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Let {A_n}_(n=0)^infty be a sequence of events occurring with a certain probability distribution, and let A be the event consisting of the occurrence of a finite number of ...

If X is any space, then there is a CW-complex Y and a map f:Y->X inducing isomorphisms on all homotopy, homology, and cohomology groups.

Let A be a unital Banach algebra. If a in A and ||1-a||<1, then a^(-1) can be represented by the series sum_(n=0)^(infty)(1-a)^n. This criterion for checking invertibility of ...

Classical algebraic geometry is the study of algebraic varieties, both affine varieties in C^n and projective algebraic varieties in CP^n. The original motivation was to ...

A function f(x) is completely convex in an open interval (a,b) if it has derivatives of all orders there and if (-1)^kf^((2k))(x)>=0 for k=0, 1, 2, ... in that interval ...

A complex vector bundle is a vector bundle pi:E->M whose fiber bundle pi^(-1)(x) is a complex vector space. It is not necessarily a complex manifold, even if its base ...

A complex vector space is a vector space whose field of scalars is the complex numbers. A linear transformation between complex vector spaces is given by a matrix with ...

Every compact 3-manifold is the connected sum of a unique collection of prime 3-manifolds.

A continuous vector bundle is a vector bundle pi:E->M with only the structure of a topological manifold. The map pi is continuous. It has no smooth structure or bundle metric.

R^n is homeomorphic to R^m iff n=m. This theorem was first proved by Brouwer.