Ciro Santilli @cirosantilli 37

take a scalar field as input. Otherwise, how do you decide which vector is larger than the other?
output a vector field that contains the direction in which the scalar increases fastest locally at each point. This has to give out vectors, since we are talking about directions

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Domain, codomain and image by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Discrete Fourier transform by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Input: a sequence of

N

complex numbers

x_{k}

Output: another sequence of

N

complex numbers

X_{k}

such that:

x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{kn}{N}}

(1)

Intuitively, this means that we are braking up the complex signal into

N

sinusoidal frequencies:

$X_{0}$ : is kind of magic and ends up being a constant added to the signal because $e^{i 2 π \frac{kn}{N}} = e^{0} = 1$
$X_{1}$ : sinusoidal that completes one cycle over the signal. The larger the $N$ , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
$X_{2}$ : sinusoidal that completes two cycles over the signal
...
$X_{N - 1}$ : sinusoidal that completes $N - 1$ cycles over the signal

and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:

compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise

In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.

Sample software implementations:

numpy.fft, notably see the example: numpy/fft.py

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ with 25 points
. This is a simple example of a discrete Fourier transform for a real input signal. It illustrates how the DFT takes N complex numbers as input, and produces N complex numbers as output. It also illustrates how the discrete Fourier transform of a real signal is symmetric around the center point.

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Mathematical proof by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Theorem by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Slang by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Alternating and direct current by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Electronics vendor by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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E-learning websites must keep content free, only charge for certification by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Charging for certification is fine. Creating exams and preventing cheating has a cost.

Another thing that is fine charging for is dedicated 1-to-1 tutor time. This is something Udacity is doing as of 2022.

www.investopedia.com/articles/investing/042815/how-coursera-works-makes-money.asp has a good mention:

MOOCs were first created by people with utopian visions for the internet. This means the idea for platforms like Coursera was likely conceived without a business plan in mind. Nonetheless, Coursera has managed to monetize its platform. It is worth noting, however, that monetization has lead to the effective elimination of the original MOOC idea, which is predicated on ideals like free and open access, as well as the building of online communities.
Coursera users must pay to engage with the material in a meaningful way and take courses for individualistic purposes. This has been a consistent trend among all major online education platforms.

and it links to: www.freecodecamp.org/news/massive-open-online-courses-started-out-completely-free-but-where-are-they-now-1dd1020f59/, very good article!

That is a fundamental guiding principle of OurBigBook.com. The educational content must be licensed CC BY-SA!

Perhaps the most reliable way of reaching this state is E-learning websites must allow students to create learning content.

Bibliography:

academia.stackexchange.com/questions/86179/is-it-financially-worth-it-to-teach-a-mooc-e-g-coursera Is it financially worth it to teach a MOOC (e.g. Coursera)?
www.classcentral.com/about amazing, they can make money just from ads! I wouldn't expect that they could scale like TripAdvisor, because travelling means very local knowledge, I would expect there to be much fewer MOOCs and for them to be more easily findable on Google. Good thing though, this website.

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Steve Jobs' 2005 Stanford Commencement Address by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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Ciro feels that this resonates a lot with his OurBigBook.com.

Supercut:

The minute I dropped out I could stop taking the required classes that didn't interest me, and begin dropping in on the ones that looked far more interesting.
And much of what I stumbled into by following my curiosity and intuition turned out to be priceless later on.
Because I had dropped out and didn't have to take the normal classes, I decided to take a calligraphy class to learn how to do this.
None of this had even a hope of any practical application in my life.
If I had never dropped in on that single course in college, the Mac would have never had multiple typefaces or proportionally spaced fonts.
Of course it was impossible to connect the dots looking forward when I was in college. But it was very, very clear looking backward 10 years later.
Again, you can't connect the dots looking forward; you can only connect them looking backward. So you have to trust that the dots will somehow connect in your future. You have to trust in something — your gut, destiny, life, karma, whatever. This approach has never let me down, and it has made all the difference in my life.

Then:

You've got to find what you love.
And that is as true for your work as it is for your lovers.
Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work.
And the only way to do great work is to love what you do. If you haven't found it yet, keep looking.
Don't settle. As with all matters of the heart, you'll know when you find it.
And, like any great relationship, it just gets better and better as the years roll on.
So keep looking until you find it.
Don't settle.

And:

When I was 17, I read a quote that went something like: "If you live each day as if it was your last, someday you'll most certainly be right."
It made an impression on me, and since then, for the past 33 years, I have looked in the mirror every morning and asked myself: "If today were the last day of my life, would I want to do what I am about to do today?"
And whenever the answer has been "No" for too many days in a row, I know I need to change something.

Mirror and morning are not required though, a computer screen will do just fine: www.reddit.com/r/depression/comments/6jtamj/im_at_work_just_staring_at_my_computer_screen/

And then he quotes form the Whole Earth Catalog, a paper Atlas from the '70s he admired:

Stay Hungry. Stay Foolish

Video 1. Source.

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Lie group by

Ciro Santilli 37  Updated 2025-07-11  +Created 1970-01-01

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The key and central motivation for studying Lie groups and their Lie algebras appears to be to characterize symmetry in Lagrangian mechanics through Noether's theorem, just start from there.

Notably local symmetries appear to map to forces, and local means "around the identity", notably: local symmetries of the Lagrangian imply conserved currents.

More precisely: local symmetries of the Lagrangian imply conserved currents.

TODO Ciro Santilli really wants to understand what all the fuss is about:

Oh, there is a low dimensional classification! Ciro is a sucker for classification theorems! en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras

The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of continuous problems are simpler than discrete ones.

Bibliography: