A generalization of the Pythagorean triple infinity question.

Dual vectors are the members of a dual space.

In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:The dual basis vectors $e^i$ are defined to "pick the corresponding coordinate" out of elements of V. E.g.:By expanding into the basis, we can put this more succinctly with the Kronecker delta as:

Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector $f$ as:

In the context of quantum mechanics, the bra notation is also used for dual vectors.

A good definition is that the sparse matrix has non-zero entries proportional the number of rows. Therefore this is Big O notation less than something that has $N^2$ non zero entries. Of course, this only makes sense when generalizing to larger and larger matrices, otherwise we could take the constant of proportionality very high for one specific matrix.

Of course, this only makes sense when generalizing to larger and larger matrices, otherwise we could take the constant of proportionality very high for one specific matrix.

A member of the underlying field of a vector space. E.g. in ${\mathbb{R}}^3$, the underlying field is $\mathbb{R}$, and a scalar is a member of $\mathbb{R}$, i.e. a real number.

You just map the value (1, 1) $C_m\times C_n$ to the value 1 of $C_{mn}$, and it works out. E.g. for $C_2\times C_3$, the group generated by of (1, 1) is:

Suppose we have a given permutation group that acts on a set of n elements.

If we pick k elements of the set, the stabilizer subgroup of those k elements is a subgroup of the given permutation group that keeps those elements unchanged.

Note that an analogous definition can be given for non-finite groups. Also note that the case for all finite groups is covered by the permutation definition since all groups are isomorphic to a subgroup of the symmetric group

TODO existence and uniqueness. Existence is obvious for the identity permutation, but proper subgroup likely does not exist in general.

Bibliography:

Oh, and if it weren't enough, mathematicians have a separate name for the damned nabla symbol : "del" instead of "nabla".

TODO why is it called "Del"? Is is because it is an inverted uppercase delta?

Just use limit instead, please. The French are particularly guilty of this.

The non-regular version of the hypercube.

Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".

Furthermore, the non-compact part is always isomorphic to ${\mathbb{R}}^n$, only the non-compact part can have more interesting structure.

Specials sub case of the general linear group when the determinant equals exactly 1.

Group of rotations of a rigid body.

Like orthogonal group but without reflections. So it is a "special case" of the orthogonal group.

This is a subgroup of both the orthogonal group and the special linear group.