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The sum I over all internal (circular) nodes of the paths from the root of an extended binary tree to each node. For example, in the tree above, the internal path length is ...

A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base). For a right circular cone, let s be the slant height and R_1 ...

To find the motion of a rectangular membrane with sides of length L_x and L_y (in the absence of gravity), use the two-dimensional wave equation ...

A prime circle of order 2n is a free circular permutation of the numbers from 1 to 2n with adjacent pairs summing to a prime. The number of prime circles for n=1, 2, ..., are ...

Finding the height above the bottom of a horizontal cylinder (such as a cylindrical gas tank) to which the it must be filled for it to be one quarter full amounts to plugging ...

Let [arg(f(z))] denote the change in the complex argument of a function f(z) around a contour gamma. Also let N denote the number of roots of f(z) in gamma and P denote the ...

A continuous distribution defined on the range x in [0,2pi) with probability density function P(x)=(e^(bcos(x-a)))/(2piI_0(b)), (1) where I_0(x) is a modified Bessel function ...

In the technical combinatorial sense, an a-ary necklace of length n is a string of n characters, each of a possible types. Rotation is ignored, in the sense that b_1b_2...b_n ...

A real polynomial P is said to be stable if all its roots lie in the left half-plane. The term "stable" is used to describe such a polynomial because, in the theory of linear ...

In a given circle, find an isosceles triangle whose legs pass through two given points inside the circle. This can be restated as: from two points in the plane of a circle, ...