A "product integral" is any product-based counterpart of the usual sum-based integral of classical calculus. The first product integral (Type I below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Other examples of product integrals are the geometric integral (Type II below), the bigeometric integral (Type III below), and some other integrals of non-Newtonian calculus. Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals)...

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure (see below). The underlying space locally looks like the quotient space of a Euclidean space under the linear action of a finite group. Definitions of orbifold have been given several times: by Satake in the context of automorphic forms in the 1950s under the name V-manifold; by Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the...

The Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen. It is named after the University Erlangen-Nürnberg, where Klein worked. By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways: Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was...