Shows up when trying to solve 2D wave equation on a circular domain in polar coordinates with separation of variables, where we have to decompose the initial condition in termes of a fourier-Bessel series, exactly like the Fourier series appears when solving the wave equation in linear coordinates.
For the same fundamental reasons, also appears when calculating the Schrödinger equation solution for the hydrogen atom.
eigenvalue problem of Laplace's equation.
Video 1. Alegria, Alegria by Caetano Veloso (1968)Source. Critique of Military dictatorship in Brazil.
- 1972
- Transa (1972) album. Literally: "The Fuck", good old seventies. Caetano himself later mentions that this is one of his own favorite albums.[ref] The album was composed when he was living in London.
Video 2. Triste Bahia by Caetano Veloso (1972)Source. Inspired by (or more likely: actually is) capoeira music.
Video 4. Você Não Entende Nada by Caetano Veloso (1970)Source.Amazing performance at Coliseu dos Recreios, Lisbon. 1981. Discogs says original album is Legal (1970), but wiki page and photos of back disagree...This song talks about a man's mixed desires to remain with his partner and also escape to adventure. It contains one of the most amazing sexual innuendo ever recorded: the man describes foods that his partner will serve him, and the he endlessly eats and eats and eats:translation:eu como, eu como, eu como, você [... huge intentional pause ...] não está entendendo quase nada do que eu digo.
which makes it unclear "you" is part of "I eat you" (to eat (comer) is a slang for fucking), or if "you" is part of the next sentence.I eat, I eat, I eat, you [... huge intentional pause ...] don't understand anything that I'm saying.
Video 5. Partido Alto by Caetano Veloso (1972)Source. Atheism song (theodicy)! Lyrics by Chico Buarque, but Ciro Santilli prefers this interpretation. From the 1972 "Caetano E Chico Juntos E Ao Vivo" joint album with Chico Buarque.
- 1975 Qualquer coisa album
Video 7. Samba e Amor by Caetano Veloso (1975)Source. Cover from original song by Chico Buarque, original album Chico Buarque de Hollanda - Nº4. Desperately reminds Ciro of his University day weekend nights. Except that there was no Samba. And little Amor. Mostly a silent and wholesome loneliness and emulation.Video 8. Georde de Capadócia by Caetano Veloso (1975)Source.Composed by Jorge Ben, but this interpretation is remarkable. corpo fechado (closed body) style song. This idea is much linked to Capoeira/African religion idea. E.g. a more traditional capoeira corpo fechado song: www.youtube.com/watch?v=tfd9j6XFmSg
- 1976 Doces Bárbaros (1976) album
Video 10. Um Índio by Caetano Veloso (1976)Source. Recording from 1992. Also appeared in the Bicho (1977) album. - 1977 Bicho (1977)
- 1978 album Muito (Dentro da Estrela Azulada)
Video 12. Sampa by Caetano Veloso (1978)Source."Sampa" is an affectionate slang for São Paulo City. The song perfectly captures the city, and reminds Ciro so badly of his University days there.Video 13. Terra by Caetano Veloso (1978)Source.Ciro used to watch a television nature show called "Planeta Terra" in the legendary TV Cultura with his parents in the couch when he was young, and under a duvet when it was a bit cold. Those days were the best. The narrator's lady voice was particularly soothing, and would easily put you in a kind of sleepy trance, her name is Valéria GrilloTODO what was the original show exactly? Here is a sample: www.youtube.com/watch?v=vNwfYEMdrRU Very likely just a translation of some British nature show with a custom Brazilian intro and presenter. Credits at end mention English narrator: "Eugene Fraser", and "Thirteen WNET Nature" production, which produced Nature (1982) that ran since 1982, making that a likely candidate. - 1980
- 1984 Velô (1984)
As if Greek letters weren't enough, physicists and mathematicians also like to make up tons of symbols, some of which look like the could actually be Greek letters!
And of course, 4chan just takes that to a whole new level, usually closing on the same day, and then getting deleted within a week. Why would anyone contribute non-illegal content to that king of system?!
Ridiculous, so when new information comes out, we just duplicate all the old comments on a new thread again?
Remember, Ciro Santilli is the Necromancer God.
Generally, if something is labelled as "e-learning", it's not a good sign, as it implies that it adheres to the "teacher"/"student" separation which Ciro Santilli much despises: E-learning websites must allow students to create learning content.
Furthermore, TODO confirm it is possible that a solution does not exist at all if and aren't sufficiently small.
This formula is likely the basis for the Lie group-Lie algebra correspondence. With it, we express the actual group operation in terms of the Lie algebra operations.
Notably, remember that a algebra over a field is just a vector space with one extra product operation defined.
Vector spaces are simple because all vector spaces of the same dimension on a given field are isomorphic, so besides the dimension, once we define a Lie bracket, we also define the corresponding Lie group.
Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the Baker-Campbell-Hausdorff formula, we are basically done defining the group in terms of the algebra.
The one parameter subgroup of a Lie group for a given element of its Lie algebra is a subgroup of given by:
Intuitively, is a direction, and is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of , the rotation group.
One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.
Can be calculated efficiently with the Extended Euclidean algorithm.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact