We consider the oriented area function A on the moduli space M(P) of mechanical linkage P representing a planar multiple pendulum. For generic lengths of the sides of P, it is proved that A is a Morse function on M(P) and its critical points are given by the cyclic configurations of P satisfying an
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open po
In 1895 Henri Poincaré published his topological work 'Analysis Situs'. A new subdiscipline in mathematics was born. Analysis Situs was an inspiration to new fields like algebraic topology, Morse theory and cobordism. With use of today’s knowledge and notation, Dirk Siersma views back to this histor
We initiate a classification of polynomials f : Cn - C of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may have at most one line singularity of Morse transversal typ
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open po
We consider the (minimal) distance function of a point in the plane to a set P of N points in the plane. The locus of non-dierentiability of this distance function consists (besides of the points of P) exactly of the Voronoi diagram of P. We show that the number of minima (m), maxima (M) and `saddle
During the last years there is an increasing interest in the behaviour of polynomials at innity. In studying the family of levels curves f(x; y) = t one wants to know e.g the topological type of generic bres, the set of bifurcation values, the change of topology of the bre near bifurcation values, t
We study vanishing cycles of meromorphic functions This gives a new and unitary point of view extending the study of the topology of holomorphic germs as initiated by Milnor in the sixties and of the global topology of polynomial functions which has been advanced more recently We dene singularitie
This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane established by Siersma In the present case a conflict surface arises equidistant from the given convex sets The Gaussian Mean Curvatures and the location of Umbilic Points o
This paper studies the smoothness and the curvature of conict sets of the distance function in the plane Conict sets are also well known as bisectors We prove smoothness in the case of two convex sets and give a formula for the curvature We generalize moreover to weighted distance functions the soca
Go to page top
Go back to contents
Go back to site navigation