SUMMARY / RELATED TOPICS

Uniform matroid

In mathematics, a uniform matroid is a matroid in which every permutation of the elements is a symmetry. The uniform matroid U n r is defined over a set of n elements. A subset of the elements is only if it contains at most r elements. A subset is a basis if it has r elements, it is a circuit if it has r + 1 elements; the rank of a subset S is min and the rank of the matroid is r. A matroid of rank r is uniform if and only if all of its circuits have r + 1 elements; the matroid U n 2 is called the n -point line. The dual matroid of the uniform matroid U n r is another uniform matroid U n n − r. A uniform matroid is self-dual if and only if r = n / 2; every minor of a uniform matroid is uniform. Restricting a uniform matroid U n r by one element produces the matroid U n − 1 r and contracting it by one element produces the matroid U n − 1 r − 1; the uniform matroid U n r may be represented as the matroid of affinely independent subsets of n points in general position in r -dimensional Euclidean space, or as the matroid of linearly independent subsets of n vectors in general position in an -dimensional real vector space.

Every uniform matroid may be realized in projective spaces and vector spaces over all sufficiently large finite fields. However, the field must be large enough to include enough independent vectors. For instance, the n -point line U n 2 can be realized only over finite fields of n − 1 or more elements: U 4 2 is not a binary matroid, U 5 2 is not a ternary matroid, etc. For this reason, uniform matroids play an important role in Rota's conjecture concerning the forbidden minor characterization of the matroids that can be realized over finite fields; the problem of finding the minimum-weight basis of a weighted uniform matroid is well-studied in computer science as the selection problem. It may be solved in linear time. Any algorithm that tests whether a given matroid is uniform, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, therefore cannot take polynomial time. Unless r ∈, a uniform matroid U n r is connected: it is not the direct sum of two smaller matroids.

The direct sum of a family of uniform matroids is called a partition matroid. Every uniform matroid is a transversal matroid and a strict gammoid. Not every uniform matroid is graphic, the uniform matroids provide the smallest example of a non-graphic matroid, U 4 2; the uniform matroid U n 1 is the graphic matroid of an n -edge dipole graph, the dual uniform matroid U n n − 1 is the graphic matroid of its dual graph, the n -edge cycle graph. U n 0 is the graphic matroid of a graph with n self-loops, U n n is the graphic matroid of an n -edge forest. Other than these examples, every uniform matroid U n r with 1 < r < n − 1 contains U 4 2 as a minor and therefore is not graphic. The n -point line provides an example of a Sylvester matroid, a matroid in which every line contains three or more points. K-set

Shaun David Barker

Shaun David Barker is an Australian born talent manager, marketing executive and executive music producer. He manages and markets Australian DJ’s including Timmy Trumpet, Stafford Brothers and tyDi, who have all been voted the number 1 DJ’s in Australia at the ITM Awards. Barker has managed marketing campaigns for over 20 singles including the three times platinum selling and ARIA nominated ‘Hello’ by Stafford Brothers feat. Lil Wayne & Christina Milian, the 5 times platinum selling ‘Freaks’ by Timmy Trumpet & Savage. In 2012, he appeared as himself in Season Two of the FOX8 reality TV series The Stafford Brothers. Barker is the Creative Director at TPH Media in Los Angeles and is involved in the development and production of feature films based around music content. Barker was born in Brisbane, Australia, to parents David Barker, a musician and scientist, Leanne Barker. Barker served as head of production for a series of albums, mix compilations and singles under the GeneratioNext banner from 2007 to 2010.

Barker is the only music manager to have three DJ’s named the number 1 DJ in Australia, six times in total. Barker managed the marketing campaigns for singles featuring artists including Stafford Brothers, Lil Wayne, Christina Milian, T. I. Eva Simons, Timmy Trumpet, Lil Jon, Erick Morillo, Nadia Ali, Carnage. From 2012 - 2015, Barker was the marketing manager for the Stafford Brothers. In 2012 and 2013, Barker was a guest speaker and panelist at EMC on the subject of brand marketing in music. Shaun David Barker on IMDb Shaun David Barker discography at Discogs