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In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73-77): 1. The locus of the center of a variable circle, tangent to a fixed circle and ...

A plane curve is a curve that lies in a single plane. A plane curve may be closed or open. Curves which are interesting for some reason and whose properties have therefore ...

The blow-up lemma essentially says that regular pairs in Szemerédi's regularity lemma behave like complete bipartite graphs from the point of view of embedding bounded degree ...

There are a number of meanings for the word "arc" in mathematics. In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length. ...

The phrase dependent percolation is used in two-dimensional discrete percolation to describe any general model in which the states of the various graph edges (in the case of ...

The nullity of a linear transformation f:V->W of vector spaces is the dimension of its null space. The nullity and the map rank add up to the dimension of V, a result ...

Let G=(V,E) be a finite graph, let Omega be the set Omega={0,1}^E whose members are vectors omega=(omega(e):e in E), and let F be the sigma-algebra of all subsets of Omega. A ...

The self-intersection of a one-sided surface. The word "cross-cap" is sometimes also written without the hyphen as the single word "crosscap." The cross-cap can be thought of ...

A lattice polygon consisting of a closed self-avoiding walk on a square lattice. The perimeter, horizontal perimeter, vertical perimeter, and area are all well-defined for ...

A connex is a geometric form introduced by Clebsch (1872) that included as special cases the curve considered as a point locus and the curve considered as a line envelope ...