The Fourier series of an function (i.e. the function generated from the infinite sum of weighted sines) converges to the function pointwise almost everywhere.
The theorem also seems to hold (maybe trivially given the transform result) for the Fourier series (TODO if trivially, why trivially).
Only proved in 1966, and known to be a hard result without any known simple proof.
This theorem of course implies that Fourier basis is complete for , as it explicitly constructs a decomposition into the Fourier basis for every single function.
TODO vs Riesz-Fischer theorem. Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?
One of the many fourier inversion theorems.
Original 1931 experiment by Raman and Bhagavantam: dspace.rri.res.in/bitstream/2289/2123/1/1931%20IJP%20V6%20p353.pdf
Since Ciro Santilli is Brazilian, this is understandably a common conversation opener.
And rightly so, since soccer in particular is truly ridiculously popular in Brazil, where "what is your local soccer team?" is just as valid a conversation starter as "Which city are you from?".
So here goes Ciro's 2020 cynic answer:
I currently root actively against Brazil.
The ironic reason is simple: maybe is Brazil loses more on this useless art, then maybe people will get tired of it, and instead invest on more useful and beautiful arts.
Notably, what Ciro really wants people to root for are:
- the number of Brazilian Nobel Prizes, which is zero, yes, zero, as of 2020, despite a population of 210 million people. But thank God for our one Field Medal, what an epic start, even though Mathematics is useless.
- the number of high tech companies that have a global impact, which is likely extremely low as of 2020, and must contain only a few mammoths that dominate some local commodity market and therefore got enough money from that to expand a bit of technology worldwide. But they were mostly not classic tech startups that did world innovation from the start.
- how low your country's Gini coefficient is
Don't get Ciro wrong.
Observing professionals who do it amazingly can be beautiful.
But why the F do you have to root for a team unless your wife or children are playing in it (and even then..., how will that help?)?
What will you get from that?
Even if it is your national team, why does it matter if they win or lose?
Hooliganism just takes that uselessness to a hole new level.
Now some confessions.
A five year old Ciro will never forget when the feeling of Brazil won the 1994 World Cup on the penalties and everyone went mad that evening.
A nine year old Ciro stopped watching the 1998 World Cup Final of Brazil vs France half way during the 3-0 massacre and went to his front garden to kick his soccer ball on the metallic fence gate which represented a goal.
After that, Ciro went through puberty he guesses, and noticed that the natural sciences are just cooler than this soccer watching bullshit.
Football, Football, Football by Mitchell and Webb
. Source. Why football is the most popular sport in the world by Desmond Morris
. Source. Interview by Web of Stories. His thesis is that football is the sport that looks most like a hunt, the scoring of a goal being the kill of the prey, and thus appeals to people's Paleolithic hunting habits the most. He feels it is more like hunting than war, and that the opposing team is just there to add some difficulty to scoring. He mentions that soccer has all the fundamental aspects of hunting: running after something, aiming, and hunting in a pack of allies. Ciro agrees with this dude. Ciro also adds that the fact that each soccer match has few goals, e.g. as opposed to basketball, makes it much more like a hunt, where you score few large kills per hunt.
Well summarized as "the branch of mathematics that deals with limits".
The best place to get answers to programming questions as of 2019. Google into Stack Overflow is always the best bet.
An overview of Ciro Santilli's Stack Overflow contribution can be found at: Ciro Santilli's Stack Overflow contributions.
Stack Overflow in a nutshell
. Source. As the "WTF look at my points" guy, Ciro Santilli approves of this meme. A few more elements could be added, notably deletion of the last link-only answer, but good enough.
By the profile image, the "Grammar Nazi" editor is actually appropriately the notorious serial editor Peter Mortensen. Ciro Santilli welcomes grammar fixes, but more subjective style fixes can be a bit annoying.
Catching mice by Nakanoart
. Possible non-canonical source: twitter.com/nixcraft/status/1376023938749190147
How stack overflow hires meme using Welcome Aboard! meme template
. Source. The elliptic curve group of an elliptic curve is a group in which the elements of the group are points on an elliptic curve.
The group operation is called elliptic curve point addition.
Ahh, this dude is just like Ciro Santilli, trying to create the ultimate natural sciences encyclopedia!
____
In 1995, Weisstein converted a Microsoft Word document of over 200 pages to hypertext format and uploaded it to his webspace at Caltech under the title Eric's Treasure Trove of Sciences.
____
In 1995, Weisstein converted a Microsoft Word document of over 200 pages to hypertext format and uploaded it to his webspace at Caltech under the title Eric's Treasure Trove of Sciences.
____
Conservation of the square amplitude in the Schrodinger equation Updated 2025-01-10 +Created 1970-01-01
Proof that the probability 1 is conserved by the time evolution:
It can be derived directly from the Schrödinger equation.
Bibliography:
- That proof also mentions that if the potential
Vis not real, then there is no conservation of probability! Therefore the potential must be real valued!
Lowercase looks like a lowercase letter N for some reason.
The equation that allows us to calculate stuff in special relativity!
Take two observers with identical rules and stopwatch, and aligned axes, but one is on a car moving at towards the direction at speed .
TODO image.
When both observe an event, if we denote:It is of course arbitrary who is standing and who is moving, we will just use the term "standing" for the one without primes.
- the observation of the standing observer
- the observation of the ending observer on a car
Note that if tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:
This explains why we don't observe special relativity in our daily lives: macroscopic objects move too slowly compared to light, and is almost zero.
Unlisted articles are being shown, click here to show only listed articles.