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A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into ...

In 1657, Fermat posed the problem of finding solutions to sigma(x^3)=y^2, and solutions to sigma(x^2)=y^3, where sigma(n) is the divisor function (Dickson 2005). The first ...

A fibered category F over a topological space X consists of 1. a category F(U) for each open subset U subset= X, 2. a functor i^*:F(U)->F(V) for each inclusion i:V↪U, and 3. ...

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 ...

For a field K with multiplicative identity 1, consider the numbers 2=1+1, 3=1+1+1, 4=1+1+1+1, etc. Either these numbers are all different, in which case we say that K has ...

Let the squares square ABCD and square AB^'C^'D^' share a common polygon vertex A. The midpoints Q and S of the segments B^'D and BD^' together with the centers of the ...

Let X(x)=X(x_1,x_2,...,x_n) be a random vector in R^n and let f_X(x) be a probability distribution on X with continuous first and second order partial derivatives. The Fisher ...

A point related to the construction and properties of conic sections. Hyperbolas and noncircular ellipses have two distinct foci and two associated conic section directrices, ...

Given three coins of possibly different sizes which are arranged so that each is tangent to the other two, find the coin which is tangent to the other three coins. The ...

The fourth group isomorphism theorem, also called the lattice group isomorphism theorem, lets G be a group and let N⊴G, where N⊴G indicates that N is a normal subgroup of G. ...