Estimate of halo ellipticity as a function of radius with flexions
  The cold dark matter theory predicts triaxial dark matter haloes. The radial
distribution of halo ellipticity depends on baryonic processes and the nature
of dark matter particles (collisionless or collisional). Here we show that we
can use lensing flexion ratios to measure the halo ellipticity as a function of
radius. We introduce a weight function and study the relationship between the
first and second order statistics of flexion ratios, both of which can be used
to reduce the bias in the estimate of ellipticity. we perform numerical tests
for our method, and demonstrate that it can reduce the bias and determine the
halo ellipticity as a function of radius. We also point out that the minimum
mean flexion ratio can be used to trace the centres of galaxy clusters.
