Bandwidth of the cartesian product of two connected graphs

AbstractThe bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)|:xy∈E(G)} taken over all injective integer numberings f of G. The cartesian product of two graphs G and H, written as G×H, is the graph with vertex set V(G)×V(H) and with (u1,v1) adjacent to (u2,v2) if either u1 is adjacent to u2 in G and v1=v2 or u1=u2 and v1 is adjacent to v2 in H. In this paper we investigate the bandwidth of the cartesian product of two connected graphs. For a graph G, we denote the diameter of G and the connectivity of G by D(G) and κ(G), respectively. Let G and H be two connected graphs. Among other results, we show that if B(H)=κ(H) and |V(H)|⩾2B(H)D(G)−min{1,D(G)−1}, then B(G×H)=B(H)|V(G)|. Moreover, the order condition in this result is sharp

Meteorite search by JARE-39 in 1998-99 season

A meteorite search party composed of eight men of the 39th Japanese Antarctic Research Expedition (JARE) collected 4136 meteorites on bare ice areas around the Yamato Mountains and the Belgica Mountains in 1998-99 field season. Approximately 4100 meteorites have been collected around the Yamato Mountains. They include many kinds of rare meteorites. By classification in the field, 2 lunar meteorites, 3 irons, 4 stony irons, 15 ureilites, 37 diogenites, 63 eucrites, 29 unclassified achondrites and 160 carbonaceous chondrites are distinguished. The number of irons is very small for such a great number of the meteorites. We also searched for meteorites around the Belgica Mountains. Twenty-one meteorites have been collected. All of them are ordinary chondrites

Molecular Basis on Nitrogen Utilization in Rice(Recent Topics of the Agricultunal Biological Science in Tohoku University)

Rice (Oryza sativa L.) is the major provision for half of the world population and is the important model crop in terms of synteny. Nitrogen is a massive prerequisite element for rice during its life span. During evolutionary processes, rice has acquired strategic systems of nitrogen metabolism for the survival, i.e., the highly efficient ammonium assimilation in roots and nitrogen remobilization (nitrogen recycling). In our laboratory, research is underway to elucidate molecular mechanisms, cellular functions and the communication mechanisms in nitrogen metabolisms, especially ammonium assimilation in roots and nitrogen recycling, in rice. In this article, aim and overview of our research projects, and some recent research topics are shown

Experimental Verification of a One-Dimensional Diffraction-Limit Coronagraph

We performed an experimental verification of a coronagraph. As a result, we confirmed that, at the focal region where the planetary point spread function exists, the coronagraph system mitigates the raw contrast of a star-planet system by at least $1\times10^{-5}$ even for the 1-$\lambda/D$ star-planet separation. In addition, the verified coronagraph keeps the shapes of the off-axis point spread functions when the setup has the source angular separation of 1$\lambda/D$. The low-order wavefront error and the non-zero extinction ratio of the linear polarizer may affect the currently confirmed contrast. The sharpness of the off-axis point spread function generated by the sub-$\lambda/D$ separated sources is promising for the fiber-based observation of exoplanets. The coupling efficiency with a single mode fiber exceeds 50% when the angular separation is greater than 3--4$\times 10^{-1}\lambda/D$. For sub-$\lambda/D$ separated sources, the peak positions (obtained with Gaussian fitting) of the output point spread functions are different from the angular positions of sources; the peak position moved from about $0.8\lambda/D$ to $1.0\lambda/D$ as the angular separation of the light source varies from $0.1\lambda/D$ to $1.0\lambda/D$. The off-axis throughput including the fiber-coupling efficiency (with respect to no focal plane mask) is about 40% for 1-$\lambda/D$ separated sources and 10% for 0.5-$\lambda/D$ separated ones (excluding the factor of the ratio of pupil aperture width and Lyot stop width), where we assumed a linear-polarized-light injection. In addition, because this coronagraph can remove point sources on a line in the sky, it has another promising application for high-contrast imaging of exoplanets in binary systems.Comment: 18 pages, 10 figures, accepted for the Publications of the Astronomical Society of the Pacifi