Width and size of regular resolution proofs

This paper discusses the topic of the minimum width of a regular resolution refutation of a set of clauses. The main result shows that there are examples having small regular resolution refutations, for which any regular refutation must contain a large clause. This forms a contrast with corresponding results for general resolution refutations.Comment: The article was reformatted using the style file for Logical Methods in Computer Scienc

Super-Resolution of Positive Sources: the Discrete Setup

In single-molecule microscopy it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at frequencies capped by $f_c$, which is just about the frequency of natural light. This paper rigorously establishes that this super-resolution problem can be solved via linear programming in a stable manner. We prove that the quality of the reconstruction crucially depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within a square of side length about $1/f_c$. The theoretical performance guarantee is complemented with a converse result showing that our simple convex program convex is nearly optimal. Finally, numerical experiments illustrate our methods.Comment: 31 page, 7 figure

Optimal use of Charge Information for the HL-LHC Pixel Detector Readout

The pixel detectors for the High Luminosity upgrades of the ATLAS and CMS detectors will preserve digitized charge information in spite of extremely high hit rates. Both circuit physical size and output bandwidth will limit the number of bits to which charge can be digitized and stored. We therefore study the effect of the number of bits used for digitization and storage on single and multi-particle cluster resolution, efficiency, classification, and particle identification. We show how performance degrades as fewer bits are used to digitize and to store charge. We find that with limited charge information (4 bits), one can achieve near optimal performance on a variety of tasks.Comment: 27 pages, 20 figure