To install click the Add extension button. That's it.
The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.
How to transfigure the Wikipedia
Would you like Wikipedia to always look as professional and up-to-date? We have created a browser extension. It will enhance any encyclopedic page you visit with the magic of the WIKI 2 technology.
Try it — you can delete it anytime.
Install in 5 seconds
Yep, but later
4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
In mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid. Given a matroid , the matroid polytope is the convex hull of the indicator vectors of the bases of .
Definition
Let be a matroid on elements. Given a basis of , the indicator vector of is
where is the standard th unit vector in . The matroid polytope is the convex hull of the set
Examples
Square pyramid
Octahedron
Let be the rank 2 matroid on 4 elements with bases
That is, all 2-element subsets of except . The corresponding indicator vectors of are
Let be the rank 2 matroid on 4 elements with bases that are all 2-element subsets of . The corresponding matroid polytope is the octahedron. Observe that the polytope from the previous example is contained in .
A matroid polytope is contained in the hypersimplex, where is the rank of the associated matroid and is the size of the ground set of the associated matroid.[2] Moreover, the vertices of are a subset of the vertices of .
Every edge of a matroid polytope is a parallel translate of for some , the ground set of the associated matroid. In other words, the edges of correspond exactly to the pairs of bases that satisfy the basis exchange property: for some [2] Because of this property, every edge length is the square root of two. More generally, the families of sets for which the convex hull of indicator vectors has edge lengths one or the square root of two are exactly the delta-matroids.
Let be the rank function of a matroid . The matroid polytope can be written uniquely as a signed Minkowski sum of simplices:[3]
where is the ground set of the matroid and is the signed beta invariant of :
Related polytopes
Independence matroid polytope
The matroid independence polytope or independence matroid polytope is the convex hull of the set
The (basis) matroid polytope is a face of the independence matroid polytope. Given the rank of a matroid , the independence matroid polytope is equal to the polymatroid determined by .
Flag matroid polytope
The flag matroid polytope is another polytope constructed from the bases of matroids. A flag is a strictly increasing sequence
of finite sets.[4] Let be the cardinality of the set . Two matroids and are said to be concordant if their rank functions satisfy
Given pairwise concordant matroids on the ground set with ranks , consider the collection of flags where is a basis of the matroid and . Such a collection of flags is a flag matroid. The matroids are called the constituents of .
For each flag in a flag matroid , let be the sum of the indicator vectors of each basis in
Given a flag matroid , the flag matroid polytope is the convex hull of the set
A flag matroid polytope can be written as a Minkowski sum of the (basis) matroid polytopes of the constituent matroids:[4]
References
^Grötschel, Martin (2004), "Cardinality homogeneous set systems, cycles in matroids, and associated polytopes", The Sharpest Cut: The Impact of Manfred Padberg and His Work, MPS/SIAM Ser. Optim., SIAM, Philadelphia, PA, pp. 99–120, MR2077557. See in particular the remarks following Prop. 8.20 on p. 114.