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Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of ...

The areas of the regions illustrated above can be found from the equations A+4B+4C=1 (1) A+3B+2C=1/4pi. (2) Since we want to solve for three variables, we need a third ...

For a linear homogeneous ordinary differential equation, if y_1(x) and y_2(x) are solutions, then so is y_1(x)+y_2(x).

The swastika curve is Cundy and Rollett's (1989, p. 71) name for the quartic plane curve with Cartesian equation y^4-x^4=xy and polar equation ...

In general, there is no unique matrix solution A to the matrix equation y=Ax. Even in the case of y parallel to x, there are still multiple matrices that perform this ...

Solution of a system of second-order homogeneous ordinary differential equations with constant coefficients of the form (d^2x)/(dt^2)+bx=0, where b is a positive definite ...

A solution zeta_k=e^(2piik/d) to the cyclotomic equation x^d=1. The de Moivre numbers give the coordinates in the complex plane of the polygon vertices of a regular polygon ...

The quintic equation x^5+ax^3+1/5a^2x+b=0 (1) is sometimes known as de Moivre's quintic (Spearman and Williams 1994). It has solutions x_j=omega^ju_1+omega^(4j)u_2 (2) for ...

Given a Taylor series f(z)=sum_(n=0)^inftyC_nz^n=sum_(n=0)^inftyC_nr^ne^(intheta), (1) where the complex number z has been written in the polar form z=re^(itheta), examine ...

A number of the form n^...^_()_(n)n, where Knuth up-arrow notation has been used. The first few Ackermann numbers are 1^1=1, 2^^2=4, and ...