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The Perko pair is the pair of knots 10_(161) and 10_(162) illustrated above. For many years, they were listed as separate knots in Little (1885) and all similar tables, ...

A piecewise linear function is a function composed of some number of linear segments defined over an equal number of intervals, usually of equal size. For example, consider ...

Given a plane ax+by+cz+d=0 (1) and a point x_0=(x_0,y_0,z_0), the normal vector to the plane is given by v=[a; b; c], (2) and a vector from the plane to the point is given by ...

The Poisson sum formula is a special case of the general result sum_(-infty)^inftyf(x+n)=sum_(k=-infty)^inftye^(2piikx)int_(-infty)^inftyf(x^')e^(-2piikx^')dx^' (1) with x=0, ...

Let chi be a nonprincipal number theoretic character over Z/Zn. Then for any integer h, |sum_(x=1)^hchi(x)|<=2sqrt(n)lnn.

The volume of a polyhedron composed of N triangular faces with vertices (a_i,b_i,c_i) can be computed using the curl theorem as V=1/6sum_(i=1)^Na_i·n_i, where the normal n_i ...

A power floor prime sequence is a sequence of prime numbers {|_theta^n_|}_n, where |_x_| is the floor function and theta>1 is real number. It is unknown if, though extremely ...

Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication (Wolfram 2002, pp. 5 and 716-717). More specifically, the ...

Pythagoras's theorem states that the diagonal d of a square with sides of integral length s cannot be rational. Assume d/s is rational and equal to p/q where p and q are ...

Qubic is a generalization of tic-tac-toe in which players alternately place pieces to get four in a row (rows, columns, pillars, face diagonals, or space diagonals) on a ...