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Suppose f(x) is a function of x that is twice differentiable at a stationary point x_0. 1. If f^('')(x_0)>0, then f has a local minimum at x_0. 2. If f^('')(x_0)<0, then f ...

A number of areas of mathematics have the notion of a "dual" which can be applies to objects of that particular area. Whenever an object A has the property that it is equal ...

A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal. If K is a field, the ...

A set S and a binary operator * are said to exhibit closure if applying the binary operator to two elements S returns a value which is itself a member of S. The closure of a ...

A sheaf is a presheaf with "something" added allowing us to define things locally. This task is forbidden for presheaves in general. Specifically, a presheaf F on a ...

A fractal which can be written as a Lindenmayer system with initial string "YF", string rewriting rules "X" -> "YF+XF+Y", "Y" -> "XF-YF-X", and angle 60 degrees.

A pathwise-connected domain is said to be simply connected (also called 1-connected) if any simple closed curve can be shrunk to a point continuously in the set. If the ...

Simpson's paradox, also known as the amalgamation paradox, reversal paradox, or Yule-Simpson effect, is a paradox in which a statistical trend appears to be present when data ...

Given an ordinary differential equation y^'=f(x,y), the slope field for that differential equation is the vector field that takes a point (x,y) to a unit vector with slope ...

Another word for a C^infty (infinitely differentiable) manifold, also called a differentiable manifold. A smooth manifold is a topological manifold together with its ...