Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.Comment: v2: The bound in the main result has been improved by using the Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte

A linear-time algorithm for finding a complete graph minor in a dense graph

Let g(t) be the minimum number such that every graph G with average degree d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon > 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq (2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) \geq 2^{t-2}.Comment: 6 pages, 0 figures; Clarification added in several places, no change to arguments or result

How global biodiversity hotspots may go unrecognized: Lessons from the North American Coastal Plain

© 2014 John Wiley & Sons Ltd. Biodiversity hotspots are conservation priorities. We identify the North American Coastal Plain (NACP) as a global hotspot based on the classic definition, a region with \u3e 1500 endemic plant species and \u3e 70% habitat loss. This region has been bypassed in prior designations due to misconceptions and myths about its ecology and history. These fallacies include: (1) young age of the NACP, climatic instability over time and submergence during high sea-level stands; (2) climatic and environmental homogeneity; (3) closed forest as the climax vegetation; and (4) fire regimes that are mostly anthropogenic. We show that the NACP is older and more climatically stable than usually assumed, spatially heterogeneous and extremely rich in species and endemics for its range of latitude, especially within pine savannas and other mostly herbaceous and fire-dependent communities. We suspect systematic biases and misconceptions, in addition to missing information, obscure the existence of similarly biologically significant regions world-wide. Potential solutions to this problem include (1) increased field biological surveys and taxonomic determinations, especially within grassy biomes and regions with low soil fertility, which tend to have much overlooked biodiversity; (2) more research on the climatic refugium role of hotspots, given that regions of high endemism often coincide with regions with low velocity of climate change; (3) in low-lying coastal regions, consideration of the heterogeneity in land area generated by historically fluctuating sea levels, which likely enhanced opportunities for evolution of endemic species; and (4) immediate actions to establish new protected areas and implement science-based management to restore evolutionary environmental conditions in newly recognized hotspots

Calibration of remotely sensed, coarse resolution NDVI to CO2 fluxes in a sagebrush–steppe ecosystem

The net ecosystem exchange (NEE) of carbon flux can be partitioned into gross primary productivity (GPP) and respiration (R). The contribution of remote sensing and modeling holds the potential to predict these components and map them spatially and temporally. This has obvious utility to quantify carbon sink and source relationships and to identify improved land management strategies for optimizing carbon sequestration. The objective of our study was to evaluate prediction of 14-day average daytime CO2 fluxes ( Fday) and nighttime CO2 fluxes (Rn) using remote sensing and other data. Fday and Rn were measured with a Bowen ratio–energy balance (BREB) technique in a sagebrush (Artemisia spp.)–steppe ecosystem in northeast Idaho, USA, during 1996–1999. Micrometeorological variables aggregated across 14-day periods and time-integrated Advanced Very High Resolution Radiometer (AVHRR) Normalized Difference Vegetation Index (iNDVI) were determined during four growing seasons (1996–1999) and used to predict Fday and Rn. We found that iNDVI was a strong predictor of Fday (R2= 0.79, n = 66, P \u3c 0.0001). Inclusion of evapotranspiration in the predictive equation led to improved predictions of Fday (R2= 0.82, n = 66, P \u3c 0.0001). Cross-validation indicated that regression tree predictions of Fday were prone to overfitting and that linear regression models were more robust. Multiple regression and regression tree models predicted Rn quite well (R2 = 0.75–0.77, n = 66) with the regression tree model being slightly more robust in cross-validation. Temporal mapping of Fday and Rn is possible with these techniques and would allow the assessment of NEE in sagebrush–steppe ecosystems. Simulations of periodic Fday measurements, as might be provided by a mobile flux tower, indicated that such measurements could be used in combination with iNDVI to accurately predict Fday. These periodic measurements could maximize the utility of expensive flux towers for evaluating various carbon management strategies, carbon certification, and validation and calibration of carbon flux models

Infrared heater system for warming tropical forest understory plants and soils

The response of tropical forests to global warming is one of the largest uncertainties in predicting the future carbon balance of Earth. To determine the likely effects of elevated temperatures on tropical forest understory plants and soils, as well as other ecosystems, an infrared (IR) heater system was developed to provide in situ warming for the Tropical Responses to Altered Climate Experiment (TRACE) in the Luquillo Experimental Forest in Puerto Rico. Three replicate heated 4‐m‐diameter plots were warmed to maintain a 4°C increase in understory vegetation compared to three unheated control plots, as sensed by IR thermometers. The equipment was larger than any used previously and was subjected to challenges different from those of many temperate ecosystem warming systems, including frequent power surges and outages, high humidity, heavy rains, hurricanes, saturated clayey soils, and steep slopes. The system was able to maintain the target 4.0°C increase in hourly average vegetation temperatures to within ± 0.1°C. The vegetation was heterogeneous and on a 21° slope, which decreased uniformity of the warming treatment on the plots; yet, the green leaves were fairly uniformly warmed, and there was little difference among 0–10 cm depth soil temperatures at the plot centers, edges, and midway between. Soil temperatures at the 40–50 cm depth increased about 3°C compared to the controls after a month of warming. As expected, the soil in the heated plots dried faster than that of the control plots, but the average soil moisture remained adequate for the plants. The TRACE heating system produced an adequately uniform warming precisely controlled down to at least 50‐cm soil depth, thereby creating a treatment that allows for assessing mechanistic responses of tropical plants and soil to warming, with applicability to other ecosystems. No physical obstacles to scaling the approach to taller vegetation (i.e., trees) and larger plots were observed

Various spin-polarization states beyond the maximum-density droplet: a quantum Monte Carlo study

Using variational quantum Monte Carlo method, the effect of Landau-level mixing on the lowest-energy--state diagram of small quantum dots is studied in the magnetic field range where the density of magnetic flux quanta just exceeds the density of electrons. An accurate analytical many-body wave function is constructed for various angular momentum and spin states in the lowest Landau level, and Landau-level mixing is then introduced using a Jastrow factor. The effect of higher Landau levels is shown to be significant; the transition lines are shifted considerably towards higher values of magnetic field and certain lowest-energy states vanish altogether.Comment: 4 pages, 2 figures. Submitted to Phys. Rev.