Distances on the tropical line determined by two points

Let $p',q'\in R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $R^n/\sim$ uniquely determine a tropical line $L(p,q)$, passing through them, and stable under small perturbations. This line is a balanced unrooted semi--labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p'$ and $q'$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also prove that $L(p,q)$ is caterpillar. We prove that every vertex in $L(p,q)$ belongs to the tropical linear segment joining $p$ and $q$. A vertex, denoted $pq$, closest (w.r.t tropical distance) to $p$ exists in $L(p,q)$. Same for $q$. The distances between pairs of adjacent vertices in $L(p,q)$ and the distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the matrix $|F|$. In addition, if $p$ and $q$ are generic, then the tree $L(p,q)$ is trivalent. The entries of $F$ are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of $A$.Comment: New corrected version. 31 pages and 9 figures. The main result is theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7 was obtained with A. Jim\'enez; see Arxiv 1205.416

Tropical conics for the layman

We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining polynomial. Finally, we characterize those degree--two tropical polynomials which are reducible and factorize them. We show that there exist irreducible degree--two tropical polynomials giving rise to pairs of tropical lines.Comment: 19 pages, 4 figures. Major rewriting of formerly entitled paper "Metric invariants of tropical conics and factorization of degree--two homogeneous tropical polynomials in three variables". To appear in Idempotent and tropical mathematics and problems of mathematical physics (vol. II), G. Litvinov, V. Maslov, S. Sergeev (eds.), Proceedings Workshop, Moscow, 200

Collective oscillations in quantum rings: a broken symmetry case

We present calculations within density functional theory of the ground state and collective electronic oscillations in small two-dimensional quantum rings. No spatial symmetries are imposed to the solutions and, as in a recent contribution, a transition to a broken symmetry solution in the intrinsic reference frame for an increasingly narrow ring is found. The oscillations are addressed by using real-time simulation. Conspicuous effects of the broken symmetry solution on the spectra are pointed out.Comment: ReVTeX, 5 embedded eps, two gifs. Accepted in EPJ

Improving edge finite element assembly for geophysical electromagnetic modelling on shared-memory architectures

This work presents a set of node-level optimizations to perform the assembly of edge finite element matrices that arise in 3D geophysical electromagnetic modelling on shared-memory architectures. Firstly, we describe the traditional and sequential assembly approach. Secondly, we depict our vectorized and shared-memory strategy which does not require any low level instructions because it is based on an interpreted programming language, namely, Python. As a result, we obtained a simple parallel-vectorized algorithm whose runtime performance is considerably better than sequential version. The set of optimizations have been included to the work-flow of the Parallel Edge-based Tool for Geophysical Electromagnetic Modelling (PETGEM) which is developed as open-source at the Barcelona Supercomputing Center. Finally, we present numerical results for a set of tests in order to illustrate the performance of our strategy.This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 644202. The research leading to these results has received funding from the European Union's Horizon 2020 Programme (2014-2020) and from Brazilian Ministry of Science, Technology and Innovation through Rede Nacional de Pesquisa (RNP) under the HPC4E Project (www.hpc4e.eu), grant agreement No. 689772. Authors gratefully acknowledge the support from the Mexican National Council for Science and Technology (CONACYT). All numerical tests were performed on the MareNostrum supercomputer of the Barcelona Supercomputing Center - Centro Nacional de Supercomputación (www.bsc.es).Peer ReviewedPostprint (author's final draft