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Partial label partial derivative notation by

Ciro Santilli 40 Updated 2025-07-16

Total derivative by

Ciro Santilli 40 Updated 2025-07-16

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:

T o t a l d er i v a t i v e = D [f (x_{0})] (x) = f (x_{0}) + \frac{\partial f}{\partial x} (x_{0}) \times x

and in 2D:

D [f (x_{0}, y_{0})] (x, y) = f (x_{0}, y_{0}) + \frac{\partial f}{\partial x} (x_{0}, y_{0}) \times x + \frac{\partial f}{\partial y} (x_{0}, y_{0}) \times y

(2)

Volume by

Ciro Santilli 40 Updated 2025-07-16

3D area.

Riesz-Fischer theorem by

Ciro Santilli 40 Updated 2025-07-16

A measurable function defined on a closed interval is square integrable (and therefore in

L^{2}

) if and only if Fourier series converges in

L^{2}

norm the function:

lim_{N \to \infty} ∥ S_{N} f - f ∥_{2} = 0

Mordell's theorem by

Ciro Santilli 40 Updated 2025-07-16

The elliptic curve group of all elliptic curve over the rational numbers is always a finitely generated group.

The number of points may be either finite or infinite. But when infinite, it is still a finitely generated group.

For this reason, the rank of an elliptic curve over the rational numbers is always defined.

TODO example.

Rank of an elliptic curve over the rational numbers by

Ciro Santilli 40 Updated 2025-07-16

Mordell's theorem guarantees that the rank (number of elements in the generating set of the group) is always well defined for an elliptic curve over the rational numbers. But as of 2023 there is no known algorithm which calculates the rank of any curve!

It is not even known if there are elliptic curves of every rank or not: Largest known ranks of an elliptic curve over the rational numbers, and it has proven extremely hard to find new ones over time.

TODO list of known values and algorithms? The Birch and Swinnerton-Dyer conjecture would immediately provide a stupid algorithm for it.

Birch and Swinnerton-Dyer conjecture by

Ciro Santilli 40 Updated 2026-08-21

The BSD conjecture states that if your name is long enough, it will always count as two letters on a famous conjecture.

Maybe also insert a joke about BSD Operating Systems if you're into that kind of stuff.

The conjecture states that Equation 1. "BSD conjecture" holds for every elliptic curve over the rational numbers (which is defined by its constants

a

and

b

)

lim_{x \to \infty} \prod_{p \leq x} \frac{N _{p}}{p} = C lo g (x)^{r}

Equation 1.

BSD conjecture

. Where the following numbers are defined for the elliptic curve we are currently considering, defined by its constants

a

and

b

$N_{p}$ : number of elements of the elliptic curve over the finite field, where the finite field comes from the reduction of an elliptic curve from $E (Q)$ to $E (F_{p}) mod p$ . $N_{p}$ can be calculated algorithmically with Schoof's algorithm in polynomial time
$r$ : rank of the elliptic curve over the rational numbers. We don't really have a good general way to calculate this besides this conjecture (?).
$C$ : a constant

The conjecture, if true, provides a (possibly inefficient) way to calculate the rank of an elliptic curve over the rational numbers, since we can calculate the number of elements of an elliptic curve over a finite field by Schoof's algorithm in polynomial time. So it is just a matter of calculating

N_{p}

like that up to some point at which we are quite certain about

r

The Wikipedia page of the this conecture is the perfect example of why it is not possible to teach natural sciences on Wikipedia. A million dollar problem, and the page is thoroughly incomprehensible unless you already know everything!

Video 1.

Birch and Swinnerton-Dyer conjecture by Kinertia (2020)

Source.

Video 2.

The $1,000,000 Birch and Swinnerton-Dyer conjecture by Absolutely Uniformly Confused (2022)

Source. A respectable 1 minute attempt. But will be too fast for most people. The sweet spot is likely 2 minutes.

Winston Churchill by

Ciro Santilli 40 Updated 2025-07-16

BSD conjecture bibliography by

Ciro Santilli 40 Updated 2025-07-16

Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli by

Ciro Santilli 40 Updated 2025-07-16

Summary:

overview of the formula of the BSD conjecture
definition of elliptic curve
domain of an elliptic curve. Prerequisite: field
elliptic curve group. Prerequisite: group
Mordell's theorem lets us define the rank of an elliptic curve over the rational numbers, which is the $r$ . Prerequisite: generating set of a group
reduction of an elliptic curve from $E (Q)$ to $E (F_{p}) mod p$ lets us define $N_{r}$ as the number of elements of the generated finite group

Video 1. Source.

Elliptic curve over a finite field by

Ciro Santilli 40 Updated 2025-07-16

The Equation "Definition of the elliptic curves" and definitions on elliptic curve point addition both hold directly.

Schoof's algorithm by

Ciro Santilli 40 Updated 2025-07-16

Chemical bond by

Ciro Santilli 40 Updated 2025-07-16

Calculus by

Ciro Santilli 40 Updated 2025-07-16

Well summarized as "the branch of mathematics that deals with limits".

Limit (mathematics) by

Ciro Santilli 40 Updated 2025-07-16

The fundamental concept of calculus!

The reason why the epsilon delta definition is so venerated is that it fits directly into well known methods of the formalization of mathematics, making the notion completely precise.

Convergent series by

Ciro Santilli 40 Updated 2025-07-16

Solving partial differential equations with the Fourier series by

Ciro Santilli 40 Updated 2025-07-16

See: 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!

Other basis besides the Fourier series show up for other equations, e.g.:

Chain rule by

Ciro Santilli 40 Updated 2025-07-16

Here's an example of the chain rule. Suppose we want to calculate:

\frac{d}{d x} e^{2 x}

So we have:

f (x) = e^{x} g (x) = 2 x

(2)

and so:

f^{'} (x) = e^{x} g^{'} (x) = 2

(3)

Therefore the final result is:

f^{'} (g (x)) g^{'} (x) = e^{2 x} 2 = 2 e^{2 x}

(4)

Normalized DFT by

Ciro Santilli 40 Updated 2025-07-16

There are actually two possible definitions for the DFT:

1/N, given as "the default" in many sources:
$x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{kn}{N}}$
(1)
$1/ N$ , known as the "normalized DFT" by some sources: www.dsprelated.com/freebooks/mdft/Normalized_DFT.html, definition which we adopt:
$x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{kn}{N}}$
(2)

The

1/ N

is nicer mathematically as the inverse becomse more symmetric, and power is conserved between time and frequency domains.