A note on the deformed Hermitian Yang-Mills PDE

We prove a priori estimates for a generalised Monge-Amp\`ere PDE with "non-constant coefficients" thus improving a result of Sun in the K\"ahler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins-Sz\`ekelyhidi on toric varieties and use it to address a conjecture of Collins-Jacob-Yau.Comment: Final version. 14 pages. To appear in Complex Variables and Elliptic equation

Computing Teichm\"{u}ller Maps between Polygons

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichm\"{u}ller maps. Although there are many efficient ways to compute or approximate conformal maps, there is currently no such algorithm for extremal quasiconformal maps. This paper studies the problem of computing extremal quasiconformal maps both in the continuous and discrete settings. We provide the first constructive method to obtain the extremal quasiconformal map in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly to the unique extremal map. To get to within $\epsilon$ of the dilatation of the extremal map, our method uses $O(1/\epsilon^{4})$ iterations. Every step of the iteration involves convex optimization and solving differential equations, and guarantees a decrease in the dilatation. Our method uses a reduction of the polygon mapping problem to that of the punctured sphere problem, thus solving a more general problem. We also discretize our procedure. We provide evidence for the fact that the discrete procedure closely follows the continuous construction and is therefore expected to converge quickly to a good approximation of the extremal quasiconformal map.Comment: 28 pages, 6 figure

Intensification-induced Degradation of Irrigated Infrastructure: The Case of Waterlogging and Salinity in Pakistan

Water and land development, use, and distribution has played a vital role in agricultural development in Pakistan. The country's canal irrigation system is the largest contiguous irrigation system in the world-consisting of 40,000 miles of canals and over 80,000 water courses, field channels and ditches running for another million miles [Qureshi and Zakir (1994)]. This irrigation network covers more than 70 percent of Pakistan's agriculture. Private investment has also contributed significantly to the irrigation system in the form of private tubewells. About 32 percent of farm-gate available water is supplied by the private tubewells, [Government of Pakistan (1988)]. These develoPlIlents have not only brought new land under cultivation but also permitted a considerable increase in cropping intensities

Factors affecting the sticking of insects on modified aircraft wings

The adhesion of insects to aircraft wings is studied. Insects were collected in road tests in past studies and a large experimental error was introduced caused by the variability of insect flux. The presence of such errors has been detected by studying the insect distribution across an aluminum-strip covered half-cylinder mounted on the top of a car. After a nonuniform insect distribution (insect flux) was found from three road tests, a new arrangement of samples was developed. The feasibility of coating aircraft wing surfaces with polymers to reduce the number of insects sticking onto the surfaces was studied using fluorocarbon elastomers, styrene butadiene rubbers, and Teflon