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Find necessary and sufficient conditions that determine when the integral curve of two periodic functions kappa(s) and tau(s) with the same period L is a closed curve.

The underlying set of the fundamental group of X is the set of based homotopy classes from the circle to X, denoted [S^1,X]. For general spaces X and Y, there is no natural ...

Two open manifolds M and M^' are cobordant if there exists a manifold with boundary W^(n+1) such that an acceptable restrictive relationship holds.

The converse of Fisher's theorem.

Two lines that lie on top of one another are called coincident lines.

An abbreviation for the cokernel.

The logarithm of the reciprocal of a number, equal to the negative of the logarithm of the number itself, cologx=log(1/x)=-logx.

An m×1 matrix [a_(11); a_(21); |; a_(m1)].

The value of b, where a=b (mod m), taken to be nonnegative and smaller than m.

A ring is commutative if the multiplication operation is commutative.