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A manifold with a Riemannian metric that has zero curvature is a flat manifold. The basic example is Euclidean space with the usual metric ds^2=sum_(i)dx_i^2. In fact, any ...

Given two groups G and H, there are several ways to form a new group. The simplest is the direct product, denoted G×H. As a set, the group direct product is the Cartesian ...

Let P be a class of (universal) algebras. Then an algebra A is a local P-algebra provided that every finitely generated subalgebra F of A is a member of the class P. Note ...

The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...

Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication (Wolfram 2002, pp. 5 and 716-717). More specifically, the ...

A topological space X is semilocally simply connected (also called semilocally 1-connected) if every point x in X has a neighborhood U such that any loop L:[0,1]->U with ...

The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a ...

An Abelian variety which is canonically attached to an algebraic variety which is the solution to a certain universal problem. The Albanese variety is dual to the Picard ...

Suppose that X^~,X are arcwise-connected and locally arcwise-connected topological spaces. Then (X^~,p) is said to be a covering space of X if p:X^~->X is a surjective ...

The distance p (sometimes also denoted k) from the focus to the conic section directrix of a conic section. The following table gives the focal parameter for the different ...