17 research outputs found
Matroids with at least two regular elements
For a matroid , an element such that both and
are regular is called a regular element of . We determine completely the
structure of non-regular matroids with at least two regular elements. Besides
four small size matroids, all 3-connected matroids in the class can be pieced
together from or and a regular matroid using 3-sums. This result
takes a step toward solving a problem posed by Paul Seymour: Find all
3-connected non-regular matroids with at least one regular element [5, 14.8.8]
Intersections of circuits and cocircuits in binary matroids
AbstractOxley has shown that if, for some k ⩾ 4, a matroid M has a k-element set that is the intersection of a circuit and a cocircuit, then M has a 4-element set that is the intersection of a circuit and a cocircuit. We prove that, under the above hypothesis, for k ⩾ 6, a binary matroid will also have a 6-element set that is the intersection of a circuit and a cocircuit. In addition, we determine explicitly the regular matroids which do not have a 6-element set that is the intersection of a circuit and cocircuit. Finally, we prove that in the case of graphs, if for some k ⩾ 4, a circuit and a cocircuit intersect in k elements, then there must be a circuit and a cocircuit that intersect in (k − 2) elements
A decomposition theorem for binary matroids with no prism minor
The prism graph is the dual of the complete graph on five vertices with an
edge deleted, . In this paper we determine the class of binary
matroids with no prism minor. The motivation for this problem is the 1963
result by Dirac where he identified the simple 3-connected graphs with no minor
isomorphic to the prism graph. We prove that besides Dirac's infinite families
of graphs and four infinite families of non-regular matroids determined by
Oxley, there are only three possibilities for a matroid in this class: it is
isomorphic to the dual of the generalized parallel connection of with
itself across a triangle with an element of the triangle deleted; it's rank is
bounded by 5; or it admits a non-minimal exact 3-separation induced by the
3-separation in . Since the prism graph has rank 5, the class has to
contain the binary projective geometries of rank 3 and 4, and ,
respectively. We show that there is just one rank 5 extremal matroid in the
class. It has 17 elements and is an extension of , the unique splitter
for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's
result identifying the binary internally 4-connected matroids with no prism
minor [5]
Low- direct-photon production in AuAu collisions at and 62.4 GeV
The measurement of direct photons from AuAu collisions at and 62.4 GeV in the transverse-momentum range Gev/ is presented by the PHENIX collaboration at the Relativistic Heavy Ion Collider. A significant direct-photon yield is observed in both collision systems. A universal scaling is observed when the direct-photon spectra for different center-of-mass energies and for different centrality selections at GeV is scaled with for . This scaling also holds true for direct-photon spectra from AuAu collisions at GeV measured earlier by PHENIX, as well as the spectra from PbPb at GeV published by ALICE. The scaling power seems to be independent of , center of mass energy, and collision centrality. The spectra from different collision energies have a similar shape up to of 2 GeV/. The spectra have a local inverse slope increasing with of GeV/ in the range GeV/ and increasing to GeV/ for GeV/. The observed similarity of low- direct-photon production from to 2760 GeV suggests a common source of direct photons for the different collision energies and event centrality selections, and suggests a comparable space-time evolution of direct-photon emission
Low- direct-photon production in AuAu collisions at and 62.4 GeV
The measurement of direct photons from AuAu collisions at and 62.4 GeV in the transverse-momentum range Gev/ is presented by the PHENIX collaboration at the Relativistic Heavy Ion Collider. A significant direct-photon yield is observed in both collision systems. A universal scaling is observed when the direct-photon spectra for different center-of-mass energies and for different centrality selections at GeV is scaled with for . This scaling also holds true for direct-photon spectra from AuAu collisions at GeV measured earlier by PHENIX, as well as the spectra from PbPb at GeV published by ALICE. The scaling power seems to be independent of , center of mass energy, and collision centrality. The spectra from different collision energies have a similar shape up to of 2 GeV/. The spectra have a local inverse slope increasing with of GeV/ in the range GeV/ and increasing to GeV/ for GeV/. The observed similarity of low- direct-photon production from to 2760 GeV suggests a common source of direct photons for the different collision energies and event centrality selections, and suggests a comparable space-time evolution of direct-photon emission