A Ring can be seen as a generalization of a field where:
- multiplication is not necessarily commutative. If this is satisfied, we can call it a commutative ring.
- multiplication may not have inverse elements. If this is satisfied, we can call it a division ring.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.
The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that is not a ring because you can by addition reach the zero matrix.
- we know that 2x2 matrix multiplication is non-commutative in general
- some 2x2 matrices have a multiplicative inverse, but others don't
Linear combination of a Dirichlet boundary condition and Neumann boundary condition at each point of the boundary.
Examples:
- In this case, the normal derivative at the boundary is proportional to the difference between the temperature of the boundary and the fixed temperature of the external environment.The result as time tends to infinity is that the temperature of the plaque tends to that of the environment.
Collection of coordinate charts.
Atomic and laser Physics subdepartment of the University of Oxford Updated 2025-06-17 +Created 1970-01-01
Some criticisms:
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 2025-06-17 +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 2025-06-17 +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
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