Rearrangement inequalty :(BOMC)

Discussion on Bangladesh National Math Camp
sourav das
Posts:461
Joined:Wed Dec 15, 2010 10:05 am
Location:Dhaka
Contact:
Rearrangement inequalty :(BOMC)

Unread post by sourav das » Fri Oct 28, 2011 9:26 am

Consider two collections of real numbers in increasing order,
$a_1 \leq a_2 \leq a_3··· \leq a_n$ and $b_1 \leq b_2 \leq···\leq b_n$.
For any permutation $({a_1}',{a_2}',...,{a_n}')$of$(a_1,a_2,...,a_n)$, it happens that
$a_1 b_1 + a_2 b_2 + ··· + a_n b_n \geq {a_1}'b_1 + {a_2}'b_2+..+{a_n}'b_n \geq$
$\geq a_nb_1 + a_n b_2 + ··· + a_1b_n$
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

willpower
Posts:30
Joined:Tue Nov 01, 2011 6:30 pm
Location:Pakistan

Re: Rearrangement inequalty :(BOMC)

Unread post by willpower » Thu Nov 03, 2011 11:08 pm

If \[{a_{1}}', {a_{2}}', ...{a_{n}}'\] is any permutation of \[a_{1}, a_{2}, ..., a_{n}\], how can you express \[{a_{1}}'\] in terms of \[a_{1}\]?
Everybody is a genius; but if you judge a fish on its ability to climb a tree, it will live its entire life believing that it is stupid. - Albert Einstein

User avatar
nafistiham
Posts:829
Joined:Mon Oct 17, 2011 3:56 pm
Location:24.758613,90.400161
Contact:

Re: Rearrangement inequalty :(BOMC)

Unread post by nafistiham » Thu Nov 03, 2011 11:34 pm

it means ${a_n}'$ can be any of $(a_1,a_2,...,a_n)$
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Introduction:
Nafis Tiham
CSE Dept. SUST -HSC 14'
http://www.facebook.com/nafistiham
nafistiham@gmail

Hasib
Posts:238
Joined:Fri Dec 10, 2010 11:29 am
Location:খুলনা, বাংলাদেশ
Contact:

Re: Rearrangement inequalty :(BOMC)

Unread post by Hasib » Thu Nov 03, 2011 11:38 pm

@Willpower:

note that it isnt bound to be $a_i=a_i'$

it may be $a_1=a_5'$ for example.
A man is not finished when he's defeated, he's finished when he quits.

willpower
Posts:30
Joined:Tue Nov 01, 2011 6:30 pm
Location:Pakistan

Re: Rearrangement inequalty :(BOMC)

Unread post by willpower » Thu Nov 03, 2011 11:59 pm

All right! Thank you.
Everybody is a genius; but if you judge a fish on its ability to climb a tree, it will live its entire life believing that it is stupid. - Albert Einstein

Post Reply