Find a permutation $sigma$ maximizing the sum $sum_{i=1}^n {a_i over sigma(i)}$.
0
$begingroup$
Given $$a_1< a_2<dots < a_n,$$
find a permutation $sigma$ maximizing the sum >$$sum_{i=1}^n {a_i over sigma(i)}.$$
I can't figure our where to begin. I know that the solution is $sigma=e$, but I cannot prove it.
real-analysis sequences-and-series permutations
edited Dec 22 '18 at 12:19
Shaun
9,366113684
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
$endgroup$
add a comment |
0
$begingroup$
Given $$a_1< a_2<dots < a_n,$$
find a permutation $sigma$ maximizing the sum >$$sum_{i=1}^n {a_i over sigma(i)}.$$
I can't figure our where to begin. I know that the solution is $sigma=e$, but I cannot prove it.
real-analysis sequences-and-series permutations
edited Dec 22 '18 at 12:19
Shaun
9,366113684
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
$endgroup$
2
$begingroup$
Maybe worth trying: without loss of generality, assume the $a_i$'s are non-decreasing, and use the rearrangement inequality?
$endgroup$
– Clement C.
Jul 16 '16 at 14:48
1
$begingroup$
After the edit, and to provide more details to my previous comment: en.m.wikipedia.org/wiki/Rearrangement_inequality
$endgroup$
– Clement C.
Jul 16 '16 at 14:56
3
$begingroup$
For the answer to be $sigma=e$, I think you need $a_1>a_2>cdots>a_n$.
$endgroup$
– Thomas Andrews
Jul 16 '16 at 14:57
$begingroup$
Sorry for the lack of information. I edited it.
$endgroup$
– Razvan Paraschiv
Jul 16 '16 at 14:59
2
$begingroup$
Imagine $a_n$ is huge compared to the other terms. Then the maximum sum will certainly have $sigma(n)=1$
$endgroup$
– Joffan
Jul 16 '16 at 15:08
add a comment |
0
0
0
$begingroup$
Given $$a_1< a_2<dots < a_n,$$
find a permutation $sigma$ maximizing the sum >$$sum_{i=1}^n {a_i over sigma(i)}.$$
I can't figure our where to begin. I know that the solution is $sigma=e$, but I cannot prove it.
real-analysis sequences-and-series permutations
edited Dec 22 '18 at 12:19
Shaun
9,366113684
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
$endgroup$
Given $$a_1< a_2<dots < a_n,$$
find a permutation $sigma$ maximizing the sum >$$sum_{i=1}^n {a_i over sigma(i)}.$$
I can't figure our where to begin. I know that the solution is $sigma=e$, but I cannot prove it.
real-analysis sequences-and-series permutations
real-analysis sequences-and-series permutations
edited Dec 22 '18 at 12:19
Shaun
9,366113684
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
edited Dec 22 '18 at 12:19
Shaun
9,366113684
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
edited Dec 22 '18 at 12:19
Shaun
9,366113684
edited Dec 22 '18 at 12:19
Shaun
9,366113684
edited Dec 22 '18 at 12:19
Shaun
9,366113684
9,366113684
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
asked Jul 16 '16 at 14:43
Razvan ParaschivRazvan Paraschiv
861615
861615
2
$begingroup$
Maybe worth trying: without loss of generality, assume the $a_i$'s are non-decreasing, and use the rearrangement inequality?
$endgroup$
– Clement C.
Jul 16 '16 at 14:48
1
$begingroup$
After the edit, and to provide more details to my previous comment: en.m.wikipedia.org/wiki/Rearrangement_inequality
$endgroup$
– Clement C.
Jul 16 '16 at 14:56
3
$begingroup$
For the answer to be $sigma=e$, I think you need $a_1>a_2>cdots>a_n$.
$endgroup$
– Thomas Andrews
Jul 16 '16 at 14:57
$begingroup$
Sorry for the lack of information. I edited it.
$endgroup$
– Razvan Paraschiv
Jul 16 '16 at 14:59
2
$begingroup$
Imagine $a_n$ is huge compared to the other terms. Then the maximum sum will certainly have $sigma(n)=1$
$endgroup$
– Joffan
Jul 16 '16 at 15:08
add a comment |
2
$begingroup$
Maybe worth trying: without loss of generality, assume the $a_i$'s are non-decreasing, and use the rearrangement inequality?
$endgroup$
– Clement C.
Jul 16 '16 at 14:48
1
$begingroup$
After the edit, and to provide more details to my previous comment: en.m.wikipedia.org/wiki/Rearrangement_inequality
$endgroup$
– Clement C.
Jul 16 '16 at 14:56
3
$begingroup$
For the answer to be $sigma=e$, I think you need $a_1>a_2>cdots>a_n$.
$endgroup$
– Thomas Andrews
Jul 16 '16 at 14:57
$begingroup$
Sorry for the lack of information. I edited it.
$endgroup$
– Razvan Paraschiv
Jul 16 '16 at 14:59
2
$begingroup$
Imagine $a_n$ is huge compared to the other terms. Then the maximum sum will certainly have $sigma(n)=1$
$endgroup$
– Joffan
Jul 16 '16 at 15:08
2
2
$begingroup$
Maybe worth trying: without loss of generality, assume the $a_i$'s are non-decreasing, and use the rearrangement inequality?
$endgroup$
– Clement C.
Jul 16 '16 at 14:48
$begingroup$
Maybe worth trying: without loss of generality, assume the $a_i$'s are non-decreasing, and use the rearrangement inequality?
$endgroup$
– Clement C.
Jul 16 '16 at 14:48
1
1
$begingroup$
After the edit, and to provide more details to my previous comment: en.m.wikipedia.org/wiki/Rearrangement_inequality
$endgroup$
– Clement C.
Jul 16 '16 at 14:56
$begingroup$
After the edit, and to provide more details to my previous comment: en.m.wikipedia.org/wiki/Rearrangement_inequality
$endgroup$
– Clement C.
Jul 16 '16 at 14:56
3
3
$begingroup$
For the answer to be $sigma=e$, I think you need $a_1>a_2>cdots>a_n$.
$endgroup$
– Thomas Andrews
Jul 16 '16 at 14:57
$begingroup$
For the answer to be $sigma=e$, I think you need $a_1>a_2>cdots>a_n$.
$endgroup$
– Thomas Andrews
Jul 16 '16 at 14:57
$begingroup$
Sorry for the lack of information. I edited it.
$endgroup$
– Razvan Paraschiv
Jul 16 '16 at 14:59
$begingroup$
Sorry for the lack of information. I edited it.
$endgroup$
– Razvan Paraschiv
Jul 16 '16 at 14:59
2
2
$begingroup$
Imagine $a_n$ is huge compared to the other terms. Then the maximum sum will certainly have $sigma(n)=1$
$endgroup$
– Joffan
Jul 16 '16 at 15:08
$begingroup$
Imagine $a_n$ is huge compared to the other terms. Then the maximum sum will certainly have $sigma(n)=1$
$endgroup$
– Joffan
Jul 16 '16 at 15:08
add a comment |
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2
$begingroup$
Maybe worth trying: without loss of generality, assume the $a_i$'s are non-decreasing, and use the rearrangement inequality?
$endgroup$
– Clement C.
Jul 16 '16 at 14:48
1
$begingroup$
After the edit, and to provide more details to my previous comment: en.m.wikipedia.org/wiki/Rearrangement_inequality
$endgroup$
– Clement C.
Jul 16 '16 at 14:56
3
$begingroup$
For the answer to be $sigma=e$, I think you need $a_1>a_2>cdots>a_n$.
$endgroup$
– Thomas Andrews
Jul 16 '16 at 14:57
$begingroup$
Sorry for the lack of information. I edited it.
$endgroup$
– Razvan Paraschiv
Jul 16 '16 at 14:59
2
$begingroup$
Imagine $a_n$ is huge compared to the other terms. Then the maximum sum will certainly have $sigma(n)=1$
$endgroup$
– Joffan
Jul 16 '16 at 15:08