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3h
revised Algorithm Involving Quadratic Diophantine Equation
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asked Algorithm Involving Quadratic Diophantine Equation
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comment Entropy dominance of certain restricted sequenes
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accepted Entropy dominance of certain restricted sequenes
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revised Entropy dominance of certain restricted sequenes
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asked Entropy difference dominance of decreasing sequences
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revised Entropy dominance of certain restricted sequenes
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revised Entropy dominance of certain restricted sequenes
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comment 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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revised Entropy dominance of certain restricted sequenes
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comment 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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revised Entropy dominance of certain restricted sequenes
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comment Entropy dominance
Added mathoverflow.net/questions/191397/….
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asked Entropy dominance of certain restricted sequenes
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accepted Entropy dominance
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revised Entropy dominance
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revised Entropy dominance
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comment 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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comment 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?