Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $\mathbb{Z}^d$. The construction must be {\em consistent} (that is, satisfy the natural extension of the Euclidean axioms) while resembling them as much as possible. Previous work has shown asymptotically tight results in two dimensions with $\Theta(\log N)$ error, where resemblance between segments is measured with the Hausdorff distance, and $N$ is the $L_1$ distance between the two points. This construction was considered tight because of a $\Omega(\log N)$ lower bound that applies to any consistent construction in $\mathbb{Z}^2$. In this paper we observe that the lower bound does not directly extend to higher dimensions. We give an alternative argument showing that any consistent construction in $d$ dimensions must have $\Omega(\log^{1/(d-1)} N)$ error. We tie the error of a consistent construction in high dimensions to the error of similar {\em weak} constructions in two dimensions (constructions for which some points need not satisfy all the axioms). This not only opens the possibility for having constructions with $o(\log N)$ error in high dimensions, but also opens up an interesting line of research in the tradeoff between the number of axiom violations and the error of the construction. In order to show our lower bound, we also consider a colored variation of the concept of discrepancy of a set of points that we find of independent interest

Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in Zd. The construction must be consistent (that is, satisfy the natural extension of the Euclidean axioms) while resembling them as much as possible. Previous work has shown asymptotically tight results in two dimensions with Θ(logN) error, where resemblance between segments is measured with the Hausdorff distance, and N is the L1 distance between the two points. This construction was considered tight because of a Ω(logN) lower bound that applies to any consistent construction in Z2. In this paper we observe that the lower bound does not directly extend to higher dimensions. We give an alternative argument showing that any consistent construction in d dimensions must have Ω(log1/(d−1)N) error. We tie the error of a consistent construction in high dimensions to the error of similar weak constructions in two dimensions (constructions for which some points need not satisfy all the axioms). This not only opens the possibility for having constructions with o(logN) error in high dimensions, but also opens up an interesting line of research in the tradeoff between the number of axiom violations and the error of the construction. A side result, that we find of independent interest, is the introduction of the bichromatic discrepancy: a natural extension of the concept of discrepancy of a set of points. In this paper, we define this concept and extend known results to the chromatic setting

High Quality Consistent Digital Curved Rays via Vector Field Rounding

We consider the consistent digital rays (CDR) of curved rays, which approximates a set of curved rays emanating from the origin by the set of rooted paths (called digital rays) of a spanning tree of a grid graph. Previously, a construction algorithm of CDR for diffused families of curved rays to attain an O(?{n log n}) bound for the distance between digital ray and the corresponding ray is known [Chun et al., 2019]. In this paper, we give a description of the problem as a rounding problem of the vector field generated from the ray family, and investigate the relation of the quality of CDR and the discrepancy of the range space generated from gradient curves of rays. Consequently, we show the existence of a CDR with an O(log ^{1.5} n) distance bound for any diffused family of curved rays

IceHEP High Energy Physics at the South Pole

With the solar and SN87 neutrino observations as proofs of concepts, the kilometer-scale neutrino experiment IceCube will scrutinize its data for new particle physics. In this paper we review the prospects for the realization of such a program. We begin with a short overview of the detector response and discuss the reach of ``beam'' luminosity. After that we discuss the potential of IceCube to probe deviations of neutrino-nucleon cross sections from the Standard Model predictions at center-of-mass energies well beyond those accessible in man-made accelerators. Then we review the prospects for extremely long-baseline analyses and discuss the sensitivity to measure tiny deviations of the flavor mixing angle, expected to be induced by quantum gravity effects. Finally we discuss the potential to uncover annihilation of dark matter particles gravitationally trapped at the center of the Sun, as well as processes occurring in the early Universe at energies close to the Grand Unification scale.Comment: Typos corrected and references added. Version with high resolution figures available at http://www.hep.physics.neu.edu/staff/doqui/icehep_rev6.p

The IceCube Neutrino Observatory - Contributions to ICRC 2015 Part II: Atmospheric and Astrophysical Diffuse Neutrino Searches of All Flavors

Papers on atmospheric and astrophysical diffuse neutrino searches of all flavors submitted to the 34th International Cosmic Ray Conference (ICRC 2015, The Hague) by the IceCube Collaboration.Comment: 66 pages, 36 figures, Papers submitted to the 34th International Cosmic Ray Conference, The Hague 2015, v2 has a corrected author lis