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

Breve reflexión ante una eventual constitución de la UE : de las soberanías nacionales al derecho comunitario

A finales del siglo XV, principio del XVI, se llevó a cabo en Europa la formación de los estados nacionales y apareció el Derecho internacional moderno. Dos fenómenos coexistentes, inseparables el uno del otro. Esta dualidad explica la doble legitimidad en la que se inserta el poder político estatal de los Estados nacionales. Cada uno establece el suyo propio aunque reconoce un orden superior que enlaza a todos los elementos nacionales. Pero en el caso de la UE, hay otra línea de legitimidad más clara y eficaz. Ha sido el genio creador de una generación de hombres clarividentes el que puso en marcha en Europa occidental hace medio siglo, un fenómeno de integración política que puede culminar en una Unión Política desde la diversidad de soberanías estatales insolidarias. Esta singularidad se manifiesta de modo multiforme: en los propios tratados comunitarios, en la naturaleza y características del Derecho comunitario y de modo muy especial en la relación entre el Derecho de la Unión y el particular de sus Estados miembros en la que prevalece el primero de ellos. Es así como la dinámica de integración y el correspondiente Derecho que las conduce han entrado en un proceso de aceleración que puede llevar lejos. ________________________________________At the end of the Fifteen century and beginning of the Sixteenth, national estates were created in Europe and modern International Law appeared. These are two coexisting and inseparable phenomena. This duality explains why the political power of national Estates is based on a double legitimacy: each Estate sets its own political power, but a the same time it recognises a higher system that binds all the national elements. Regarding the European Union, however, there is a clearer and more efficient legitimacy. Half a century ago, the creative talent of a generation of clear-sighted men was the driving force of the start in Western Europe of a political integration phenomenon that can culminate in a Political Union in spite of the differences between the several national estates. The singularity of this process becomes evident in different ways: in the community treaties, in the nature and features of Community Law and especially in the relationship between Community Law and the law of the different Member Estates. In this relationship is Community Law the one that prevails. Integration dynamics and its corresponding law have entered an acceleration process that may go far

Classical resolution of black hole singularities via wormholes

In certain extensions of General Relativity, wormholes generated by spherically symmetric electric fields can resolve black hole singularities without necessarily removing curvature divergences. This is shown by studying geodesic completeness, the behavior of time-like congruences going through the divergent region, and by means of scattering of waves off the wormhole. This provides an example of the logical independence between curvature divergences and space-time singularities, concepts very often identified with each other in the literature.ns of curvature divergences in the context of space-time singularities.Comment: 6 pages, 4 figures; several improvements in main body and abstract; final version to appear in Eur. Phys. J.