Geometric residue theorems for bundle maps

Geometric Residue Theorems for Bundle Maps

In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CRsingularities, finite singularities and singularities of odd forms as spinor bundle maps.

Shortcut Node Classification for Membrane Residue Curve Maps

comNode classification within Membrane Residue Curves (M-RCMs) currently hinges on Lyapunov’s Theorem and therefore the computation of mathematically complex eigenvalues. This paper presents an alternative criterion for the classification of nodes within M-RCMs based on the total membrane flux at node compositions. This paper demonstrates that for a system exhibiting simple permeation behaviour...

Geometric Residue

Compactness Theorems for Geometric Packings

Moser asked whether the collection of rectangles of dimensions 1 × 12 , 12 × 13 , 1 3 × 14 , . . . , whose total area equals 1, can be packed into the unit square without overlap, and whether the collection of squares of side lengths 12 , 1 3 , 1 4 , . . . can be packed without overlap into a rectangle of area π 2 6 − 1. Computational investigations have been made into packing these collections...

Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3, 5-quadrangulation. The vertices of a 2, 4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4, 8-triangulations and 2, 6-quadrangulations. We prove these results, of which the first two are known and the others seem t...