A Geometric Proof of Rosenbrock's Theorem on Pole Assignment

A geometric proof of Rosenbrock's theorem on pole assignment

On Harnack’s Theorem and Extensions: a Geometric Proof and Applications

Harnack’s theorem states that the fixed points of an orientation reversing involution of a compact orientable surface of genus g are a set of k disjoint simple closed geodesic where 0 ≤ k ≤ g+1. The first goal of this article is to give a purely geometric, complete and self-contained proof of this fact. In the case where the fixed curves of the involution do not separate the surface, we prove a...

Proof of Poincare ' S Geometric Theorem

In a paper recently published in the Rendiconti del Circolo Matemático di Palermo (vol. 33, 1912, pp. 375-407) and entitled Sur un théorème de Géométrie, Poincaré enunciated a theorem of great importance, in particular for the restricted problem of three bodies; but, having only succeeded in treating a variety of special cases after long-continued efforts, he gave out the theorem for the consid...

Sen's theorem: geometric proof, new interpretations

Sen’s classic social choice result supposedly demonstrates a conflict between Pareto and even minimal forms of liberalism. By providing the first direct mathematical proof of this seminal result, we underscore a significantly different interpretation: rather than conflicts among rights, Sen’s result occurs because the liberalism assumption negates the assumption that voters have transitive pref...

A Geometric Proof of the Colored Tverberg Theorem

The colored Tverberg theorem asserts that for every d and r there exists t = t(d, r) such that for every set C ⊂ R of cardinality (d + 1)t, partitioned into t-point subsets C1, C2, . . . , Cd+1 (which we think of as color classes; e.g., the points of C1 are red, the points of C2 blue, etc.), there exist r disjoint sets R1, R2, . . . , Rr ⊆ C that are rainbow, meaning that |Ri ∩ Cj| ≤ 1 for ever...