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The Heronian mean of two numbers a and b is defined as HM(a,b) = 1/3(2A+G) (1) = 1/3(a+sqrt(ab)+b), (2) where A is the arithmetic mean and G the geometric mean. It arises in ...

The following three pieces of information completely determine the homeomorphic type of a surface (Massey 1996): 1. Orientability, 2. Number of boundary components, 3. Euler ...

A point p on a regular surface M in R^3 is said to be hyperbolic if the Gaussian curvature K(p)<0 or equivalently, the principal curvatures kappa_1 and kappa_2, have opposite ...

A 4-hyperboloid has negative curvature, with R^2=x^2+y^2+z^2-w^2 (1) 2x(dx)/(dw)+2y(dy)/(dw)+2z(dz)/(dw)-2w=0. (2) Since r=xx^^+yy^^+zz^^, (3) it follows that ...

A function of the coordinates which is constant along a trajectory in phase space. The number of degrees of freedom of a dynamical system such as the Duffing differential ...

The condition for isoenergetic nondegeneracy for a Hamiltonian H=H_0(I)+epsilonH_1(I,theta) is |(partial^2H_0)/(partialI_ipartialI_j) (partialH_0)/(partialI_i); ...

A bistable eight-sided polyhedron discovered by Wunderlich and Schwabe (1986).

A method for constructing magic squares of singly even order n>=6.

The ordinary differential equation y^('')+(1-|y|)y^'+y=0.

Second and higher derivatives of the metric tensor g_(ab) need not be continuous across a surface of discontinuity, but g_(ab) and g_(ab,c) must be continuous across it.