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The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not ...

The rectilinear crossing number of a graph G is the minimum number of crossings in a straight line embedding of G in a plane. It is variously denoted rcr(G), cr^_(G) ...

Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions ...

A catenary of revolution. The catenoid and plane are the only surfaces of revolution which are also minimal surfaces. The catenoid can be given by the parametric equations x ...

The curl of a vector field, denoted curl(F) or del xF (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at ...

The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The ...

The elliptic lambda function lambda(tau) is a lambda-modular function defined on the upper half-plane by lambda(tau)=(theta_2^4(0,q))/(theta_3^4(0,q)), (1) where tau is the ...

The Fox H-function is a very general function defined by where 0<=m<=q, 0<=n<=p, alpha_j,beta_j>0, and a_j,b_j are complex numbers such that no pole of Gamma(b_j-beta_js) for ...

A fusene is a simple planar 2-connected graph embedded in the plane with all vertices of degree 2 or 3, all bounded faces (not necessarily regular) hexagons, and all vertices ...

A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight ...