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{{Orphan|date=March 2012}} | |||
{{more footnotes|date=April 2011}} | |||
This article describes the '''The Entropy Influence Conjecture'''. | |||
== The Conjecture == | |||
The | For a function <math> f: \{-1,1\}^n \to \{-1,1\} \!</math> the Entropy-Influence relates the following two quantities, both of which may be expressed in terms of the Fourier Expansion of the functions | ||
<math> f = \sum_{S \subset [n]} \hat{f}(S) x_S \!</math>, where <math> x_S = \prod_{i \in S} x_i \!</math>. | |||
The first expression is the total influence of the function defined by | |||
<math> I(f) = \sum_S |S| \hat{f}^2(S) \!</math>. The second terms is the Entropy (of the spectrum) of the function defined by | |||
<math> H(f) = - \sum_S \hat{f}^2(S) \log \hat{f}^2(S) \!</math> (where <math> x \log x = 0 \!</math> as usual). | |||
The conjecture states that there exists an absolute constant C such that for all Boolean functions <math> f: \{-1,1\}^n \to \{-1,1\} \!</math> it holds that <math> H(f) \leq C I(f) \!</math>. | |||
==See also== | |||
* [[Hilbert's problems|Hilbert's 23 problems]] | |||
== References == | |||
* [[Cameron–Erdős conjecture]] ([[Ben J. Green]], 2003, conjectured by Paul)<ref>{{citation | |||
| last = Green | first = Ben | author-link = Ben J. Green | |||
| arxiv = math.NT/0304058 | |||
| doi = 10.1112/S0024609304003650 | |||
| issue = 6 | |||
| journal = The Bulletin of the London Mathematical Society | |||
| mr = 2083752 | |||
| pages = 769–778 | |||
| title = The Cameron-Erdős conjecture | |||
| volume = 36 | |||
| year = 2004}}.</ref> | |||
==References== | |||
{{Reflist}} | |||
*[http://unsolvedproblems.org/ Unsolved Problems in Number Theory, Logic and Cryptography] | |||
* [http://cs.smith.edu/~orourke/TOPP/ The Open Problems Project (TOPP)], discrete and computational geometry problems | |||
== External links == | |||
<!--- Categories ---> | |||
{{DEFAULTSORT:Entropy Influence Conjecture}} | |||
[[Category:Articles created via the Article Wizard]] | |||
[[Category:Entropy]] | |||
Revision as of 13:15, 1 February 2014
Template:Orphan Template:More footnotes
This article describes the The Entropy Influence Conjecture.
The Conjecture
For a function the Entropy-Influence relates the following two quantities, both of which may be expressed in terms of the Fourier Expansion of the functions , where . The first expression is the total influence of the function defined by . The second terms is the Entropy (of the spectrum) of the function defined by (where as usual).
The conjecture states that there exists an absolute constant C such that for all Boolean functions it holds that .
See also
References
- Cameron–Erdős conjecture (Ben J. Green, 2003, conjectured by Paul)[1]
References
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- Unsolved Problems in Number Theory, Logic and Cryptography
- The Open Problems Project (TOPP), discrete and computational geometry problems
External links
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