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In common usage, an ordinal number is an adjective which describes the numerical position of an object, e.g., first, second, third, etc. In formal set theory, an ordinal ...

Starting with the circle P_1 tangent to the three semicircles forming the arbelos, construct a chain of tangent circles P_i, all tangent to one of the two small interior ...

There are two kinds of power sums commonly considered. The first is the sum of pth powers of a set of n variables x_k, S_p(x_1,...,x_n)=sum_(k=1)^nx_k^p, (1) and the second ...

A roulette is a curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the foci of ...

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds. Before stating Thurston's geometrization conjecture in ...

A determinant which arises in the solution of the second-order ordinary differential equation x^2(d^2psi)/(dx^2)+x(dpsi)/(dx)+(1/4h^2x^2+1/2h^2-b+(h^2)/(4x^2))psi=0. (1) ...

A spheroidal harmonic is a special case of an ellipsoidal harmonic that satisfies the differential equation d/(dx)[(1-x^2)(dS)/(dx)]+(lambda-c^2x^2-(m^2)/(1-x^2))S=0 on the ...

The m+1 ellipsoidal harmonics when kappa_1, kappa_2, and kappa_3 are given can be arranged in such a way that the rth function has r-1 zeros between -a^2 and -b^2 and the ...

The plots above show the values of the function obtained by taking the natural logarithm of the gamma function, lnGamma(z). Note that this introduces complicated branch cut ...

Consider the probability Q_1(n,d) that no two people out of a group of n will have matching birthdays out of d equally possible birthdays. Start with an arbitrary person's ...