Inversive semirings

What is inversive geometry?

Suppose ↵, are concentric. Let a and b denote the radii of ↵ and respectively, and assume a > b. Clearly, such an arrangement of circles i exists only when the centers C1, . . . , Cn of the n circles 1, . . . n are the vertices of a regular n-gon, whose center is the common center O of ↵ and . Consider then a point of tangency T between two circles i and i+1, and the triangle 4OCiT . The edge O...

Inversive Difference Fields

It may happen that the functions of an analytic difference field admit not only the operation of replacing z by s + 1, but also its inverse. Since these two operations have essentially the same properties it is to be expected that to each theorem concerning difference equations there will be a corresponding theorem, valid in fields of the type we have just described, in which the rôles of the h...

Tropical Semirings

It is a well-known fact that the boolean calculus is one of the mathematical foundations of electronic computers. This explains the important role of the boolean semiring in computer science. The aim of this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our ...

Inversive and Conformal Convexity

Predicting the Inversive Generator

Let p be a prime and let a and b be integers modulo p. The inversive congruential generator (ICG) is a sequence (un) of pseudorandom numbers defined by the relation un+1 ≡ au−1 n + b mod p. We show that if b and sufficiently many of the most significant bits of three consecutive values un of the ICG are given, one can recover in polynomial time the initial value u0 (even in the case where the c...