Furthermore, the fields are required to span the ring of functions.
2.
Commutative algebra makes great use of rings of functions of algebraic geometric objects.
3.
Such action induces one on the ring of functions.
4.
Thus, the ring of functions can be taken to be the ring of polynomial functions.
5.
The fields ( or operators ) A ^ i ( x ) are required to span the ring of functions.
6.
There have also been recent connections between these invariants, the motivic Hall algebra, and the ring of functions on the quantum torus.
7.
One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold see the discussion at the end of this article.
8.
The above can be viewed as a requirement that is imposed on a ring of functions; imposing this requirement on the fields of a conformal field theory is known as the conformal bootstrap.
9.
Two classes of localizations occur commonly in commutative algebra and algebraic geometry and are used to construct the rings of functions on open subsets in Zariski topology of the spectrum of a ring, Spec ( " R " ).
10.
Positive answers to analogues of B�chi's Problem in various other rings of functions have been obtained since then ( in the case of rings of functions, one adds the hypothesis that not all " x " " n " are constant ).
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