Tangles, tree-decompositions, and grids in matroids

A tangle in a matroid is an obstruction to small branch-width. In particular, the maximum order of a tangle is equal to the branch-width. We prove that: (i) there is a tree-decomposition of a matroid that “displays” all of the maximal tangles, and (ii) when M is representable over a finite field, each tangle of sufficiently large order “dominates” a large grid-minor. This extends results of Robertson and Seymour concerning Graph Minors

Branch-width and well-quasi-ordering in matroids and graphs

AbstractWe prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off

Quasi-graphic matroids

Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, quasi-graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every 3-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a rame matroid

On inequivalent representations of matroids over non-prime fields

For each finite field $F$ of prime order there is a constant $c$ such that every 4-connected matroid has at most $c$ inequivalent representations over $F$. We had hoped that this would extend to all finite fields, however, it was not to be. The $(m,n)$-mace is the matroid obtained by adding a point freely to $M(K_{m,n})$. For all $n \geq 3$, the $(3,n)$-mace is 4-connected and has at least $2n$ representations over any field $F$ of non-prime order $q \geq 9$. More generally, for $n \geq m$, the $(m,n)$-mace is vertically $(m+1)$-connected and has at least $2n$ inequivalent representations over any finite field of non-prime order $q\geq m^m$

Systematic analysis of the impact of mixing locality on Mixing-DAC linearity for multicarrier GSM

In an RF transmitter, the function of the mixer and the DAC can be combined in a single block: the Mixing-DAC. For the generation of multicarrier GSM signals in a basestation, high dynamic linearity is required, i.e. SFDR>85dBc, at high output signal frequency, i.e. ƒout ˜ 4GHz. This represents a challenge which cannot be addressed efficiently by current available hardware or state-of-the-art published solutions. Mixing locality indicates if the mixing operation is executed locally in each DAC unit cell or globally on the combined DAC output signal. The mixing locality is identified as one of the most important aspects of the Mixing-DAC architecture with respect to linearity. Simulations of a current steering Mixing-DAC show that local mixing with a local output cascode can result in the highest linearity, i.e. IMD3<-88dBc at ƒout=4GHz