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For a braid with M strands, R components, P positive crossings, and N negative crossings, {P-N<=U_++M-R if P>=N; P-N<=U_-+M-R if P<=N, (1) where U_+/- are the smallest number ...

For a curve with radius vector r(t), the unit tangent vector T^^(t) is defined by T^^(t) = (r^.)/(|r^.|) (1) = (r^.)/(s^.) (2) = (dr)/(ds), (3) where t is a parameterization ...

A left Hilbert Algebra A whose involution is an antilinear isometry is called a unimodular Hilbert algebra. The involution is usually denoted xi|->xi^*.

An ordered vector basisv_1,...,v_n for a finite-dimensional vector space V defines an orientation. Another basis w_i=Av_i gives the same orientation if the matrix A has a ...

Newton's method, also called the Newton-Raphson method, is a root-finding algorithm that uses the first few terms of the Taylor series of a function f(x) in the vicinity of a ...

Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and ...

A braid with M strands and R components with P positive crossings and N negative crossings satisfies |P-N|<=2U+M-R<=P+N, where U is the unknotting number. While the second ...

Let f:M|->N be a map between two compact, connected, oriented n-dimensional manifolds without boundary. Then f induces a homomorphism f_* from the homology groups H_n(M) to ...

A category consists of three things: a collection of objects, for each pair of objects a collection of morphisms (sometimes call "arrows") from one to another, and a binary ...

A chain complex is a sequence of maps ...-->^(partial_(i+1))C_i-->^(partial_i)C_(i-1)-->^(partial_(i-1))..., (1) where the spaces C_i may be Abelian groups or modules. The ...