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A trail is a walk v_0, e_1, v_1, ..., v_k with no repeated edge. The length of a trail is its number of edges. A u,v-trail is a trail with first vertex u and last vertex v, ...

An exchange of two elements of an ordered list with all others staying the same. A transposition is therefore a permutation of two elements. For example, the swapping of 2 ...

The height of a tree g is defined as the vertex height of its root vertex, where the vertex height of a vertex v in a tree g is the number of edges on the longest downward ...

Triangle geometry is the study of the properties of triangles, including associated triangle centers, triangle lines, central circles, triangle cubics, and many others. These ...

A transformation x^'=Ax is unimodular if the determinant of the matrix A satisfies det(A)=+/-1. A necessary and sufficient condition that a linear transformation transform a ...

In an integral domain R, the decomposition of a nonzero noninvertible element a as a product of prime (or irreducible) factors a=p_1...p_n, (1) is unique if every other ...

The vertex height of a vertex v in a rooted tree is the number of edges on the longest downward path between v and a tree leaf. The height of the root vertex of a rooted tree ...

The Voronov-Neopryatnaya-Dergachev graphs are two graphs on 372 and 972 vertices which have unit-distance embeddings with all vertices on a sphere and chromatic number 5. The ...

Let X_1, ..., X_N be a sequence of independent observations of a random variable X, and let the number of observations N itself be chosen at random. Then Wald's equation ...

There are at least two theorems known as Weierstrass's theorem. The first states that the only hypercomplex number systems with commutative multiplication and addition are ...