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The second knot polynomial discovered. Unlike the first-discovered Alexander polynomial, the Jones polynomial can sometimes distinguish handedness (as can its more powerful ...

Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles. The Malfatti configuration ...

The smallest number of times u(K) a knot K must be passed through itself to untie it. Lower bounds can be computed using relatively straightforward techniques, but it is in ...

Pre-Calculus

The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, illustrated above in four embeddings. The left embedding ...

The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) generalize the Legendre polynomials P_l(x) and are solutions to the associated Legendre differential equation, ...

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 ...

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of ...

The partial differential equation u_t+u_(xxx)-6uu_x=0 (1) (Lamb 1980; Zwillinger 1997, p. 175), often abbreviated "KdV." This is a nondimensionalized version of the equation ...

The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions ...