Ciro Santilli @cirosantilli 37

TODO find a concrete numerical example of doing calculus on a differentiable manifold and visualizing it. Likely start with a boring circle. That would be sweet...

Based on the fact that we don't have a P algorithm for the discrete logarithm of the cyclic group as of 2020, but we do have an efficient algorithm for modular exponentiation. But nor do we have proof that one does not exist! Living on the edge as usual for public-key cryptography.

Ahh, this brings good memories of Ciro Santilli's musical formative teenage years scouring the web for the best art humanity had ever produced in certain generes. And it still is a valuable resource as of the 2020's!

Reaches 2 mK^[ref]. youtu.be/upw9nkjawdy?t=487 from Video "Building a quantum computer with superconducting qubits by Daniel Sank (2019)" mentions that 15 mK are widely available.

Used for example in some times of quantum computers, notably superconducting quantum computers. As mentioned at: youtu.be/uPw9nkJAwDY?t=487, in that case we need to go so low to reduce thermal noise.

Polynomial (possibly a multivariate polynomial) with integer coefficients.

Sometimes systems of Diophantine equations are considered.

Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.

The general problem is known to be undecidable: Hilbert's tenth problem.

The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.

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:

where:

Remember that $\psi$ is a 4-vetor, gamma matrices are 4x4 matrices, so the whole thing comes down to a dot product of two 4-vectors, with a modified $\psi$ by matrix multiplication/derivatives, and the result is a scalar, as expected for a Lagrangian.

Like any other Lagrangian, you can then recover the Dirac equation, which is the corresponding equations of motion, by applying the Euler-Lagrange equation to the Lagrangian.

Specifies fixed values.

Can be used for elliptic partial differential equations and parabolic partial differential equations.

Numerical examples:

Discord is useless if you want to participate in more than one large group because of this. It is impossible to get email notification for selected threads you care about.

No way to get email notifications for missed activity? support.discord.com/hc/en-us/community/posts/360041806392-Can-we-get-an-email-notification-option-for-messages-

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

Can we do better than "wrong password implies random bytes"?

Can the last disk access times be checked via forensic methods?

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.