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Euler integration was defined by Schanuel and subsequently explored by Rota, Chen, and Klain. The Euler integral of a function f:R->R (assumed to be piecewise-constant with ...

Since |(a+ib)(c+id)| = |a+ib||c+di| (1) |(ac-bd)+i(bc+ad)| = sqrt(a^2+b^2)sqrt(c^2+d^2), (2) it follows that (a^2+b^2)(c^2+d^2) = (ac-bd)^2+(bc+ad)^2 (3) = e^2+f^2. (4) This ...

A continuous real function L(x,y) defined on the tangent bundle T(M) of an n-dimensional smooth manifold M is said to be a Finsler metric if 1. L(x,y) is differentiable at ...

Any collineation from P(V) to P(V), where V is a three-dimensional vector space, is associated with a semilinear map from V to V.

The standard Gauss measure of a finite dimensional real Hilbert space H with norm ||·||_H has the Borel measure mu_H(dh)=(sqrt(2pi))^(-dim(H))exp(1/2||h||_H^2)lambda_H(dh), ...

The four-dimensional version of the gradient, encountered frequently in general relativity and special relativity, is del _mu=[1/cpartial/(partialt); partial/(partialx); ...

A special case of a flag manifold. A Grassmann manifold is a certain collection of vector subspaces of a vector space. In particular, g_(n,k) is the Grassmann manifold of ...

A one-dimensional transform which makes use of the Haar functions.

Consider a one-dimensional Hamiltonian map of the form H(p,q)=1/2p^2+V(q), (1) which satisfies Hamilton's equations q^. = (partialH)/(partialp) (2) p^. = ...

A handlebody of type (n,k) is an n-dimensional manifold that is attained from the standard n-disk by attaching only k-D handles.