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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 positive integers are the numbers 1, 2, 3, ... (OEIS A000027), sometimes called the counting numbers or natural numbers, denoted Z^+. They are the solution to the simple ...

Given a topological vector space X and a neighborhood V of 0 in X, the polar K=K(V) of V is defined to be the set K(V)={Lambda in X^*:|Lambdax|<=1 for every x in V} and where ...

If V and W are Banach spaces and T:V->W is a bounded linear operator, the T is said to be a compact operator if it maps the unit ball of V into a relatively compact subset of ...

A two-dimensional piecewise linear map defined by x_(n+1) = 1-y_n+|x_n| (1) y_(n+1) = x_n. (2) The map is chaotic in the filled region above and stable in the six hexagonal ...

For the hyperbolic partial differential equation u_(xy) = F(x,y,u,p,q) (1) p = u_x (2) q = u_y (3) on a domain Omega, Goursat's problem asks to find a solution u(x,y) of (3) ...

The so-called rule of three is an educational tool utilized historically to verbalize the process of solving basic linear equations with four terms where three of the terms ...

A tesseral harmonic is a spherical harmonic of the form cos; sin(mphi)P_l^m(costheta). These harmonics are so named because the curves on which they vanish are l-m parallels ...

A zonal harmonic is a spherical harmonic of the form P_l(costheta), i.e., one which reduces to a Legendre polynomial (Whittaker and Watson 1990, p. 302). These harmonics are ...