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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 14:15, 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).
![f=\sum _{{S\subset [n]}}{\hat {f}}(S)x_{S}\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/496ce061e94f65a69f0150207b397e8b17553f79)
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
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- Unsolved Problems in Number Theory, Logic and Cryptography
- The Open Problems Project (TOPP), discrete and computational geometry problems
External links
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