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The Gegenbauer polynomials C_n^((lambda))(x) are solutions to the Gegenbauer differential equation for integer n. They are generalizations of the associated Legendre ...

Keller conjectured that tiling an n-dimensional space with n-dimensional hypercubes of equal size yields an arrangement in which at least two hypercubes have an entire ...

A homogeneous ideal I in a graded ring R= direct sum A_i is an ideal generated by a set of homogeneous elements, i.e., each one is contained in only one of the A_i. For ...

Consider the circle map. If K is nonzero, then the motion is periodic in some finite region surrounding each rational Omega. This execution of periodic motion in response to ...

The maximum number of regions that can be created by n cuts using space division by planes, cube division by planes, cylinder cutting, etc., is given by N_(max)=1/6(n^3+5n+6) ...

The complex structure of a point x=x_1,x_2 in the plane is defined by the linear map J:R^2->R^2 J(x_1,x_2)=(-x_2,x_1), (1) and corresponds to a counterclockwise rotation by ...

A coordinate chart is a way of expressing the points of a small neighborhood, usually on a manifold M, as coordinates in Euclidean space. An example from geography is the ...

The average number of regions into which n randomly chosen planes divide a cube is N^_(n)=1/(324)(2n+23)n(n-1)pi+n+1 (Finch 2003, p. 482). The maximum number of regions is ...

The maximum number of pieces into which a cylinder can be divided by n oblique cuts is given by f(n) = (n+1; 3)+n+1 (1) = 1/6(n+1)(n^2-n+6) (2) = 1/6(n^3+5n+6), (3) where (a; ...

A discrete group is a topological group with the discrete topology. Often in practice, discrete groups arise as discrete subgroups of continuous Lie groups acting on a ...