Ciro Santilli @cirosantilli 37

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Types:

A Ring can be seen as a generalization of a field where:

Addition however has to be commutative and have inverses, i.e. it is an Abelian group.

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.

A polynomial ring is another example with the same properties as the integers.

The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that $GL(n)$ is not a ring because you can by addition reach the zero matrix.

Linear combination of a Dirichlet boundary condition and Neumann boundary condition at each point of the boundary.

Examples:

Collection of coordinate charts.

The key element in the definition of a manifold.

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:

Every Lie algebra corresponds to a single simply connected Lie group.

The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.

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: