On extreme points of the diffusion polytope
| Autor: | Jeremy Schiff, Nathaniel J. Fisch, Michael J. Hay |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
Discrete mathematics medicine.medical_specialty Mathematics::Combinatorics Birkhoff polytope Linear programming Polyhedral combinatorics Complete graph FOS: Physical sciences Uniform k 21 polytope Mathematical Physics (math-ph) Condensed Matter Physics 01 natural sciences Physics - Plasma Physics 010305 fluids & plasmas Combinatorics Plasma Physics (physics.plasm-ph) 0103 physical sciences Convex polytope Cross-polytope medicine 010306 general physics Vertex enumeration problem Mathematical Physics Mathematics |
| DOI: | 10.48550/arxiv.1604.08573 |
| Popis: | We consider a class of diffusion problems defined on simple graphs in which the populations at any two vertices may be averaged if they are connected by an edge. The diffusion polytope is the convex hull of the set of population vectors attainable using finite sequences of these operations. A number of physical problems have linear programming solutions taking the diffusion polytope as the feasible region, e.g. the free energy that can be removed from plasma using waves, so there is a need to describe and enumerate its extreme points. We review known results for the case of the complete graph $K_n$, and study a variety of problems for the path graph $P_n$ and the cyclic graph $C_n$. We describe the different kinds of extreme points that arise, and identify the diffusion polytope in a number of simple cases. In the case of increasing initial populations on $P_n$ the diffusion polytope is topologically an $n$-dimensional hypercube. Comment: 10 pages, 8 figures |
| Databáze: | OpenAIRE |
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