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Let F_0 and F_1 denote two directly similar figures in the plane, where P_1 in F_1 corresponds to P_0 in F_0 under the given similarity. Let r in (0,1), and define ...

Let five circles with concyclic centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. ...

Two polygons are congruent by dissection iff they have the same area. In particular, any polygon is congruent by dissection to a square of the same area. Laczkovich (1988) ...

Trigonometric functions of pi/p for p prime have an especially complicated Galois-minimal representation. In particular, the case cos(pi/23) requires approximately 500 MB of ...

Newton's method for finding roots of a complex polynomial f entails iterating the function z-[f(z)/f^'(z)], which can be viewed as applying the Euler backward method with ...

Given the statement "if P, then Q," or P=>Q, the converse is "if Q, then P." For example, the converse of "If a thing is a dog then it is a mammal" is "If a thing is a mammal ...

The operator partial^_ is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative d takes a function and yields a one-form. It decomposes ...

Let P be the set of primes, and let Q_p and Z_p(t) be the fields of p-adic numbers and formal power series over Z_p=(0,1,...,p-1). Further, suppose that D is a "nonprincipal ...

Let f(x,y) be a homogeneous function of order n so that f(tx,ty)=t^nf(x,y). (1) Then define x^'=xt and y^'=yt. Then nt^(n-1)f(x,y) = ...

A number D that possesses no common divisor with a prime number p is either a quadratic residue or nonresidue of p, depending whether D^((p-1)/2) is congruent mod p to +/-1.