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A ring is called left (respectively, right) Noetherian if it does not contain an infinite ascending chain of left (respectively, right) ideals. In this case, the ring in ...

According to many authors (e.g., Kelley 1955, p. 112; Joshi 1983, p. 162; Willard 1970, p. 99) a normal space is a topological space in which for any two disjoint closed sets ...

The prescription that a trigonometry identity can be converted to an analogous identity for hyperbolic functions by expanding, exchanging trigonometric functions with their ...

The parallelogram law gives the rule for vector addition of vectors A and B. The sum A+B of the vectors is obtained by placing them head to tail and drawing the vector from ...

If a function has a Fourier series given by f(x)=1/2a_0+sum_(n=1)^inftya_ncos(nx)+sum_(n=1)^inftyb_nsin(nx), (1) then Bessel's inequality becomes an equality known as ...

A group that coincides with its commutator subgroup. If G is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group. It follows that ...

An operator which describes the time evolution of densities in phase space. The operator can be defined by rho_(n+1)=L^~rho_n, where rho_n are the natural invariants after ...

A piecewise linear function is a function composed of some number of linear segments defined over an equal number of intervals, usually of equal size. For example, consider ...

Let u and v be any functions of a set of variables (q_1,...,q_n,p_1,...,p_n). Then the expression ...

A second-order partial differential equation arising in physics, del ^2psi=-4pirho. If rho=0, it reduces to Laplace's equation. It is also related to the Helmholtz ...