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.
"I can't draw" is almost (but not quite) as common a misconception as "I can't do #maths". Prove yourself wrong for one of these at Olivia's free drop-in #drawing workshop at @NPGLondon on 21st June:
A couple of weeks ago, I posted an #animation of a point on a circle generating a #cycloid.
If you turn the curve "upside down", you get the #BrachistochroneCurve. This curve provides the shortest travel time starting from one cusp to any other point on the curve for a ball rolling under uniform #gravity. It is always faster than the straight-line travel time.
Anyway, the #animation took a bit of thought as it requires a bit of #Mechanics, some #Integration and is made a bit more tricky as the curve is multi-valued and so you need to treat different branches separately. The #AnimatedGif was produce with #WxMaxima.
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.
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].
@sjb Ah... Klein-Nishina, the QFT formula related to Compton scattering - I was getting a Compton scattering prac up and running in a 3rd Year Uni teaching lab and found a mistake in relation to calculations using Klein-Nishina for this prac in the classic Melissinos "Experiments in Modern Physics" (if I remember correctly the end result is OK... possibly two mistakes cancelling out... but it was a while back). I emailed the publisher but got no response - oh well.
Early on in my hobby I came to the realization that cryptographic prowess has no viable market price point. More's the pity. Yet I think one day I may change that with my secrecy sauce.
"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater ; not because the pleasure it gives (although very pure) is comparable [...] to that of music [...]" – Bertrand Russell (1872–1970) #quote#mathematics#art#maths#math
Started listening to the audiobook edition of Everything Is Predictable. How Bayes' Remarkable Theorem Explains the World, written and read by Tom Chivers.
Il reprochait à #DidierRaoult des erreurs "niveau brevet des collèges" : accusé de diffamation, un prof de #maths relaxé
Attaqué en #justice par Didier Raoult pour diffamation et injure publique, Guillaume Limousin a été relaxé, mardi 14 mai [...]. Sur Twitter, ce professeur de mathématiques isérois, reprochait à l'infectiologue une série d'erreurs de "niveau brevet des collèges". Didier Raoult devra lui verser 2000€, au titre des frais de justice.