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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 ...

Poinsot's spirals are the two polar curves with equations r = acsch(ntheta) (1) r = asech(ntheta). (2)

A relation "<=" is called a preorder (or quasiorder) on a set S if it satisfies: 1. Reflexivity: a<=a for all a in S. 2. Transitivity: a<=b and b<=c implies a<=c. A preorder ...

The evolute of the prolate cycloid x = at-bsint (1) y = a-bcost (2) (with b>a) is given by x = a[t+((bcost-a)sint)/(acost-b)] (3) y = (a(a-bcost)^2)/(b(acost-b)). (4)

A method to obtain a signal C_l(z) with a flat spectrum c(theta;z) (such as a pulse), but having a smaller amplitude than the pulse. ...

For a delta function at (x_0,y_0), R(p,tau) = int_(-infty)^inftyint_(-infty)^inftydelta(x-x_0)delta(y-y_0)delta[y-(tau+px)]dydx (1) = ...

The constant ratio by which all distances are increased (or decreased) in a similarity. A similarity with ratio of magnification equal to 1 is called an isometry.

Consider a countable subgroup H with elements h_i and an element x not in H, then h_ix for i=1, 2, ... constitute the right coset of the subgroup H with respect to x.

The nonlinear three-dimensional map X^. = -(Y+Z) (1) Y^. = X+aY (2) Z^. = b+XZ-cZ (3) whose strange attractor is show above for a=0.2, b=0.2, and c=8.0.

Specifying three sides uniquely determines a triangle whose area is given by Heron's formula, K=sqrt(s(s-a)(s-b)(s-c)), (1) where s=1/2(a+b+c) (2) is the semiperimeter of the ...