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math.stackexchange.com/questions/1786964/is-it-possible-to-construct-a-smooth-flat-top-bump-function

GF(4) by

Ciro Santilli 40 Updated 2025-07-16

Ciro Santilli tried to add this example to Wikipedia, but it was reverted, so here we are, see also: Section "Deletionism on Wikipedia".

This is a good first example of a field of a finite field of non-prime order, this one is a prime power order instead.

4 = 2^{2}

, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over GF(2):

0X + 0
0X + 1
1X + 0
1X + 1

Note that we refer in this definition to anther field, but that is fine, because we only refer to fields of prime order such as GF(2), because we are dealing with prime powers only. And we have already defined fields of prime order easily previously with modular arithmetic.

Over GF(2), there is only one irreducible polynomial of degree 2:

X^{2} + X + 1

Addition is defined element-wise with modular arithmetic modulo 2 as defined over GF(2), e.g.:

(1 X + 0) + (1 X + 1) = (1 + 1) X + (0 + 1) = 0 X + 1

(2)

Multiplication is done modulo

X^{2} + X + 1

, which ensures that the result is also of degree 1.

For example first we do a regular multiplication:

(1 X + 0) \times (1 X + 1) = (1 \times 1) X^{2} + (1 \times 1) X + (0 \times 1) X + (0 \times 1) = 1 X^{2} + 1 X + 0

(3)

Without modulo, that would not be one of the elements of the field anymore due to the

1 X^{2}

So we take the modulo, we note that:

1 X^{2} + 1 X + 0 = 1 (X^{2} + X + 1) + (0 X + 1)

(4)

and by the definition of modulo:

(1 X^{2} + 1 X + 0) mod (X^{2} + X + 1) = (0 X + 1)

(5)

which is the final result of the multiplication.

TODO show how taking a reducible polynomial for modulo fails. Presumably it is for a similar reason to why things fail for the prime case.

Quadratically closed field by

Ciro Santilli 40 Updated 2025-07-16

Derivative test by

Ciro Santilli 40 Updated 2025-07-16

Saddle point by

Ciro Santilli 40 Updated 2025-07-16

Division algebra by

Ciro Santilli 40 Updated 2025-07-16

An algebra over a field where division exists.

Newton dot notation by

Ciro Santilli 40 Updated 2025-07-16

Partial derivative symbol by

Ciro Santilli 40 Updated 2025-07-16

Nope, it is not a Greek letter, notably it is not a lowercase delta. It is just some random made up symbol that looks like a letter D. Which is of course derived from delta, which is why it is all so damn confusing.

I think the symbol is usually just read as "D" as in "d f d x" for

\frac{\partial F ( x , y , z )}{\partial x}

Partial label partial derivative notation by

Ciro Santilli 40 Updated 2025-07-16

Algebraic geometry by

Ciro Santilli 40 Updated 2025-07-16

Partial index partial derivative notation by

Ciro Santilli 40 Updated 2025-07-16

This notation is not so common in basic mathematics, but it is so incredibly convenient, especially with Einstein notation as shown at Section "Einstein notation for partial derivatives":

\partial_{0} F (x, y, z) = \frac{\partial F ( x , y , z )}{\partial x} \partial_{1} F (x, y, z) = \frac{\partial F ( x , y , z )}{\partial y} \partial_{2} F (x, y, z) = \frac{\partial F ( x , y , z )}{\partial x}

This notation is similar to partial label partial derivative notation, but it uses indices instead of labels such as

x

y

, etc.

Monatomic gas by

Ciro Santilli 40 Updated 2025-07-16

Elliptic curve by

Ciro Santilli 40 Updated 2025-07-16

An elliptic curve is defined by numbers

a

and

b

. The curve is the set of all points

(x, y)

of the real plane that satisfy the Equation 1. "Definition of the elliptic curves"

y^{2} = x^{3} + a x + b

Equation 1.

Definition of the elliptic curves

Figure 1.
Plots of real elliptic curves for various values of $a$ and $b$
. 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.

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)

Area by

Ciro Santilli 40 Updated 2025-07-16

Volume by

Ciro Santilli 40 Updated 2025-07-16

3D area.

Riemann integral by

Ciro Santilli 40 Updated 2025-07-16

The easy and less generic integral. The harder one is the Lebesgue integral.

Lebesgue integral vs Riemann integral by

Ciro Santilli 40 Updated 2025-07-16

Advantages over Riemann:

Lebesgue integral of $L^{p}$ is complete but Riemann isn't.
youtu.be/PGPZ0P1PJfw?t=710 you are able to switch the order of integrals and limits of function sequences on non-uniform convergence. TODO why do we care? This is linked to the Fourier series of course, but concrete example?

Video 1.

Riemann integral vs. Lebesgue integral by The Bright Side Of Mathematics (2018)

Source.

youtube.com/watch?v=PGPZ0P1PJfw&t=808 shows how Lebesgue can be visualized as a partition of the function range instead of domain, and then you just have to be able to measure the size of pre-images.

One advantage of that is that the range is always one dimensional.

But the main advantage is that having infinitely many discontinuities does not matter.

Infinitely many discontinuities can make the Riemann partitioning diverge.

But in Lebesgue, you are instead measuring the size of preimage, and to fit infinitely many discontinuities in a finite domain, the size of this preimage is going to be zero.

So then the question becomes more of "how to define the measure of a subset of the domain".

Which is why we then fall into measure theory!

Real world applications of the Lebesgue integral by

Ciro Santilli 40 Updated 2025-07-16

In "practice" it is likely "useless", because the functions that it can integrate that Riemann can't are just too funky to appear in practice :-)

Its value is much more indirect and subtle, as in "it serves as a solid basis of quantum mechanics" due to the definition of Hilbert spaces.

Bibliography:

Lebesgue measurable by

Ciro Santilli 40 Updated 2025-07-16