We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings [4]. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almos...

A mathematical model is presented to investigate the effects of sandiness, irregular boundary interfaces, heterogeneity and viscoelasticity on the phase velocity of Love waves. Geometry of the problem is consisting of an initially stressed viscoelastic layer with corrugated irregular boundaries, which is sandwiched between heterogeneous orthotropic semi-infinite half-space with initial stress a...

Let A be a finite subset of an abelian group G and let P be a closed half-plane of the complex plane, containing zero. We show that (unless A possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of A which belongs to P . In other words, there exists a non-trivial character χ ∈ Ĝ such that ∑ a∈A χ(a) ∈ P . 1. Summary of res...

We present a distribution-free model for incremental learning when concepts vary with time. Concepts are caused to change by an adversary while an incremental learning algorithm attempts to track the changing concepts by minimizing the error between the current target concept and the hypothesis. For a single halfplane and the intersection of two half-planes, we show that the average mistake rat...

The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly di...

For the standard class S of normalized univalent functions f analytic in the unit disk U , we consider a problem on the minimal area of the image f(U) concentrated in any given half-plane. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regards minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect ...

Efforts at modelling the propagation of seismic waves in half-spaces with continuously varying properties have mostly been focused on shear-horizontal waves. Here, a sagittally polarized (Rayleigh type) wave travels along a symmetry axis (and is attenuated along another) of an orthotropic material with stiffnesses and mass density varying in the same exponential manner with depth. In contrast t...

Recent morphological studies have demonstrated a laminar (sheet) organization of ventricular myofibers. Multiaxial measurements of orthotropic myocardial constitutive properties have not been reported, but regional distributions of three-dimensional diastolic and systolic strains relative to fiber and sheet axes have recently been measured in the dog heart by Takayama et al. [30]. A three-dimen...