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The theta series of a lattice is the generating function for the number of vectors with norm n in the lattice. Theta series for a number of lattices are implemented in the ...

The Tschirnhausen cubic is a plane curve given by the polar equation r=asec^3(1/3theta). (1) Letting theta=3tan^(-1)t gives the parametric equations x = a(1-3t^2) (2) y = ...

Just as the ratio of the arc length of a semicircle to its radius is always pi, the ratio P of the arc length of the parabolic segment formed by the latus rectum of any ...

Geometry

A Pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a,b and hypotenuse c. By the Pythagorean theorem, this is ...

An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r_1 and r_2 from two fixed points F_1 and F_2 (the foci) separated by a distance ...

Replacing the logistic equation (dx)/(dt)=rx(1-x) (1) with the quadratic recurrence equation x_(n+1)=rx_n(1-x_n), (2) where r (sometimes also denoted mu) is a positive ...

The Bessel functions of the first kind J_n(x) are defined as the solutions to the Bessel differential equation x^2(d^2y)/(dx^2)+x(dy)/(dx)+(x^2-n^2)y=0 (1) which are ...

Bouwer graphs, a term coined here for the first time, are a family of regular graphs which includes members that are symmetric but not arc-transitive. Such graphs are termed ...

The braced square problem asks, given a hinged square composed of four equal rods (indicated by the red lines above), how many more hinged rods must be added in the same ...