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In a cochain complex of modules ...->C^(i-1)->^(d^(i-1))C^i->^(d^i)C^(i+1)->..., the module B^i of i-coboundaries is the image of d^(i-1). It is a submodule of C^i and is ...

The polar angle on a sphere measured from the north pole instead of the equator. The angle phi in spherical coordinates is the colatitude. It is related to the latitude delta ...

In mathematics, the term "collection" is generally used to mean a multiset, i.e., a set in which order is ignored but multiplicity is significant.

If the parameters of a Lie group vary over a closed interval, them the Lie group is said to be compact. Every representation of a compact group is equivalent to a unitary ...

Let suma_k and sumb_k be a series with positive terms and suppose a_1<=b_1, a_2<=b_2, .... 1. If the bigger series converges, then the smaller series also converges. 2. If ...

Two complex numbers z=x+iy and z^'=x^'+iy^' are added together componentwise, z+z^'=(x+x^')+i(y+y^'). In component form, (x,y)+(x^',y^')=(x+x^',y+y^') (Krantz 1999, p. 1).

A function f(x) is said to be concave on an interval [a,b] if, for any points x_1 and x_2 in [a,b], the function -f(x) is convex on that interval (Gradshteyn and Ryzhik 2000).

A concordance between knots K_0 and K_1 in S^3 is a locally flat cylinder C=S^1×[0,1] embedded in S^3×[0,1] in such a way that the ends S^1×{1} are embedded in S^3×{i} as ...

A topological structure which interpolates between contact structures and codimension-one foliations.

A polynomial P(x) that, when evaluated over each x in the domain of definition, results in the same value. The simplest example is P(x)=c for x in R and c a constant.