On the structure of nonarchimedean analytic curves

Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space X^an. We define semistable vertex sets of X^an and their associated skeleta, which are essentially finite metric graphs embedded in X^an. We prove a folklore theorem which states that semistable vertex sets of X are in natural bijective correspondence with semistable models of X, thus showing that our notion of skeleton coincides with the standard definition of Berkovich. We use the skeletal theory to define a canonical metric on H(X^an) := X^an - X(K), and we give a proof of Thuillier's nonarchimedean Poincar\'e-Lelong formula in this language using results of Bosch and L\"utkebohmert.Comment: 23 pages. This an expanded version of section 5 of arXiv:1104.0320 which appears in the conference proceedings "Tropical and Non-Archimedean Geometry

Innovation, Intellectual Property, and Development: A Better Set of Approaches for the 21st Century

This paper aims to provide an intellectual basis to think about the relationship between development, intellectual property and innovation; where we currently are and what alternatives are available. For the most part, we are concerned less with the implications of current IP laws for the advanced countries as we are with their impact on developing countries. We focus here not only on the current pathologies of the system and on potential alternative ways to tackle its most egregious excesses; but on a more positive note, on what kind of "system" would best promote development and well-being in the developing world. We are looking for a world with new and better rules for intellectual property. Just as some have begun to think about re-writing the rules of the American economy to ensure a more just and efficient system, the time is ripe for doing the same for the global economy, especially with regard to the IP system

The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves

Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of these results, we deduce that \Lambda-rational principal divisors on \Gamma, in the sense of tropical geometry, are exactly the retractions of principal divisors on X. We actually prove a more precise result which says that, although zeros and poles of divisors can cancel under the retraction map, in order to lift a \Lambda-rational principal divisor on \Gamma to a principal divisor on X it is never necessary to add more than g extra zeros and g extra poles. Our results imply that a continuous function F:\Gamma -> R is the restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X if and only if F is a \Lambda-rational tropical meromorphic function, and we use this fact to prove that there is a rational map f : X --> P^3 whose tropicalization, when restricted to \Gamma, is an isometry onto its image.Comment: 21 pages, 1 figur

Probing Late Neutrino Mass Properties with Supernova Neutrinos

Models of late-time neutrino mass generation contain new interactions of the cosmic background neutrinos with supernova relic neutrinos (SRNs) through exchange of the on-shell light boson, leading to significant modification of the differential SRN flux observed at earth. We consider Abelian U(1) model for generating neutrino masses at low scales and we show that there is a large parameter space in this model for which the changes induced in the flux by the exchange of the light bosons might allow one to distinguish between neutrinos being Majorana or Dirac particles, the type of neutrino mass hierarchy (normal or inverted or quasi-degenerate), and could also possibly determine the absolute values of the neutrino masses. Measurements of the presence of these effects would be possible at the next-generation water Cerenkov detectors enriched with Gadolinium, or a large 100 kton liquid argon detector.Comment: 29 pages latex, 15 figures included. Version to be published in Phys. Rev. D., added discussion of signal detection for water Cerenkov and liquid argon detectors, and discussion of non-adiabatic vs adiabatic neutrino evolution, new figures added, references updated. Results unchange

Lifting harmonic morphisms II: tropical curves and metrized complexes

In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.Comment: 35 pages, 18 figures. This article used to be the second half of arXiv:1303.4812, and is now its seque

What Lies Beneath: Treatment of Canvas-backed Pennsylvania Coal Mining Maps for Digitization

An ongoing program to preserve approximately seven hundred oversized, canvas-backed, coal mining maps from the CONSOL Energy Mining Map Collection was initiated by the University of Pittsburgh (Pitt) in 2007, supported by funding from the United States Department of the Interior Office of Surface Mining and Reclamation (OSM) and the Pennsylvania Department of Environmental Protection (PA-DEP). The main goal of this project is to stabilize and clean the mining maps for digitization at the OSM National Mine Map Repository (NMMR) located in Pittsburgh, Pennsylvania. The digitized data of the underground mines will be incorporated into Geographical Information Systems relative to mine safety, land reclamation, current mining operations, and new development

Crowdsourcing for Identification of Polyp-Free Segments in Virtual Colonoscopy Videos

Virtual colonoscopy (VC) allows a physician to virtually navigate within a reconstructed 3D colon model searching for colorectal polyps. Though VC is widely recognized as a highly sensitive and specific test for identifying polyps, one limitation is the reading time, which can take over 30 minutes per patient. Large amounts of the colon are often devoid of polyps, and a way of identifying these polyp-free segments could be of valuable use in reducing the required reading time for the interrogating radiologist. To this end, we have tested the ability of the collective crowd intelligence of non-expert workers to identify polyp candidates and polyp-free regions. We presented twenty short videos flying through a segment of a virtual colon to each worker, and the crowd was asked to determine whether or not a possible polyp was observed within that video segment. We evaluated our framework on Amazon Mechanical Turk and found that the crowd was able to achieve a sensitivity of 80.0% and specificity of 86.5% in identifying video segments which contained a clinically proven polyp. Since each polyp appeared in multiple consecutive segments, all polyps were in fact identified. Using the crowd results as a first pass, 80% of the video segments could in theory be skipped by the radiologist, equating to a significant time savings and enabling more VC examinations to be performed