Ciro Santilli @cirosantilli 40

Ideally can be thought of as a one-way ticket gate that only lets electrons go in one direction with zero resistance! Real devices do have imperfections however, so there is some resistance.

First they were made out of vacuum tubes, but later semiconductor diodes were invented and became much more widespread.

Given jointly by the department of Physics of the University of Oxford and the mathematics institute of the University of Oxford.

Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

Experiments not explained: those that quantum electrodynamics explains like:See also: Dirac equation vs quantum electrodynamics.

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

Predicts fine structure.

Bibliography:

TODO: in high level terms, why is QED more general than just solving the Dirac equation, and therefore explaining quantum electrodynamics experiments?

Also, is it just a bunch of differential equation (like the Dirac equation itself), or does it have some other more complicated mathematical formulation, as seems to be the case? Why do we need something more complicated than

The main high level insight seems to be that The Dirac equation does not work for more than one electron.

Bibliography:

Generalize function to allow adding some useful things which people wanted to be classical functions but which are not,

It therefore requires you to redefine and reprove all of calculus.

For this reason, most people are tempted to assume that all the hand wavy intuitive arguments undergrad teachers give are true and just move on with life. And they generally are.

One notable example where distributions pop up are the eigenvectors of the position operator in quantum mechanics, which are given by Dirac delta functions, which is most commonly rigorously defined in terms of distribution.

Distributions are also defined in a way that allows you to do calculus on them. Notably, you can define a derivative, and the derivative of the Heaviside step function is the Dirac delta function.

Ciro Santilli's discord ID: cirosantilli#8921. See also: how to contact Ciro Santilli.

You gotta be born after the year 2000 to understand it.

This is becoming more and more popular as a group chat with channels and threads possibility as of 2020.

Very similar to Slack.

No user URLs? support.discord.com/hc/en-us/community/posts/360041519131-UserProfilesLinks

They force your username to have 4 random digits? www.reddit.com/r/discordapp/comments/43kjdl/whats_the_number_next_to_the_username/

Not possible to anonymously join just one server without creating a new account? What's the point of servers then! www.reddit.com/r/discordapp/comments/6gmjl7/changing_nick_before_joining_a_new_server/ Oh, also nicks don't hide your username from the server in any way, you can get the original username by just clicking on the person's username.

No proper threaded discussion without creating new channels? As of 2022 there is kind of a way, but it was a bit obtuse.

As of 2022 they also have a school hub: support.discord.com/hc/en-us/articles/4406046651927-Discord-Student-Hubs-FAQ which auto creates groups by university email access. Good idea, and shows popularity amongst that user group.

Servers don't have an ID to join them? www.reddit.com/r/discordapp/comments/b9zdt6/join_discord_server_from_id/

Can only make public servers if you have 1000 members?? support.discord.com/hc/en-us/articles/360023968311 Why so much bullshit?? www.reddit.com/r/discordapp/comments/6jouf8/how_do_i_make_my_server_public/

Also asked at: webapps.stackexchange.com/questions/163441/how-to-create-a-public-discord-server-that-anyone-can-join-without-an-invite

Something that is very not continuous.

Notably studied in discrete mathematics.

Input: a sequence of $N$ complex numbers $x_k$.

Output: another sequence of $N$ complex numbers $X_k$ such that:Intuitively, this means that we are braking up the complex signal into $N$ sinusoidal frequencies:and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.

Sample software implementations:

See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal

The transform still has complex numbers.

Summary:Therefore, we only need about half of $X_k$ to represent the signal, as the other half can be derived by conjugation.

"Representation in terms of sines and cosines" from www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of $X_k$.

NumPy for example has "Real FFTs" for this: numpy.org/doc/1.24/reference/routines.fft.html#real-ffts

This is the discrete logarithm problem where the group is a cyclic group.

In this case, the problem becomes equivalent to reversing modular exponentiation.

This computational problem forms the basis for Diffie-Hellman key exchange, because modular exponentiation can be efficiently computed, but no known way exists to efficiently compute the reverse function.