An elliptic curve is defined by numbers and . The curve is the set of all points of the real plane that satisfy the Equation 1. "Definition of the elliptic curves"
Equation 1.
Definition of the elliptic curves
. 
Equation 1. "Definition of the elliptic curves" definies elliptic curves over any field, it doesn't have to the real numbers. Notably, the definition also works for finite fields, leading to elliptic curve over a finite fields, which are the ones used in Elliptic-curve Diffie-Hellman cyprotgraphy.
"More complex and general" integral. Matches the Riemann integral for "simple functions", but also works for some "funkier" functions that Riemann does not work for.
Ciro Santilli sometimes wonders how much someone can gain from learning this besides the beauty of mathematics, since we can hand-wave a Lebesgue integral on almost anything that is of practical use. The beauty is good reason enough though.
Solving partial differential equations with the Fourier series by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16See: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from heat equation solution with Fourier series.
Separation of variables of certain equations like the heat equation and wave equation are solved immediately by calculating the Fourier series of initial conditions!
A set of theorems that prove under different conditions that the Fourier transform has an inverse for a given space, examples:
Technique to solve partial differential equations
Naturally leads to the Fourier series, see: solving partial differential equations with the Fourier series, and to other analogous expansions:
One notable application is the solution of the Schrödinger equation via the time-independent Schrödinger equation.
As mentioned at: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from solving partial differential equations with the Fourier series citing courses.maths.ox.ac.uk/node/view_material/1720, analogously to the heat equation, the wave linear equation can be be solved nicely with separation of variables.
Far field approximation to Kirchhoff's diffraction formula, i.e. when the plane of observation is far from the object diffracting.
This basically adds one more ingredient to partial differential equations: a function that we can select.
And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?
Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.
A single exponential map is not enough to recover a simple Lie group from its algebra by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16We can almost reach the Lie algebra of any isometry group in a single go. For every in the Lie algebra we must have:because has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".
Bibliography:
What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16We looking at the definition the orthogonal group is the group of all matrices that preserve the dot product, we notice that the dot product is one example of positive definite symmetric bilinear form, which in turn can also be represented by a matrix as shown at: Section "Matrix representation of a symmetric bilinear form".
By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".
- : another positive definite symmetric bilinear form such as ?
- what if we drop the positive definite requirement, e.g. ?
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 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. 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.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
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