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Since (2a)/(a+b)=(2ab)/((a+b)b), (1) it follows that a/((a+b)/2)=((2ab)/(a+b))/b, (2) so a/A=H/b, (3) where A and H are the arithmetic mean and harmonic mean of a and b. This ...

The portion of a surface left when an open disk is removed from it.

The smallest radial distance of an ellipse as measured from a focus. Taking v=0 in the equation of an ellipse r=(a(1-e^2))/(1+ecosv) gives the periapsis distance r_-=a(1-e). ...

The interval (generally, the smallest interval) over which the values of a periodic function recur. Functions may have one or more periods over time and in space.

A characteristic of some systems making a transition to chaos. Doubling is followed by quadrupling, etc. An example of a map displaying period doubling is the logistic map.

Li and Yorke (1975) proved that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as ...

All elementary functions can be extended to the complex plane. Such definitions agree with the real definitions on the x-axis and constitute an analytic continuation.

The "perp dot product" a^_|_·b for a and b vectors in the plane is a modification of the two-dimensional dot product in which a is replaced by the perpendicular vector ...

If all elements a_(ij) of an irreducible matrix A are nonnegative, then R=minM_lambda is an eigenvalue of A and all the eigenvalues of A lie on the disk |z|<=R, where, if ...

If mu=(mu_1,mu_2,...,mu_n) is an arbitrary set of positive numbers, then all eigenvalues lambda of the n×n matrix a=a_(ij) lie on the disk |z|<=m_mu, where ...