Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion. The motivation to ask here if the inequality below holds is to avoid a length analysis in this paper. Let $0\leq i<j\leq N$ be integers and $\alpha \in (3,4]$ is there a constant $C>0$ independent of $i,j$ and $N$ such that $$ \frac{C}{|i-j|^{\alpha-2}}\leq \sum_{0\leq k\leq i< j\leq r \leq N} \frac{1}{|k-r|^{\alpha}} $$ holds ?

Remark: similar upper bound holds, to get it I simply compared this sum with a double integral. For the lower bound it seems that such comparison does not work.

share|improve this question
2  
If you take $i=0,j=N$ then the inequality becomes $C/N^{a-2}\le 1/N^a$, i.e $C \le N^{-2}$. Therefore I doubt if $C$ can be independent of all the values of $i,j,N$. –  Ralph May 18 '12 at 7:51
    
Thanks for the observation Ralph, you right such lower bound can not exists. –  Leandro May 18 '12 at 8:58
    
@Ralph: why not augment your comment into an answer so that the question can be officially laid to rest? –  Suvrit May 18 '12 at 15:44
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.