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Let f be analytic on the unit disk, and assume that 1. |f(z)|<=1 for all z and 2. f(0)=0. Then |f(z)|<=|z| and |f^'(0)|<=1. If either |f(z)|=|z| for some z!=0 or if ...

A family composed of countably many discrete families.

A smooth map f:S^1->R^3 whose image has singularities. In particular, in the theory of Vassiliev's knot invariants, singular knots with a finite number of ordinary double ...

Let a, b, and c be the sides of a spherical triangle, then the spherical defect is defined as D=2pi-(a+b+c).

A space which is isomorphic to a Borel subset B of a Polish space equipped with its sigma-algebra of Borel sets.

Defined for samples x_i, i=1, ..., N by alpha_r=1/Nsum_(i=1)^Nz_i^r=(mu_r)/(sigma^r), (1) where z_i=(x_i-x^_)/(s_x). (2) The first few are alpha_1 = 0 (3) alpha_2 = 1 (4) ...

The symbol ∼ used to denote similarity, equivalence relations, or asymptosy.

A symmetry group is a group of symmetry-preserving operations, i.e., rotations, reflections, and inversions (Arfken 1985, p. 245).

In nonstandard analysis, the transfer principle is the technical form of the following intuitive idea: "Anything provable about a given superstructure V by passing to a ...

An ordering of permutations in which each two adjacent permutations differ by the transposition of two elements. For the permutations of {1,2,3} there are two listings which ...