2,973 research outputs found
Fork-decompositions of matroids
For the abstract of this paper, please see the PDF file
On matroids of branch-width three
For the abstract of this paper, please see the PDF file
The structure of the 3-separations of 3-connected matroids II
The authors showed in an earlier paper that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid. The purpose of this paper is to show that if certain natural conditions are imposed on the tree, then it has a uniqueness property. In particular; suppose that, from every pair of edges that meet at a degree-2 vertex and have their other ends of degree at least three, one edge is contracted. Then the resulting tree is unique
The structure of the 3-separations of 3-connected matroids
Special Issue Dedicated to Professor W.T. TutteTutte defined a k-separation of a matroid M to be a
partition (A,B) of the ground set of M such that ā£Aā£,ā£Bā£ ā„ k and
r(A) + r(B) ā r(M) < k. If, for all m < n, the matroid M has no
m-separations, then M is n-connected. Earlier, Whitney showed that
(A,B) is a 1-separation of M if and only if A is a union of 2-connected
components of M. When M is 2-connected, Cunningham and Edmonds
gave a tree decomposition of M that displays all of its 2-separations.
When M is 3-connected, this paper describes a tree decomposition of
M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M
Computer model calibration with large non-stationary spatial outputs: application to the calibration of a climate model
Bayesian calibration of computer models tunes unknown input parameters by
comparing outputs with observations. For model outputs that are distributed
over space, this becomes computationally expensive because of the output size.
To overcome this challenge, we employ a basis representation of the model
outputs and observations: we match these decompositions to carry out the
calibration efficiently. In the second step, we incorporate the non-stationary
behaviour, in terms of spatial variations of both variance and correlations, in
the calibration. We insert two integrated nested Laplace
approximation-stochastic partial differential equation parameters into the
calibration. A synthetic example and a climate model illustration highlight the
benefits of our approach
Recoiling Black Holes in Quasars
Recent simulations of merging black holes with spin give recoil velocities
from gravitational radiation up to several thousand km/s. A recoiling
supermassive black hole can retain the inner part of its accretion disk,
providing fuel for a continuing QSO phase lasting millions of years as the hole
moves away from the galactic nucleus. One possible observational manifestation
of a recoiling accretion disk is in QSO emission lines shifted in velocity from
the host galaxy. We have examined QSOs from the Sloan Digital Sky Survey with
broad emission lines substantially shifted relative to the narrow lines. We
find no convincing evidence for recoiling black holes carrying accretion disks.
We place an upper limit on the incidence of recoiling black holes in QSOs of 4%
for kicks greater than 500 km/s and 0.35% for kicks greater than 1000 km/s
line-of-sight velocity.Comment: 4 pages, 4 figures, uses emulateapj, Submitted to ApJ Letter
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
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 ādisplaysā all of the maximal tangles, and (ii) when M is representable over a ļ¬nite ļ¬eld, each tangle of suļ¬ciently large order ādominatesā 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
On inequivalent representations of matroids over non-prime ļ¬elds
For each ļ¬nite ļ¬eld of prime order there is a constant such that every 4-connected matroid has at most inequivalent representations over . We had hoped that this would extend to all ļ¬nite ļ¬elds, however, it was not to be. The -mace is the matroid obtained by adding a point freely to . For all , the -mace is 4-connected and has at least representations over any ļ¬eld of non-prime order . More generally, for , the -mace is vertically -connected and has at least inequivalent representations over any ļ¬nite ļ¬eld of non-prime order
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