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A diagram lemma which states that every short exact sequence of chain complexes and chain homomorphisms 0-->C-->^phiD-->^psiE-->0 gives rise to a long exact sequence in ...

In the usual diagram of inclusion homomorphisms, if the upper two maps are injective, then so are the other two. More formally, consider a space X which is expressible as the ...

A complex function is said to be analytic on a region R if it is complex differentiable at every point in R. The terms holomorphic function, differentiable function, and ...

Let X be a normed space, M and N be algebraically complemented subspaces of X (i.e., M+N=X and M intersection N={0}), pi:X->X/M be the quotient map, phi:M×N->X be the natural ...

A group G is said to act on a set X when there is a map phi:G×X->X such that the following conditions hold for all elements x in X. 1. phi(e,x)=x where e is the identity ...

In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively ...

There are a number of formulas variously known as Hurwitz's formula. The first is zeta(1-s,a)=(Gamma(s))/((2pi)^s)[e^(-piis/2)F(a,s)+e^(piis/2)F(-a,s)], where zeta(z,a) is a ...

The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The ...

A field automorphism of a field F is a bijective map sigma:F->F that preserves all of F's algebraic properties, more precisely, it is an isomorphism. For example, complex ...

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 ...