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Let gamma(t) be a smooth curve in a manifold M from x to y with gamma(0)=x and gamma(1)=y. Then gamma^'(t) in T_(gamma(t)), where T_x is the tangent space of M at x. The ...

If L^~ is a linear operator on a function space, then f is an eigenfunction for L^~ and lambda is the associated eigenvalue whenever L^~f=lambdaf. Renteln and Dundes (2005) ...

A metric topology induced by the Euclidean metric. In the Euclidean topology of the n-dimensional space R^n, the open sets are the unions of n-balls. On the real line this ...

Consider a finite collection of points p=(p_1,...,p_n), p_i in R^d Euclidean space (known as a configuration) and a graph G whose graph vertices correspond to pairs of points ...

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), ...

An algebraic ring which appears in treatments of duality in algebraic geometry. Let A be a local Artinian ring with m subset A its maximal ideal. Then A is a Gorenstein ring ...

Let X be a metric space, A be a subset of X, and d a number >=0. The d-dimensional Hausdorff measure of A, H^d(A), is the infimum of positive numbers y such that for every ...

Let Gamma be an algebraic curve in a projective space of dimension n, and let p be the prime ideal defining Gamma, and let chi(p,m) be the number of linearly independent ...

On the class of topological spaces, a homeomorphism class is an equivalence class under the relation of being homeomorphic. For example, the open interval (-pi/2,pi/2) and ...

A function of the coordinates which is constant along a trajectory in phase space. The number of degrees of freedom of a dynamical system such as the Duffing differential ...