In 1971, investigators showed that a wide range of these very hard problems were mathematically equivalent: a solution to one would solve all.

Moreover, the following have been shown to be mathematically equivalent:

In these cases, the above three conditions are not mathematically equivalent.

Thus, these two definitions lead to mathematically equivalent classes of objects.

Directed pseudoforests and functions are in some sense mathematically equivalent.

The two forms of Gauss's law for gravity are mathematically equivalent.

Apollonius demonstrated that these two models were in fact mathematically equivalent.

In the ideal continuous case, the two approaches are mathematically equivalent.

They are mathematically equivalent in 3 dimensions, though the approaches differ.

Such a point would be mathematically equivalent to the sum of all other points in time and space.