We consider various hitting and piercing problems for the family of axis-parallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type results typically bound the maximum number...

A common subproblem of $DNF$ approximate counting and derandomizing $RL$ is the discrepancy problem for combinatorial rectangles. We explicitly construct a $poly(n)$size sample space that approximates the volume of any combinatorial rectangle in $[n]^n$ to within $o(1)$ error (improving on the constructions of [EGLNV92]). The construction extends the techniques of [LLSZ95] for the analogous hit...

The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric objects, often turns out to be exactly the vertex cover problem on restricted classes of graphs. In this work we explore a particular instance of such a phenomenon...

EDITOR’S NOTE Geopersia is published biannually by the College of Science, University of Tehran, Iran. The College of Science published “Journal of Science”, which covered all fields of basic science, from 1968 to 2010. It was then decided that the College’s journals should be more specialized, and hence publication of Geopersia was launched. Geopersia aims to publish peer-reviewed original...

The 46km Livingstone Creek Catchment in south-eastern Australia was flown with a passive microwave airborne remote sensor (Polarimetric L-band Multibeam Radiometer, PLMR) as part of the National Airborne Field Experiment in 2006 with a spatial resolution of ~ 200m. The catchment was experiencing extreme drought conditions leading up to the experiment and as a result ground cover in the catchmen...