #optics #math #maths

dmm, 2 months ago to space

This piece [1] is a bit older but still interesting on the origin of the famous Drake equation, which estimates the number of advanced civilizations likely to exist in the Milky Way galaxy.

References

[1] "The Origin of the Drake Equation", https://astrosociety.org/file_download/inline/58ee6041-5f61-4f88-8b15-d2d3d22ab83d

[2] "Drake equation", https://en.wikipedia.org/wiki/Drake_equation

#drakeequation #seti #extraterrestriallife #nasa

dmm, 2 months ago to nature

Weighing between 0.05 and 0.07 ounces with a head-to-body length of 1.14 to 1.29 inches and a wingspan of 5.1 to 5.7 inches, the bumblebee bat (aka Kitti’s hog-nosed bat) is the world's smallest mammal.

https://en.wikipedia.org/wiki/Kitti's_hog-nosed_bat

#nature #bats #bumblebeebat #biology

The bumblebee bat (aka Kitti’s hog-nosed bat)

dmm, 3 months ago to science

Remember COVID and the coronavirus?

There is an interesting discussion about bats, pangolins, and SARS-CoV-2 in the Nature Communications paper "Evidence for SARS-CoV-2 related coronaviruses circulating in bats and pangolins in Southeast Asia" [1].

The paper is also discussed in some detail on TWiV 722 [2]. The discussion starts at about 00:19:20.

More cool research on bats and SARS-CoV-2.

#bats #pangolins #sarscov2 #biology #viruses #twiv

References

[1] "Evidence for SARS-CoV-2 related coronaviruses circulating in bats and pangolins in Southeast Asia", https://www.nature.com/articles/s41467-021-21240-1

[2] "TWiV 722: More coronaviruses in the pipeline", https://www.microbe.tv/twiv/twiv-722/