| Title:
|
Distances on the tropical line determined by two points (English) |
| Author:
|
Puente, María Jesús de la |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
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1805-949X (online) |
| Volume:
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50 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
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408-435 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $p'$ and $q'$ be points in $\mathbb{R}^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $\mathbb{R}^n/\sim$ 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 $\operatorname{d}(p,pq)$, $\operatorname{d}(qp,q)$ and $\operatorname{d}(p,q)$ 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 $A$. (English) |
| Keyword:
|
tropical distance |
| Keyword:
|
integer length |
| Keyword:
|
tropical line |
| Keyword:
|
normal matrix |
| Keyword:
|
idempotent matrix |
| Keyword:
|
caterpillar tree |
| Keyword:
|
metric graph |
| MSC:
|
05C50 |
| MSC:
|
14T05 |
| MSC:
|
15A80 |
| idZBL:
|
Zbl 06357558 |
| idMR:
|
MR3245538 |
| DOI:
|
10.14736/kyb-2014-3-0408 |
| . |
| Date available:
|
2014-07-29T13:16:58Z |
| Last updated:
|
2016-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143883 |
| . |
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