Elliptic curve by Ciro Santilli 40 Updated 2025-07-16
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
.
Figure 1.
Plots of real elliptic curves for various values of and
. Source.
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.
Lebesgue integral by Ciro Santilli 40 Updated 2025-07-16
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.
A set of theorems that prove under different conditions that the Fourier transform has an inverse for a given space, examples:
Fraunhofer diffraction by Ciro Santilli 40 Updated 2025-07-16
Far field approximation to Kirchhoff's diffraction formula, i.e. when the plane of observation is far from the object diffracting.
Control theory by Ciro Santilli 40 Updated 2025-07-16
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?
It's quite fun from a mathematics point of view!
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.
Real projective line by Ciro Santilli 40 Updated 2025-07-16
Just a circle.
Take with a line at . Identify all the points that an observer
We 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".
Then:
so we reach:
With this relation, we can easily determine the Lie algebra of common isometries:
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".

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