Distances on the tropical line determined by two points

Title:

Distances on the tropical line determined by two points

Creator:

de la Puente, Mária Jesús

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:2e21e7b1-e7da-4df1-91da-405e2d114915
uuid:2e21e7b1-e7da-4df1-91da-405e2d114915

Subject:

tropical distance, integer length, tropical line, normal matrix, idempotent matrix, caterpillar tree, and metric graph

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let p′ and q′ be points in \Rn. Write p′∼q′ if p′−q′ is a multiple of (1,…,1). Two different points p and q in \Rn/∼ 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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

[408]-435

Source:

Kybernetika | 2014 Volume:50 | Number:3

Harvested from:

CDK

Metadata only:

false