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In mathematics, a '''±1–sequence''' is a [[sequence]] of numbers, each of which is either 1 or −1. One example is the sequence {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ...)}}, where {{math|1=''x''<sub>''i''</sub> = (−1)<sup>''i''+1</sup>}}. | |||
Such sequences are commonly studied in [[discrepancy theory]]. | |||
==Erdős discrepancy problem== | |||
<!--{{Further|[[Erdős discrepancy problem]]}}--> | |||
Let ''S''=(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>,...) be a ±1–sequence, where ''x''<sub>''j''</sub> denotes the ''j''<sup>th</sup> term. The ''Erdős discrepancy problem'' asks whether there exists a sequence ''S'' and an integer ''C''<sub>''S''</sub>, such that for any two positive integers ''d'' and ''k'', | |||
: <math> \left| \sum_{i=1}^k x_{id} \right| \leq C_S.</math> | |||
{{Asof|2010|October}}, this problem is currently being studied by the [[Polymath Project]]. | |||
==Barker codes== | |||
{{main|Barker code}} | |||
A '''Barker code''' is a sequence of ''N'' values of +1 and −1, | |||
:<math>a_j</math> for j = 1, 2, …, N | |||
such that | |||
:<math>\left|\sum_{j=1}^{N-v} a_j a_{j+v}\right| \le 1\,</math> | |||
for all <math>1 \le v < N</math>.<ref>{{cite book |author=Barker, R. H. |chapter=Group Synchronizing of Binary Digital Sequences |title=Communication Theory |place=London |publisher=Butterworth |pages=273–287 |year=1953}}</ref> | |||
Barker codes of length 11 and 13 are used in [[direct-sequence spread spectrum]] and [[pulse compression|pulse compression radar]] systems because of their low [[autocorrelation]] properties. | |||
==See also== | |||
*[[Binary sequence]] | |||
*[[Discrepancy of hypergraphs]] | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
*{{cite book|title=The Discrepancy Method: Randomness and Complexity|last=Chazelle|first=Bernard|author-link=Bernard Chazelle|publisher=Cambridge University Press|ISBN=0-521-77093-9}} | |||
==External links== | |||
*[http://michaelnielsen.org/polymath1/index.php?title=The_Erd%C5%91s_discrepancy_problem The Erdős discrepancy problem] – Polymath Project | |||
{{DEFAULTSORT:±1-Sequence}} | |||
[[Category:Binary sequences]] |
Latest revision as of 07:32, 1 April 2013
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Such sequences are commonly studied in discrepancy theory.
Erdős discrepancy problem
Let S=(x1, x2, x3,...) be a ±1–sequence, where xj denotes the jth term. The Erdős discrepancy problem asks whether there exists a sequence S and an integer CS, such that for any two positive integers d and k,
Template:Asof, this problem is currently being studied by the Polymath Project.
Barker codes
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A Barker code is a sequence of N values of +1 and −1,
such that
for all .[1]
Barker codes of length 11 and 13 are used in direct-sequence spread spectrum and pulse compression radar systems because of their low autocorrelation properties.
See also
Notes
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References
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External links
- The Erdős discrepancy problem – Polymath Project
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534