A new proof of the flat wall theorem

We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r >= 1 be integers, and let R = 49152t(24) (40t(2) +r). An r-wall is obtained from a 2r x r-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no Kt minor, and let W be an R-wall in G. We prove that there exist a set A subset of V(G) of size at most 12288t(24) and an r-subwall W' of W such that V(W') n A = 0 and W' is a flat wall in G A in the following sense. There exists a separation (X, Y) of G A such that X boolean AND Y is a subset of the vertex set of the cycle C' that bounds the outer face of W', V(W') subset of Y, every peg of W' belongs to X and the graph G[Y] can almost be drawn in the unit disk with the vertices X n Y drawn on the boundary of the disk in the order determined by C'. Here almost means that the assertion holds after repeatedly removing parts of the graph separated from X n Y by a cutset Z of size at most three, and adding all edges with both ends in Z. Our proof gives rise to an algorithm that runs in polynomial time even when r and t are part of the input instance. The proof is self-contained in the sense that it uses only results whose proofs can be found in textbooks. (C) 2017 The Authors. Published by Elsevier Inc

HIV/Aids epidemic in India and predicting the impact of the national response: mathematical modeling and analysis

After two phases of AIDS control activities in India, the third phase of the National AIDS Control Programme (NACP III) was launched in July 2007. Our focus here is to predict the number of people living with HIV/AIDS (PLHA) in India so that the results can assist the NACP III planning team to determine appropriate targets to be activated during the project period (2007-2012). We have constructed a dynamical model that captures the mixing patterns between susceptibles and infectives in both low-risk and high-risk groups in the population. Our aim is to project the HIV estimates by taking into account general interventions for susceptibles and additional interventions, such as targeted interventions among high risk groups, provision of anti-retroviral therapy, and behavior change among HIV-positive individuals. Continuing the current level of interventions in NACP II, the model estimates there will be 5.06 million PLHA by the end of 2011. If 50 percent of the targets in NACP III are achieved by the end of the above period then about 0.8 million new infections will be averted in that year. The current status of the epidemic appears to be less severe compared to the trend observed in the late 1990s. The projections based on the second phase and the third phase of the NACP indicate prevention programmes which are directed towards the general and high-risk populations, and HIV-positive individuals will determine the decline or stabilization of the epidemic. Model based results are derived separately for the revised HIV estimates released in 2007. We perform a Monte Carlo procedure for sensitivity analysis of parameters and model validation. We also predict a positive role of implementation of anti-retroviral therapy treatment of 90 percent of the eligible people in the country. We present methods for obtaining disease progression parameters using convolution approaches. We also extend our models to age-structured populations

Mars high-resolution mapping

A series of photomosaics of high-resolution Viking Orbiter images of Mars is being prepared and published to support the Mars 1:500,000 scale geologic mapping program. More than 100 of these photomosaics were made manually, but for the last several years they have all been made digitally. The digital mosaics are published on the Mars Transverse Mercator (MTM) system, and they are also available to the appropriate principal investigators as digital files in the mosaicked digital image model (MDIM) format. The mosaics contain Viking Orbiter images with the highest available resolution: in some areas as high as 10 m/pixel. This resolution, where it exists, will support a 1:100,000 map scale. The full resolution of a mosaic is preserved in a digital file, but conventional lithographic publication of such large-scale inset maps will be done only if required by the geologic map author. When high-resolution images do not full the neatlines of an MTM quadrangle, the medium-resolution (1/256 degrees/pixel, or 231 m/pixel) MDIM is used. The mosaics are tied by image-matching to the planetwide MDIM, in which random errors as large as 5 km (10 mm at 1:500,000 scale) are common; a few much larger, worst-case errors also occur. Because of the distribution of the errors, many large discrepancies appear along the cutlines between frames with very different resolutions. Furthermore, each block of quadrangles is compiled on its own local control system, and adjacent blocks, compiled later, are unlikely to match. Selection of areas to be mapped is based on geologic mapping proposals reviewed and recommended by the Mars 1:500,000 scale geologic mapping review panel. There is no intention to map the entire planet at this scale

Inflow and Loadings from Ground Water to the Great Bay Estuary, New Hampshire

This final report presents the results of a study to evaluate groundwater inflow and nutrient loadings to the Great Bay Estuary, New Hampshire. The evaluation of inflow was accomplished independently by two methods: one, used thermal imagery, and the other, piezometric mapping. The thermal imagery method assessed groundwater that was observed to discharge within the intertidal zone of an inland estuary. The groundwater piezometric mapping method used bedrock wells around the bay to create an overall piezometric map of the near-bay area. Groundwater discharge was evaluated with respect to flow, concentration, and ultimately nitrogen loading to coastal waters. The results represent a snapshot for these variables, examined by a thermal infrared aerial survey in the spring of 2000, and water quality, specific discharge, and piezometric surface maps in the summer of 2001. Monitoring wells upgradient of the Great Bay were analyzed for nitrogen as an indicator of potential discharge source waters. Total groundwater discharge to the estuary was calculated as 24.2 cubic feet per second (cfs) with an average of 0.81± 0.89 mg dissolved inorganic nitrogen (DIN)/L, with a maximum value of 2.7 mg DIN/L (n=20). Nutrient concentrations, averaging 0.83± 1.34 mg DIN/L, with a maximum value of 10.2 mg DIN/L, were observed in upgradient bedrock groundwater analyzed from 192 wells. Nutrient loading was calculated to be 19.3±21.2 tons of N per year for the total Great Bay Estuary, covering nearly 144 miles of shoreline. The groundwater derived nutrient loading accounts for approximately 5% of the total non-point source load to the estuary. The thermal imagery method was found to be an effective and affordable alternative to conventional groundwater exploration approaches

Design of a welded joint for robotic, on-orbit assembly of space trusses

A preliminary design for a weldable truss joint for on-orbit assembly of large space structures is described. The joint was designed for ease of assembly, for structural efficiency, and to allow passage of fluid (for active cooling or other purposes) along the member through the joint. The truss members were assumed to consist of graphite/epoxy tubes to which were bonded 2219-T87 aluminum alloy end fittings for welding on-orbit to truss nodes of the same alloy. A modified form of gas tungsten arc welding was assumed to be the welding process. The joint was designed to withstand the thermal and structural loading associated with a 120-ft diameter tetrahedral truss intended as an aerobrake for a mission to Mars