Mapping nn grid points onto a square forces an arbitrarily large Lipschitz constant

We prove that the regular $n\times n$ square grid of points in the integer lattice $\mathbb{Z}^{2}$ cannot be recovered from an arbitrary $n^{2}$-element subset of $\mathbb{Z}^{2}$ via a mapping with prescribed Lipschitz constant (independent of $n$). This answers negatively a question of Feige from 2002. Our resolution of Feige's question takes place largely in a continuous setting and is based on some new results for Lipschitz mappings falling into two broad areas of interest, which we study independently. Firstly the present work contains a detailed investigation of Lipschitz regular mappings on Euclidean spaces, with emphasis on their bilipschitz decomposability in a sense comparable to that of the well known result of Jones. Secondly, we build on work of Burago and Kleiner and McMullen on non-realisable densities. We verify the existence, and further prevalence, of strongly non-realisable densities inside spaces of continuous functions.Comment: 60 pages (43 pages of the main part, 13 pages of appendices), 10 figures. This is a revised version according to referees' comments. Our version of the proof of the theorem about bilipschitz decomposition of Lipschitz regular mappings was greatly simplified. To appear in GAF

Navigating New Trade Routes: The rise of Value chains, and the Challenges for Canadian Trade Policy

In the new paradigm of international trade, Canada needs a trade policy that recognizes both the increasing importance of global value chains and the critical role of Canada-US commercial and regulatory integration in gaining full benefit from their exploitation.border papers, international policy

Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\sigma$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.Comment: 35 pages; accepted version of the manuscript; accepted for publication in Topological Methods in Nonlinear Analysi

Evaluation of Continuous Monitoring as a Tool for Municipal Stormwater Management Programs

The purpose of this study is to evaluate the uncertainty attributable to inadequate temporal sampling of stormwater discharge and water quality, and understand its implications for meeting monitoring objectives relevant to municipal separate storm sewer systems (MS4s). A methodology is presented to evaluate uncertainty attributable to inadequate temporal sampling of continuous stormflow and water quality, and a case study demonstrates the application of the methodology to six small urban watersheds (0.8-6.8 km2) and six large rural watersheds (30-16,192 km2) in Virginia. Results indicate the necessity of high-frequency continuous monitoring for accurately capturing multiple monitoring objectives, including illicit discharges, acute toxicity events, and stormflow pollutant concentrations and loads, as compared to traditional methods of sampling. For example, 1-h sampling in small urban watersheds and daily sampling in large rural watersheds would introduce uncertainty in capturing pollutant loads of 3–46% and 10–28%, respectively. Overall, the outcomes from this study highlight how MS4s can leverage continuous monitoring to meet multiple objectives under current and future regulatory environments

Patterns of past and recent conversion of indigenous grasslands in the South Island, New Zealand

We used recent satellite imagery to quantify the extent, type, and rate of conversion of remaining indigenous grasslands in the inland eastern South Island of New Zealand in recent years. We describe the pattern of conversion in relation to national classifications of land use capability and land environments, and ecological and administrative districts and regions. We show that although large areas of indigenous grasslands remain, grassland loss has been ongoing. Indigenous grassland was reduced in the study area by 3% (70 200 ha) between 1990 and 2008. Almost two-thirds of post-1990 conversion occurred in threatened environments with less than 30% of indigenous cover remaining, primarily in the Waitaki, Mackenzie and Central Otago administrative districts. This conversion occurred primarily on non-arable land. In the Mackenzie and Waitaki districts the rate of conversion in 2001-2008 was approximately twice that in 1990-2001. Opportunities to protect more of the full range of indigenous grasslands lie with the continuing tenure review process in these districts