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A short theorem used in proving a larger theorem. Related concepts are the axiom, porism, postulate, principle, and theorem. The late mathematician P. Erdős has often been ...

A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements f, g, and h of a Lie algebra satisfy [f,f]=0 (1) [f+g,h]=[f,h]+[g,h], (2) and ...

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. ...

Smale's problems are a list of 18 challenging problems for the twenty-first century proposed by Field medalist Steven Smale. These problems were inspired in part by Hilbert's ...

Let A and B be two algebras over the same signature Sigma, with carriers A and B, respectively (cf. universal algebra). B is a subalgebra of A if B subset= A and every ...

A function f(x) is said to have bounded variation if, over the closed interval x in [a,b], there exists an M such that |f(x_1)-f(a)|+|f(x_2)-f(x_1)|+... +|f(b)-f(x_(n-1))|<=M ...

The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open ...

The following conditions are equivalent for a conservative vector field on a particular domain D: 1. For any oriented simple closed curve C, the line integral ∮_CF·ds=0. 2. ...

The operation of drilling a tubular neighborhood of a knot K in S^3 and then gluing in a solid torus so that its meridian curve goes to a (p,q)-curve on the torus boundary of ...

The exterior derivative of a function f is the one-form df=sum_(i)(partialf)/(partialx_i)dx_i (1) written in a coordinate chart (x_1,...,x_n). Thinking of a function as a ...