This is an important metric, because it takes some time for the quantum operations to propagate, and so the depth of a circuit gives you an idea of how long the coherence time a hardware needs to support a given circuit.
Bibliography:
Bibliography:
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:
AGI-complete in general? Obviously. But still, a lot can be done. See e.g.:
- The Busy Beaver Challenge deciders
Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p Updated 2025-05-23 +Created 1970-01-01
This construction takes as input:and it produces an elliptic curve over a finite field of order as output.
The constructions is used in the Birch and Swinnerton-Dyer conjecture.
To do it, we just convert the coefficients and from the Equation "Definition of the elliptic curves" from rational numbers to elements of the finite field.
For the denominator , we just use the multiplicative inverse, e.g. supposing we havewhere because , related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p
Relationship between the quotient group and direct products Updated 2025-05-23 +Created 1970-01-01
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.
This is also semi mentioned at: math.stackexchange.com/questions/1596500/when-is-a-group-isomorphic-to-the-product-of-normal-subgroup-and-quotient-group
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.
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:
n = p*q- a randomly chosen integer exponent between
1ande_max = lcm(p -1, q -1), wherelcmis the Least common multiple
Given a plaintext message This operation is called modular exponentiation can be calculated efficiently with the Extended Euclidean algorithm.
m, the encrypted ciphertext version is:c = m^e mod nThe 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.Bibliography:
- www.comparitech.com/blog/information-security/rsa-encryption/ has a numeric example
Front-end web framework integration: no native one:
- React:
- Vue.js:
- github.com/mikermcneil/ration Issue tracker disabled...
- live at: ration.io/
- selling a course at: courses.platzi.com/courses/sails-js/
- platzi.com/cursos/javascript-pro/ non-free and in Spanish pointed to from official README...
- Nuxt.js:
- github.com/mikermcneil/ration Issue tracker disabled...
TODO server-side rendering anyone??
- stackoverflow.com/questions/32412590/how-to-use-react-js-to-render-server-side-template-on-sails-js
- stackoverflow.com/questions/54217147/ssr-for-react-redux-application-with-sails
- gist.github.com/duffpod/746a660bcddfd986878c92dde1a04f06
- www.reddit.com/r/reactjs/comments/7saoqm/sailsjs_or_adonisjs_designed_for_server_side/
Since Snakes and Ladders is nothing but a Absorbing Markov chain, the results are exactly the same as for that general problem.
www.jstor.org/stable/3619261: How Long Is a Game of Snakes and Ladders? by Althoen, King and Schilling (1993), paywalled.
A very honest review of my Oxford University master's degree (theoretical physics at keble college) by alicedoesphysics (2020)
Source. Basically all her courses are from the Mathematical Institute of the University of Oxford, and therefore show up at the Moodle of the Oxford Mathematics Institute of Oxford.- qubit.guide/ HTML version od the book.
- github.com/thosgood/qubit.guide. Source code. Written in Bookdown.
- www.arturekert.org/iqis links to the lectures: www.youtube.com/@ArturEkert/playlists Well done in splitting those videos up!
- zhenyucai.com/post/intro_to_qi/
Interesting presentation cycle at Merton BTW: www.arturekert.org/teaching/merton
Basically a precise statement of "quantum entanglement is spooky".
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