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If del xF=0 (i.e., F(x) is an irrotational field) in a simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the gradient of a scalar field phi(x), ...

For the rational curve of an unperturbed system with rotation number r/s under a map T (for which every point is a fixed point of T^s), only an even number of fixed points ...

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general ...

For s_1,s_2=+/-1, lim_(epsilon_1->0; epsilon_2->0)1/(x_1-is_1epsilon_1)1/(x_2-is_2epsilon_2) =[PV(1/(x_1))+ipis_1delta(x_1)][PV(1/(x_2))+ipis_2delta(x_2)] ...

Let G denote the group of germs of holomorphic diffeomorphisms of (C,0). Then if |lambda|!=1, then G_lambda is a conjugacy class, i.e., all f in G_lambda are linearizable.

Let {y^k} be a set of orthonormal vectors with k=1, 2, ..., K, such that the inner product (y^k,y^k)=1. Then set x=sum_(k=1)^Ku_ky^k (1) so that for any square matrix A for ...

Every Lie algebra L is isomorphic to a subalgebra of some Lie algebra A^-, where the associative algebra A may be taken to be the linear operators over a vector space V.

The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold.

The metric ds^2=(dx^2+dy^2)/((1-|z|^2)^2) of the Poincaré hyperbolic disk.

In its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, ...