6,094 research outputs found
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
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
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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
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
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
MooseGuard: secure file sharing at scale in untrusted environments
Shared storage systems provide cheap, scalable, and reliable storage, but secure sharing in these systems requires users to encrypt their data and limit efficient sharing or trust a service provider to faithfully keep their data private. Current research has explored the use of trusted execution environments (TEEs) to operate on sensitive data and sharing policies in isolated execution. That work enables the utilization of untrusted shared resources to store and share sensitive data while maintaining stronger security guarantees. However, current research has limitations in scaling these solutions, as it bottlenecks both metadata and data operations within the same physical TEE, whereas a scaled file system distributes metadata and data operations to separate devices.
This paper explores the use of two TEEs specialized for metadata and data operations to provide file sharing at scale with less overhead in addition to strong security guarantees. This approach achieves scaled metadata and concurrent use by utilizing a server-side TEE for isolated execution on a master server and provides data privacy and efficient access revocation through a client-side TEE. MooseGuard is the prototype implementation of this design, utilizing Intel SGX as a TEE and extending the MooseFS distributed file system. MooseGuard's implementation details the modifications needed to provide security and shows how this approach can be applied to a typical distributed file system. An evaluation of MooseGuard demonstrates that TEEs specialized for metadata and data operations allow a secured distributed file system to maintain its scale with only constant overheads. As TEEs and secure hardware become more widely available in public clouds, enterprise, and personal devices, MooseGuard presents a way for users to get the best of both worlds in data privacy and efficient sharing when using scaled, shared storage systems
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