Ciro Santilli's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in ${\mathbb{R}}^3$".

Take a open half sphere e.g. a sphere but only the points with $z>0$.

Each point in the half sphere identifies a unique line through the origin.

Then, the only lines missing are the lines in the x-y plane itself.

For those sphere points in the circle on the x-y plane, you should think of them as magic poins that are identified with the corresponding antipodal point, also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.

Ciro likes this model because then all the magic is confined just to the $z=0$ part of the model, and everything else looks exactly like the sphere.

It is useful to contrast this with the sphere itself. In the sphere, all points in the circle $z=0$ are the same point. But this is not the case for the projective plane. You cannot instantly go to any other point on the $z=0$ by just moving a little bit, you have to walk around that circle.

www.youtube.com/watch?v=tq7sb3toTww&list=PLxBAVPVHJPcrNrcEBKbqC_ykiVqfxZgNl&index=19 mentions that it is a bit like a dot product but for a tangent vector to a manifold: it measures how much that vector derives along a given direction.

In plain English: the space has no visible holes. If you start walking less and less on each step, you always converge to something that also falls in the space.

One notable example where completeness matters: Lebesgue integral of $L^p$ is complete but Riemann isn't.

When a disconnected space is made up of several smaller connected spaces, then each smaller component is called a "connected component" of the larger space.

See for example the

Given:the norm ends up being:

E.g. in ${\mathbb{C}}^2$:

This seems like a cool dude. Besides a hardcore scientist, he also made many important contributions to the French education and research system.

As if Greek letters weren't enough, physicists and mathematicians also like to make up tons of symbols, some of which look like the could actually be Greek letters!

Nabla is one of those: it was completely made up in modern times, and just happens to look like an inverted upper case delta to make things even more confusing!

Nabla means "harp" in Greek, which looks like the symbol.

Size: 10 micrometers.

Genome:

Proteins per cell: 42m: www.cell.com/pb-assets/journals/research/cell-systems/cels_384.pdf

Division time: 100 minutes.

Minimization project: en.wikipedia.org/wiki/Saccharomyces_cerevisiae#Synthetic_yeast_genome_project | syntheticyeast.org/

Takes a scalar field as input and produces a vector field.

Mnemonic: the gradient shows the direction in which the function increases fastest.

Think of a color gradient going from white to black from left to right.

Therefore, it has to:

Conceptually the simplest mesons. All of them have neutral color charge:

Prime factor exponent notation is a way to express a number as a product of its prime factors, where each prime factor is raised to an exponent that indicates how many times that factor is used in the product. This notation is particularly useful in number theory for simplifying calculations, finding factors, and understanding the properties of numbers.

The first quantum mechanics theories developed.

Their most popular formulation has been the Schrödinger equation.