On the geometric dilation of closed curves, graphs, and point sets

The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal submission; it includes additional material from a conference submission (ref. [6] in the paper

Spatial and temporal variations in Pb concentrations and isotopic composition in road dust, farmland soil and vegetation in proximity to roads since cessation of use of leaded petrol in the UK

Results are presented for a study of spatial distributions and temporal trends in concentrations of lead (Pb) from different sources in soil and vegetation of an arable farm in central Scotland in the decade since the use of leaded petrol was terminated. Isotopic analyses revealed that in all of the samples analysed, the Pb conformed to a binary mixture of petrol Pb and Pb from industrial or indigenous geological sources and that locally enhanced levels of petrol Pb were restricted to within 10 m of a motorway and 3 m of a minor road. Overall, the dominant source of Pb was historical emissions from nearby industrial areas. There was no discernible change in concentration or isotopic composition of Pb in surface soil or vegetation over the decade since the ban on the sale of leaded petrol. There was an order of magnitude decrease in Pb concentrations in road dust over the study period, but petrol Pb persisted at up to 43% of the total Pb concentration in 2010. Similar concentrations and spatial distributions of petrol Pb and non petrol Pb in vegetation in both 2001 and 2010, with enhanced concentrations near roads, suggested that redistribution of previously deposited material has operated continuously over that period, maintaining a transfer pathway of Pb into the biosphere. The results for vegetation and soil transects near minor roads provided evidence of a non petrol Pb source associated with roads/traffic, but surface soil samples from the vicinity of a motorway failed to show evidence of such a source

Extended Aharonov-Bohm period analysis of strongly correlated electron systems

The `extended Aharonov-Bohm (AB) period' recently proposed by Kusakabe and Aoki [J. Phys. Soc. Jpn (65), 2772 (1996)] is extensively studied numerically for finite size systems of strongly correlated electrons. While the extended AB period is the system length times the flux quantum for noninteracting systems, we have found the existence of the boundary across which the period is halved or another boundary into an even shorter period on the phase diagram for these models. If we compare this result with the phase diagram predicted from the Tomonaga-Luttinger theory, devised for low-energy physics, the halved period (or shorter periods) has a one-to-one correspondence to the existence of the pairing (phase separation or metal-insulator transition) in these models. We have also found for the t-J model that the extended AB period does not change across the integrable-nonintegrable boundary despite the totally different level statistics.Comment: 26 pages, RevTex, 16 figures available on request from [email protected], to be published in J. Phys. Soc. Jpn 66 No. 7(1997), We disscus the extended AB period of strongly correlated systems more systematically by performing numerical calculation for the t-J-J' model and the extended Hubbard model in addition to the 1D t-J model and the t-J ladde

Multi-scale processes in metapopulations : Contributions of stage structure, rescue effect, and correlated extinctions

Peer reviewedPublisher PD