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The ordinary Onsager equation is the sixth-order ordinary differential equation (d^3)/(dx^3)[e^x(d^2)/(dx^2)(e^x(dy)/(dx))]=f(x) (Vicelli 1983; Zwillinger 1997, p. 128), ...

Several flavors of the open mapping theorem state: 1. A continuous surjective linear mapping between Banach spaces is an open map. 2. A nonconstant analytic function on a ...

Let S be a subset of a metric space. Then the set S is open if every point in S has a neighborhood lying in the set. An open set of radius r and center x_0 is the set of all ...

An operator A:f^((n))(I)|->f(I) assigns to every function f in f^((n))(I) a function A(f) in f(I). It is therefore a mapping between two function spaces. If the range is on ...

The operator norm of a linear operator T:V->W is the largest value by which T stretches an element of V, ||T||=sup_(||v||=1)||T(v)||. (1) It is necessary for V and W to be ...

Let T be a linear operator on a separable Hilbert space. The spectrum sigma(T) of T is the set of lambda such that (T-lambdaI) is not invertible on all of the Hilbert space, ...

The ophiuride is a cubic curve (left figure) given by the implicit equation x(x^2+y^2)+(ax-by)y=0, (1) where a>0,b>=0, or by the polar equation ...

Physicists and engineers use the phrase "order of magnitude" to refer to the smallest power of ten needed to represent a quantity. Two quantities A and B which are within ...

To solve the system of differential equations (dx)/(dt)=Ax(t)+p(t), (1) where A is a matrix and x and p are vectors, first consider the homogeneous case with p=0. The ...

Ore algebra is an algebra of noncommutative polynomials representing linear operators for functional equations such as linear differential or difference equations. Ore ...