Archive Team Updated 2025-07-16
Area of law Updated 2025-07-16
Clothing Updated 2025-07-16
Cryogenics Updated 2025-07-16
Electronics Updated 2025-07-16
Exoticism Updated 2025-07-16
Video 1.
Emmenez-moi by Charles Aznavour (1968)
Source. The ultimate ode to exoticism.
Geometry Updated 2025-07-16
Atlas (topology) Updated 2025-07-16
Collection of coordinate charts.
The key element in the definition of a manifold.
Numerical analysis Updated 2025-07-16
Techniques to get numerical approximations to numeric mathematical problems.
The entire field comes down to estimating the true values with a known error bound, and creating algorithms that make those error bounds asymptotically smaller.
Not the most beautiful field of pure mathematics, but fundamentally useful since we can't solve almost any useful equation without computers!
The solution visualizations can also provide valuable intuition however.
Important numerical analysis problems include solving:
Although quotients look a bit real number division, there are some important differences with the "group analog of multiplication" of direct product of groups.
If a group is isomorphic to the direct product of groups, we can take a quotient of the product to retrieve one of the groups, which is somewhat analogous to division: math.stackexchange.com/questions/723707/how-is-the-quotient-group-related-to-the-direct-product-group
The "converse" is not always true however: a group does not need to be isomorphic to the product of one of its normal subgroups and the associated quotient group. The wiki page provides an example:
Given G and a normal subgroup N, then G is a group extension of G/N by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and G/N. This is a special case of the extension problem. An example where the extension is not split is as follows: Let , and which is isomorphic to Z2. Then G/N is also isomorphic to Z2. But Z2 has only the trivial automorphism, so the only semi-direct product of N and G/N is the direct product. Since Z4 is different from Z2 × Z2, we conclude that G is not a semi-direct product of N and G/N.
TODO find a less minimal but possibly more important example.
I think this might be equivalent to why the group extension problem is hard. If this relation were true, then taking the direct product would be the only way to make larger groups from normal subgroups/quotients. But it's not.
RSA (cryptosystem) Updated 2025-07-16
Based on the fact that we don't have a P algorithm for integer factorization as of 2020. But nor proof that one does not exist!
The private key is made of two randomly generated prime numbers: and . How such large primes are found: how large primes are found for RSA.
The public key is made of:
Given a plaintext message m, the encrypted ciphertext version is:
c = m^e mod n
This operation is called modular exponentiation can be calculated efficiently with the Extended Euclidean algorithm.
The inverse operation of finding the private m from the public c, e and is however believed to be a hard problem without knowing the factors of n.
However, if we know the private p and q, we can solve the problem. As follows.
First we calculate the modular multiplicative inverse. TODO continue.
Russia Updated 2025-07-16
Automated theorem proving Updated 2025-07-16
AGI-complete in general? Obviously. But still, a lot can be done. See e.g.:

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