Dictionary · Wolfram MathWorld: The Web's Most Extensive Mathematics Resource

An embedding is a representation of a topological object, manifold, graph, field, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. For example, a field embedding preserves the algebraic structure of plus and times, an embedding of a topological space preserves open sets, and a graph embedding preserves connectivity. One space X is embedded in another space Y when the properties of Y restricted to X are the same as the properties of X.

The dimension of an object is a topological measure of the size of its covering properties. Roughly speaking, it is the number of coordinates needed to specify a point on the object.

(linear) An operator L^~ is said to be linear if, for every pair of functions f and g and scalar t, L^~(f+g)=L^~f+L^~g and L^~(tf)=tL^~f.

(logistic equation) The continuous version of the logistic model is described by the differential equation (dN)/(dt)=(rN(K-N))/K, (1) where r is the Malthusian parameter (rate of maximum population growth) and K is the so-called carrying capacity (i.e., the maximum sustainable population). Dividing both sides by K and defining x=N/K then gives the differential equation (dx)/(dt)=rx(1-x), (2) which is known as the logistic equation and has solution x(t)=1/(1+(1/(x_0)-1)e^(-rt)). (3) The function x(t) is sometimes known as the sigmoid function.