Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints

We propose two strategies for Presenter in the on-line interval graph coloring games. Specifically, we consider a setting in which each interval is associated with a $d$-dimensional vector of weights and the coloring needs to satisfy the $d$-dimensional bandwidth constraint, and the $k$-cardinality constraint. Such a variant was first introduced by Epstein and Levy and it is a natural model for resource-aware task scheduling with $d$ different shared resources where at most $k$ tasks can be scheduled simultaneously on a single machine. The first strategy forces any on-line interval coloring algorithm to use at least $(5m-3)\frac{d}{\log d + 3}$ different colors on an $m(\frac{d}{k} + \log{d} + 3)$-colorable set of intervals. The second strategy forces any on-line interval coloring algorithm to use at least $\lfloor\frac{5m}{2}\rfloor\frac{d}{\log d + 3}$ different colors on an $m(\frac{d}{k} + \log{d} + 3)$-colorable set of unit intervals

Subelliptic SpinC Dirac Operators, IV Proof of the Relative Index Conjecture

We prove the relative index conjecture, which in turn implies that the set of embeddable deformations of a strictly pseudoconvex CR-structure on a compact 3-manifold is closed in the C\infty-topology.Comment: 5 page

Smoothed corners and scattered waves

We introduce an arbitrary order, computationally efficient method to smooth corners on curves in the plane, as well as edges and vertices on surfaces in $\mathbb R^3$. The method is local, only modifying the original surface in a neighborhood of the geometric singularity, and preserves desirable features like convexity and symmetry. The smoothness of the final surface is an explicit parameter in the method, and the bandlimit of the smoothed surface is proportional to its smoothness. Several numerical examples are provided in the context of acoustic scattering. In particular, we compare scattered fields from smoothed geometries in two dimensions with those from polygonal domains. We observe that significant reductions in computational cost can be obtained if merely approximate solutions are desired in the near- or far-field. Provided that it is sub-wavelength, the error of the scattered field is proportional to the size of the geometry that is modified