Counting the solutions of Presburger equations without enumerating them

Counting the Solutions of Presburger Equations without Enumerating Them

Counting Circles Without Computing Them

In this paper we engineer a fast algorithm to count the number of triangles defined by three lines out of a set of n lines whose circumcircle contains the origin. The trick is not to compute any triangles or circles. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

An Automata-Theoretic Algorithm for Counting Solutions to Presburger Formulas

We present an algorithm for counting the number of integer solutions to selected free variables of a Presburger formula. We represent the Presburger formula as a deterministic finite automaton (DFA) whose accepting paths encode the standard binary representations of satisfying free variable values. We count the number of accepting paths in such a DFA to obtain the number of solutions without en...

Counting Solutions of Decomposable Form Equations

Counting solutions of perturbed harmonic map equations

In this paper we consider perturbed harmonic map equations for maps between closed Riemannian manifolds. In the case where the target manifold has negative sectional curvature we prove among other results that for a large class of semilinear and quasilinear perturbations, the perturbed harmonic map equations have solutions in any homotopy class of maps for which the Euler characteristic of the ...