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Algorithm Involving Quadratic Diophantine Equation
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Entropy dominance of certain restricted sequenes
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Entropy dominance of certain restricted sequenes
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asked  Entropy difference dominance of decreasing sequences 
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Entropy dominance of certain restricted sequenes
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Entropy dominance of certain restricted sequenes
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Entropy dominance of certain restricted sequenes
The difficulty seems to be because if you add two variables that are close enough, the entropy may not noticeably change to be always strictly increasing. That is my intuition on these finite sequence artificial probability distribution. However I cannot rule out the fact that may be we can make the entropy strictly increasing by adding new variables somehow so that the variables $a_i$ do not converge to any limit. It would be interesting design sequences that diverge whose entropy is strictly increasing while the increase becomes smaller as $n$ increases. Thought that would be interesting. 
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Entropy dominance of certain restricted sequenes
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Entropy dominance of certain restricted sequenes
The following property also seems true. If $m+1\leq n$, then let $p$ correspond to $\{a_i\}_{i=1}^{m}$, $q$ correspond to $\{a_i\}_{i=1}^{m+1}$, $r$ correspond to $\{a_i\}_{i=1}^{n}$ and $s$ correspond to $\{a_i\}_{i=1}^{n+1}$ where $a_i$ satisfy properties above. Is it true that, $H(q)H(p)>H(s)H(r)$? Should I formulate another question. 
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Entropy dominance of certain restricted sequenes
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Entropy dominance
Added mathoverflow.net/questions/191397/…. 
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Entropy dominance
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Entropy dominance
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Entropy dominance
Thank you for the wonderful insight. So this is my question actually from which the above spun out. Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define $q_i=\frac{a_i}{C+\sum_{i=1}^na_i}$ and $q_{n+1}=\frac{C}{C+\sum_{i=1}^na_i}$. Now does entropy of $q$ dominate entropy of $p$? 
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asked  Entropy dominance 
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What's your favorite equation, formula, identity or inequality?
I have not been introduced to negative binomials formally. This seems fairly advanced. Does it have algebraic, arithmetic, combinatorial or geometric meaning? Do you have a reference that could say something substantial about this interesting formulation? 