Unlabeled equivalence for matroids representable over finite fields

We present a new type of equivalence for representable matroids that uses the automorphisms of the underlying matroid. Two $r\times n$ matrices $A$ and $A'$ representing the same matroid $M$ over a field $F$ are {\it geometrically equivalent representations} of $M$ if one can be obtained from the other by elementary row operations, column scaling, and column permutations. Using geometric equivalence, we give a method for exhaustively generating non-isomorphic matroids representable over a finite field $GF(q)$, where $q$ is a power of a prime

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, $K_5\backslash e$. 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 $F_7$ 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 $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, 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]

Deletable edges in 3-connected graphs and their applications

Let $G$ and $H$ be simple 3-connected graphs such that $G$ has an $H$-minor. An edge $e$ in $G$ is called {\it $H$-deletable} if $G\backslash e$ is 3-connected and has an $H$-minor. The main result in this paper establishes that, if $G$ has no $H$-deletable edges, then there exists a sequence of simple 3-connected graphs $G_0, \dots , G_k$ with no $H$-deletable edges such that $G_0\cong H$, $G_k= G$, and for $1 \le i \le k$ one of three possibilities holds: $G_{i-1}= G_i/f$; $G_{i-1}=G_i/f \backslash e$ where $e$ and $f$ are incident to a degree 3 vertex in $G_i$; or $G_{i-1}=G_i-w$ where $w$ is a degree $3$ vertex in $G_i$. Several applications are given including a graph theoretic proof of the matroid theory result known as the Strong Splitter Theorem, a short new proof of Dirac's characterization of 3-connected graphs with no minor isomorphic to the prism graph, and an extension of a result by Halin that bounds the number of edges in a minimally 3-connected graph

On the matroids in which all hyperplanes are binary

In this paper, it is shown that, for a minor-closed class ℳ of matroids, the class of matroids in which every hyperplane is in ℳ is itself minor-closed and has, as its excluded minors, the matroids U1,1 ⊕ N such that N is an excluded minor for ℳ. This result is applied to the class of matroids of the title, and several alternative characterizations of the last class are given

Large-Scale Selective Sweep among Segregation Distorter Chromosomes in African Populations of Drosophila melanogaster

Segregation Distorter (SD) is a selfish, coadapted gene complex on chromosome 2 of Drosophila melanogaster that strongly distorts Mendelian transmission; heterozygous SD/SD+ males sire almost exclusively SD-bearing progeny. Fifty years of genetic, molecular, and theory work have made SD one of the best-characterized meiotic drive systems, but surprisingly the details of its evolutionary origins and population dynamics remain unclear. Earlier analyses suggested that the SD system arose recently in the Mediterranean basin and then spread to a low, stable equilibrium frequency (1–5%) in most natural populations worldwide. In this report, we show, first, that SD chromosomes occur in populations in sub-Saharan Africa, the ancestral range of D. melanogaster, at a similarly low frequency (∼2%), providing evidence for the robustness of its equilibrium frequency but raising doubts about the Mediterranean-origins hypothesis. Second, our genetic analyses reveal two kinds of SD chromosomes in Africa: inversion-free SD chromosomes with little or no transmission advantage; and an African-endemic inversion-bearing SD chromosome, SD-Mal, with a perfect transmission advantage. Third, our population genetic analyses show that SD-Mal chromosomes swept across the African continent very recently, causing linkage disequilibrium and an absence of variability over 39% of the length of the second chromosome. Thus, despite a seemingly stable equilibrium frequency, SD chromosomes continue to evolve, to compete with one another, or evade suppressors in the genome

Measurement of ϕ\phi-meson production in Cu++Au at sNN=200\sqrt{s_{_{NN}}}=200 GeV and U++U at sNN=193\sqrt{s_{_{NN}}}=193 GeV

The PHENIX experiment reports systematic measurements at the Relativistic Heavy Ion Collider of $\phi$-meson production in asymmetric Cu$+$Au collisions at $\sqrt{s_{_{NN}}}$=200 GeV and in U$+$U collisions at $\sqrt{s_{_{NN}}}$=193 GeV. Measurements were performed via the $\phi\rightarrow K^{+}K^{-}$ decay channel at midrapidity $|\eta|<0.35$. Features of $\phi$-meson production measured in Cu$+$Cu, Cu$+$Au, Au$+$Au, and U$+$U collisions were found to not depend on the collision geometry, which was expected because the yields are averaged over the azimuthal angle and follow the expected scaling with nuclear-overlap size. The elliptic flow of the $\phi$ meson in Cu$+$Au, Au$+$Au, and U$+$U collisions scales with second order participant eccentricity and the length scale of the nuclear overlap region (estimated with the number of participating nucleons). At moderate $p_T$, $\phi$-meson production measured in Cu$+$Au and U$+$U collisions is consistent with coalescence-model predictions, whereas at high $p_T$ the production is in agreement with expectations for in-medium energy loss of parent partons prior to their fragmentation. The elliptic flow for $\phi$ mesons measured in Cu$+$Au and U$+$U collisions is well described by a (2+1)D viscous-hydrodynamic model with specific-shear viscosity $\eta/s=1/4\pi$.Comment: 411 authors from 76 institutions, 16 pages, 12 figures, 9 tables, 2012 data. v1 is version submitted to Physical Review C. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm