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A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from A to B is an object f such that every a in A ...

Given a function f(x_1,...,x_n) defined on a domain U, the graph of f is defined as the set of points (which often form a curve or surface) showing the values taken by f over ...

Given a rectangle having sides in the ratio 1:phi, the golden ratio phi is defined such that partitioning the original rectangle into a square and new rectangle results in a ...

The volumes of any n n-dimensional solids can always be simultaneously bisected by a (n-1)-dimensional hyperplane. Proving the theorem for n=2 (where it is known as the ...

Let three similar isosceles triangles DeltaA^'BC, DeltaAB^'C, and DeltaABC^' be constructed on the sides of a triangle DeltaABC. Then DeltaABC and DeltaA^'B^'C^' are ...

The Legendre differential equation is the second-order ordinary differential equation (1-x^2)(d^2y)/(dx^2)-2x(dy)/(dx)+l(l+1)y=0, (1) which can be rewritten ...

Roughly speaking, a matroid is a finite set together with a generalization of a concept from linear algebra that satisfies a natural set of properties for that concept. For ...

For 2<=n<=32, it is possible to select 2n lattice points with x,y in [1,n] such that no three are in a straight line (where "straight line" means any line in the plane--not ...

The word "pole" is used prominently in a number of very different branches of mathematics. Perhaps the most important and widespread usage is to denote a singularity of a ...

Hardy and Littlewood (1914) proved that the sequence {frac(x^n)}, where frac(x) is the fractional part, is equidistributed for almost all real numbers x>1 (i.e., the ...