What does $mathbb{R}^E_{gt 0}$ stand for?
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I was studying an article where I encountered $mathbb{R}^E_{gt 0}$. I couldn't find out what does this notation mean exactly.
I'm sorry if my question is basic, I searched this community but I didn't find the answer to my question.
Here is a part of that article:
Given an undirected graph $G=(V, E)$ with positive edge lengths $linmathbb{R}^E_{gt 0}$ on which one desires to compute the shortest path from a vertex $s$ to $t$, ...
notation
asked Dec 4 '18 at 20:16
0100011001000110
607
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add a comment |
$begingroup$
I was studying an article where I encountered $mathbb{R}^E_{gt 0}$. I couldn't find out what does this notation mean exactly.
I'm sorry if my question is basic, I searched this community but I didn't find the answer to my question.
Here is a part of that article:
Given an undirected graph $G=(V, E)$ with positive edge lengths $linmathbb{R}^E_{gt 0}$ on which one desires to compute the shortest path from a vertex $s$ to $t$, ...
notation
asked Dec 4 '18 at 20:16
0100011001000110
607
$endgroup$
add a comment |
$begingroup$
I was studying an article where I encountered $mathbb{R}^E_{gt 0}$. I couldn't find out what does this notation mean exactly.
I'm sorry if my question is basic, I searched this community but I didn't find the answer to my question.
Here is a part of that article:
Given an undirected graph $G=(V, E)$ with positive edge lengths $linmathbb{R}^E_{gt 0}$ on which one desires to compute the shortest path from a vertex $s$ to $t$, ...
notation
asked Dec 4 '18 at 20:16
0100011001000110
607
$endgroup$
I was studying an article where I encountered $mathbb{R}^E_{gt 0}$. I couldn't find out what does this notation mean exactly.
I'm sorry if my question is basic, I searched this community but I didn't find the answer to my question.
Here is a part of that article:
Given an undirected graph $G=(V, E)$ with positive edge lengths $linmathbb{R}^E_{gt 0}$ on which one desires to compute the shortest path from a vertex $s$ to $t$, ...
notation
notation
asked Dec 4 '18 at 20:16
0100011001000110
607
asked Dec 4 '18 at 20:16
0100011001000110
607
asked Dec 4 '18 at 20:16
0100011001000110
607
asked Dec 4 '18 at 20:16
0100011001000110
607
asked Dec 4 '18 at 20:16
0100011001000110
607
607
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
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The short answer: This is just a terse way of saying that the length $l(e)$ of each edge $e$ is a nonnegative real number i.e., $l(e) in mathbb{R}_{ge 0}$. If you are satisfied with this then you can stop reading here.
A longer answer: Note that, in general, a function $f: A mapsto B$ can be thought of as a vector $f in B^A$, where for each $a in A$, the $a$-th coordinate of $f$ is $f(a)$, which is in $B$.
Here $l$ can be thought of as a function $l: E mapsto mathbb{R}_{geq 0}$ and thus a vector in $mathbb{R}^E_{ge 0}$, where for each $e in E$, the $e$-th coordinate of $l$ is $l(e)$, which is in $mathbb{R}_{ge 0}$.
answered Dec 4 '18 at 20:22
MikeMike
3,507411
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$X^Y$ is the usual notation for the set of functions $Yto X$. With $X=mathbb{R}_{>0}$ the positive reals and $Y=E$ the edges, $linmathbb{R}_{>0}^E$ is just a function associating to each edge $ein E$ a positive real number $l(e)$, its length.
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
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$begingroup$
The notation $Bbb R_{>0}^E$ usually means the set of functions from $E$ to $Bbb R_{>0}$, and $Bbb R_{>0}:=(0,infty)$.
However in the cited text it is not so clear what it means. If the edges are numbered then it is possible to interpret $EsubsetBbb N$, and then $linBbb R_{>0}^E$ is a function that assign to each $kin E$ it length $l(k)in (0,infty)$.
Where this text comes? Can you give a link to the article?
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
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$begingroup$
Here is the link to the article ;)
$endgroup$
– 01000110
Dec 4 '18 at 20:36
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The short answer: This is just a terse way of saying that the length $l(e)$ of each edge $e$ is a nonnegative real number i.e., $l(e) in mathbb{R}_{ge 0}$. If you are satisfied with this then you can stop reading here.
A longer answer: Note that, in general, a function $f: A mapsto B$ can be thought of as a vector $f in B^A$, where for each $a in A$, the $a$-th coordinate of $f$ is $f(a)$, which is in $B$.
Here $l$ can be thought of as a function $l: E mapsto mathbb{R}_{geq 0}$ and thus a vector in $mathbb{R}^E_{ge 0}$, where for each $e in E$, the $e$-th coordinate of $l$ is $l(e)$, which is in $mathbb{R}_{ge 0}$.
answered Dec 4 '18 at 20:22
MikeMike
3,507411
$endgroup$
add a comment |
$begingroup$
The short answer: This is just a terse way of saying that the length $l(e)$ of each edge $e$ is a nonnegative real number i.e., $l(e) in mathbb{R}_{ge 0}$. If you are satisfied with this then you can stop reading here.
A longer answer: Note that, in general, a function $f: A mapsto B$ can be thought of as a vector $f in B^A$, where for each $a in A$, the $a$-th coordinate of $f$ is $f(a)$, which is in $B$.
Here $l$ can be thought of as a function $l: E mapsto mathbb{R}_{geq 0}$ and thus a vector in $mathbb{R}^E_{ge 0}$, where for each $e in E$, the $e$-th coordinate of $l$ is $l(e)$, which is in $mathbb{R}_{ge 0}$.
answered Dec 4 '18 at 20:22
MikeMike
3,507411
$endgroup$
add a comment |
$begingroup$
The short answer: This is just a terse way of saying that the length $l(e)$ of each edge $e$ is a nonnegative real number i.e., $l(e) in mathbb{R}_{ge 0}$. If you are satisfied with this then you can stop reading here.
A longer answer: Note that, in general, a function $f: A mapsto B$ can be thought of as a vector $f in B^A$, where for each $a in A$, the $a$-th coordinate of $f$ is $f(a)$, which is in $B$.
Here $l$ can be thought of as a function $l: E mapsto mathbb{R}_{geq 0}$ and thus a vector in $mathbb{R}^E_{ge 0}$, where for each $e in E$, the $e$-th coordinate of $l$ is $l(e)$, which is in $mathbb{R}_{ge 0}$.
answered Dec 4 '18 at 20:22
MikeMike
3,507411
$endgroup$
The short answer: This is just a terse way of saying that the length $l(e)$ of each edge $e$ is a nonnegative real number i.e., $l(e) in mathbb{R}_{ge 0}$. If you are satisfied with this then you can stop reading here.
A longer answer: Note that, in general, a function $f: A mapsto B$ can be thought of as a vector $f in B^A$, where for each $a in A$, the $a$-th coordinate of $f$ is $f(a)$, which is in $B$.
Here $l$ can be thought of as a function $l: E mapsto mathbb{R}_{geq 0}$ and thus a vector in $mathbb{R}^E_{ge 0}$, where for each $e in E$, the $e$-th coordinate of $l$ is $l(e)$, which is in $mathbb{R}_{ge 0}$.
answered Dec 4 '18 at 20:22
MikeMike
3,507411
edited Dec 4 '18 at 22:04
answered Dec 4 '18 at 20:22
MikeMike
3,507411
answered Dec 4 '18 at 20:22
MikeMike
3,507411
answered Dec 4 '18 at 20:22
MikeMike
3,507411
3,507411
add a comment |
add a comment |
$begingroup$
$X^Y$ is the usual notation for the set of functions $Yto X$. With $X=mathbb{R}_{>0}$ the positive reals and $Y=E$ the edges, $linmathbb{R}_{>0}^E$ is just a function associating to each edge $ein E$ a positive real number $l(e)$, its length.
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
$endgroup$
add a comment |
$begingroup$
$X^Y$ is the usual notation for the set of functions $Yto X$. With $X=mathbb{R}_{>0}$ the positive reals and $Y=E$ the edges, $linmathbb{R}_{>0}^E$ is just a function associating to each edge $ein E$ a positive real number $l(e)$, its length.
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
$endgroup$
add a comment |
$begingroup$
$X^Y$ is the usual notation for the set of functions $Yto X$. With $X=mathbb{R}_{>0}$ the positive reals and $Y=E$ the edges, $linmathbb{R}_{>0}^E$ is just a function associating to each edge $ein E$ a positive real number $l(e)$, its length.
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
$endgroup$
$X^Y$ is the usual notation for the set of functions $Yto X$. With $X=mathbb{R}_{>0}$ the positive reals and $Y=E$ the edges, $linmathbb{R}_{>0}^E$ is just a function associating to each edge $ein E$ a positive real number $l(e)$, its length.
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
answered Dec 4 '18 at 20:21
user10354138user10354138
7,3772925
7,3772925
add a comment |
add a comment |
$begingroup$
The notation $Bbb R_{>0}^E$ usually means the set of functions from $E$ to $Bbb R_{>0}$, and $Bbb R_{>0}:=(0,infty)$.
However in the cited text it is not so clear what it means. If the edges are numbered then it is possible to interpret $EsubsetBbb N$, and then $linBbb R_{>0}^E$ is a function that assign to each $kin E$ it length $l(k)in (0,infty)$.
Where this text comes? Can you give a link to the article?
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
$endgroup$
$begingroup$
Here is the link to the article ;)
$endgroup$
– 01000110
Dec 4 '18 at 20:36
add a comment |
$begingroup$
The notation $Bbb R_{>0}^E$ usually means the set of functions from $E$ to $Bbb R_{>0}$, and $Bbb R_{>0}:=(0,infty)$.
However in the cited text it is not so clear what it means. If the edges are numbered then it is possible to interpret $EsubsetBbb N$, and then $linBbb R_{>0}^E$ is a function that assign to each $kin E$ it length $l(k)in (0,infty)$.
Where this text comes? Can you give a link to the article?
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
$endgroup$
$begingroup$
Here is the link to the article ;)
$endgroup$
– 01000110
Dec 4 '18 at 20:36
add a comment |
$begingroup$
The notation $Bbb R_{>0}^E$ usually means the set of functions from $E$ to $Bbb R_{>0}$, and $Bbb R_{>0}:=(0,infty)$.
However in the cited text it is not so clear what it means. If the edges are numbered then it is possible to interpret $EsubsetBbb N$, and then $linBbb R_{>0}^E$ is a function that assign to each $kin E$ it length $l(k)in (0,infty)$.
Where this text comes? Can you give a link to the article?
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
$endgroup$
The notation $Bbb R_{>0}^E$ usually means the set of functions from $E$ to $Bbb R_{>0}$, and $Bbb R_{>0}:=(0,infty)$.
However in the cited text it is not so clear what it means. If the edges are numbered then it is possible to interpret $EsubsetBbb N$, and then $linBbb R_{>0}^E$ is a function that assign to each $kin E$ it length $l(k)in (0,infty)$.
Where this text comes? Can you give a link to the article?
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
answered Dec 4 '18 at 20:21
MasacrosoMasacroso
13k41746
13k41746
$begingroup$
Here is the link to the article ;)
$endgroup$
– 01000110
Dec 4 '18 at 20:36
add a comment |
$begingroup$
Here is the link to the article ;)
$endgroup$
– 01000110
Dec 4 '18 at 20:36
$begingroup$
Here is the link to the article ;)
$endgroup$
– 01000110
Dec 4 '18 at 20:36
$begingroup$
Here is the link to the article ;)
$endgroup$
– 01000110
Dec 4 '18 at 20:36
add a comment |
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