Square Integer Heffter Arrays with Empty Cells

A Heffter array $H(m,n;s,t)$ is an $m \times n$ matrix with nonzero entries from $\mathbb{Z}_{2ms+1}$ such that $i)$ each row contains $s$ filled cells and each column contains $t$ filled cells, $ii)$ every row and column sum to 0, and $iii)$ no element from $\{x,-x\}$ appears twice. Heffter arrays are useful in embedding the complete graph $K_{2nm+1}$ on an orientable surface where the embedding has the property that each edge borders exactly one $s-$cycle and one $t-$cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be constructed in the case when $s=m$, i.e. every cell is filled. In this paper we concentrate on square arrays with empty cells where every row sum and every column sum is $0$ in $\mathbb{Z}$. We solve most of the instances of this case.Comment: 20 pages, including 2 figure

The 2HA line of Medicago truncatula has characteristics of an epigenetic mutant that is weakly ethylene insensitive

BACKGROUND The Medicago truncatula 2HA seed line is highly embryogenic while the parental line Jemalong rarely produces embryos. The 2HA line was developed from one of the rare Jemalong regenerates and this method for obtaining a highly regenerable genotype in M. truncatula is readily reproducible suggesting an epigenetic mechanism. Microarray transcriptomic analysis showed down regulation of an ETHYLENE INSENSITIVE 3-like gene in 2HA callus which provided an approach to investigating epigenetic regulation of genes related to ethylene signalling and the 2HA phenotype. Ethylene is involved in many developmental processes including somatic embryogenesis (SE) and is associated with stress responses. RESULTS Microarray transcriptomic analysis showed a significant number of up-regulated transcripts in 2HA tissue culture, including nodule and embryo specific genes and transposon-like genes, while only a few genes were down-regulated, including an EIN3-like gene we called MtEIL1. This reduced expression was associated with ethylene insensitivity of 2HA plants that was further investigated. The weak ethylene insensitivity affected root and nodule development. Sequencing of MtEIL1 found no difference between 2HA and wild-type plants. DNA methylation analysis of MtEIL1 revealed significant difference between 2HA and wild-type plants. Tiling arrays demonstrated an elevated level of miRNA in 2HA plants that hybridised to the antisense strand of the MtEIL1 gene. AFLP-like methylation profiling revealed more differences in DNA methylation between 2HA and wild-type. Segregation analysis demonstrated the recessive nature of the eil1 phenotype and the dominant nature of the SE trait. CONCLUSIONS We have demonstrated that EIL1 of Medicago truncatula (MtEIL1) is epigenetically silenced in the 2HA seed line. The possible cause is an elevated level of miRNA that targets its 3'UTR and is also associated with DNA methylation of MtEIL1. Down regulation of MtEIL1 makes it possible to form nodules in the presence of ethylene and affects root growth under normal conditions. Segregation analysis showed no association between MtEIL1 expression and SE in culture but the role and mechanism of ethylene signalling in the process of plant regeneration through SE requires further investigation. The work also suggests that epigenetic changes to a particular gene induced in culture can be fixed in regenerated plants.This work was funded by the Australian Research Council (CEO348212) through the ARC Centre of Excellence for Integrative Legume Research (CILR)

Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties

Let $S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times PG(1,\,2)$ be a Segre variety that is $N$-fold direct product of projective lines of size three. Given two geometric hyperplanes $H'$ and $H''$ of $S_{(N)}$, let us call the triple $\{H', H'', \overline{H' \Delta H''}\}$ the Veldkamp line of $S_{(N)}$. We shall demonstrate, for the sequence $2 \leq N \leq 4$, that the properties of geometric hyperplanes of $S_{(N)}$ are fully encoded in the properties of Veldkamp {\it lines} of $S_{(N-1)}$. Using this property, a complete classification of all types of geometric hyperplanes of $S_{(4)}$ is provided. Employing the fact that, for $2 \leq N \leq 4$, the (ordinary part of) Veldkamp space of $S_{(N)}$ is $PG(2^N-1,2)$, we shall further describe which types of geometric hyperplanes of $S_{(N)}$ lie on a certain hyperbolic quadric $\mathcal{Q}_0^+(2^N-1,2) \subset PG(2^N-1,2)$ that contains the $S_{(N)}$ and is invariant under its stabilizer group; in the $N=4$ case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type $LG(4,8)$, to the set of 2295 maximal subspaces of the symplectic polar space $\mathcal{W}(7,2)$.Comment: 16 pages, 8 figures and 7 table

Polymeric forms of carbon in dense lithium carbide

The immense interest in carbon nanomaterials continues to stimulate intense research activities aimed to realize carbon nanowires, since linear chains of carbon atoms are expected to display novel and technologically relevant optical, electrical and mechanical properties. Although various allotropes of carbon (e.g., diamond, nanotubes, graphene, etc.) are among the best known materials, it remains challenging to stabilize carbon in the one-dimensional form because of the difficulty to suitably saturate the dangling bonds of carbon. Here, we show through first-principles calculations that ordered polymeric carbon chains can be stabilized in solid Li$_2$C$_2$ under moderate pressure. This pressure-induced phase (above 5 GPa) consists of parallel arrays of twofold zigzag carbon chains embedded in lithium cages, which display a metallic character due to the formation of partially occupied carbon lone-pair states in \emph{sp}$^2$-like hybrids. It is found that this phase remains the most favorable one in a wide range of pressure. At extreme pressure (larger the 215 GPa) a structural and electronic phase transition towards an insulating single-bonded threefold-coordinated carbon network is predicted.Comment: 10 pages, 6 figure