Ciro Santilli @cirosantilli 35

To see that the real projective plane is not simply connected space, considering the lines through origin model of the real projective plane, take a loop that starts at $(1,0,0)$ and moves along the $y=0$ great circle ends at $(-1,0,0)$.

Note that both of those points are the same, so we have a loop.

Now try to shrink it to a point.

There's just no way!

Borrow from the Internet Archive for free: archive.org/details/supermenstory00murr

Initial chapters put good clarity on the formation of the military-industrial complex. Being backed by the military, especially just after World War II, was in itself enough credibility to start and foster a company.

It is funny to see how the first computers were very artisanal, made on a one-off basis.

Amazing how Control Data Corporation raised capital IPO style as a startup without a product. The dude was selling shares at dinner parties in his home.

Very interesting mention on page 70 of how Israel bought CDC's UNIVAC 1103 which Cray contributed greatly to design, and everyone knew that it was to make thermonuclear weapons, since that was what the big American labs like this mention should be added to: en.wikipedia.org/wiki/Nuclear_weapons_and_Israel but that's Extended Protected... the horrors of Wikipedia.

Another interesting insight is how "unintegrated" computers were back then. They were literally building computers out of individual vacuum tubes, then individual semiconducting transistors, a gate at a time. Then things got more and more integrated as time went. That is why the now outdated word "microprocessor" existed. When processors start to fit into a single integrated circuit, they were truly micro compared to the monstrosities that existed previously.

Also, because integration was so weak initially, it was important to more manually consider the length of wire signals had to travel, and try to put components closer together to reduce the critical path to be able to increase clock speeds. These constraints are also of course present in modern computer design, but they were just so much more visible in those days.

The book does unfortunately not give much detail in Crays personal life as mentioned on this book review: www.goodreads.com/review/show/1277733185?book_show_action=true. His childhood section is brief, and his wedding is described in one paragraph, and divorce in one sentence. Part of this is because he was very private about his family most likely note how Wikipedia had missed his first wedding, and likely misattribute children to the second wedding; en.wikipedia.org/wiki/Talk:Seymour_Cray section "Weddings and Children".

Crays work philosophy is is highlighted many times in the book, and it is something worthy to have in mind:

Cray's final downfall was when he opted to try to use a promising but hard to work with material gallium arsenide instead of silicon as his way to try and speed up computers, see also: gallium arsenide vs silicon. Also, he went against the extremely current of the late 80's early 90's pointing rather towards using massively parallel systems based on silicon off-the-shelf Intel processors, a current that had DARPA support, and which by far the path that won very dramatically as of 2020, see: Intel supercomputer market share.

Examples under vhdl, more details at: GHDL.

Simulate it. Just simulate it.

Let's start with the one dimensional case. Let the $q:\mathbb{R}\to \mathbb{R}$ and a Functional $I$ defined by a function of three variables $F:{\mathbb{R}}^3\to \mathbb{R}$:

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given $F$, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that $F$ such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

By $\frac{\partial F}{\partial q}$ and $\frac{\partial F}{\partial \dot{q}}$ we simply mean "the partial derivative of $F$ with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where $q$ is the unknown, since:

Now let's think about the multi-dimensional case. Instead of having $q:\mathbb{R}\to \mathbb{R}$, we now have $q:\mathbb{R}\to {\mathbb{R}}^n$. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case, $F$ is going to be a function of 5 variables $F:{\mathbb{R}}^5\to \mathbb{R}$ rather than 3 as in the one dimensional case, and the functional looks like:

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions $q_1$ and $q_2$ of order up to 2 in both variables:At this point, notation is getting a bit clunky, so people will often condense the $q_i$ vectoror just omit the arguments of $F$ entirely:

Theoretical framework on which quantum field theories are based, theories based on framework include:so basically the entire Standard Model

The basic idea is that there is a field for each particle particle type.

E.g. in QED, one for the electron and one for the photon: physics.stackexchange.com/questions/166709/are-electron-fields-and-photon-fields-part-of-the-same-field-in-qed.

And then those fields interact with some Lagrangian.

One way to look at QFT is to split it into two parts:Then interwined with those two is the part "OK, how to solve the equations, if they are solvable at all", which is an open problem: Yang-Mills existence and mass gap.

There appear to be two main equivalent formulations of quantum field theory:

RSA vs Diffie-Hellman key exchange are the dominant public-key cryptography systems as of 2020, so it is natural to ask how they compare:

As its name indicates, Diffie-Hellman key exchange is a key exchange algorithm. TODO verify: this means that in order to transmit a message, both parties must first send data to one another to reach a shared secret key. For RSA on the other hand, you can just take the public key of the other party and send encrypted data to them, the receiver does not need to send you any data at any point.

.data is section 1: