en>Archelon: /* See also */ eliminate redirect

2014-10-03T22:58:02Z

en>B-80: /* In Canonical Transformation */

2014-02-17T04:26:35Z

In Canonical Transformation

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2013-10-30T19:45:11Z

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← Older revision		Revision as of 21:45, 30 October 2013
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	~~Hello~~. ~~Allow me introduce~~ the ~~writer~~. ~~Her~~ name is ~~Refugia Shryock~~. ~~My day job~~ is a ~~meter reader~~. ~~North Dakota~~ is ~~her beginning place but she will have to move~~ 1 ~~working day or an additional~~. ~~Doing ceramics~~ is ~~what her family~~ and ~~her enjoy~~.<br><br>~~Look at my weblog~~ - [http://www.~~blaze16~~.~~com~~/~~blog~~/~~257588~~ http://~~www~~.~~blaze16~~.~~com~~]		In [[additive number theory]] and [[combinatorics]], a '''restricted sumset''' has the form

			:<math>S=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n \ \mathrm{and}\ P(a_1,\ldots,a_n)\not=0\},</math>

			where <math> A_1,\ldots,A_n</math> are finite nonempty subsets of a [[field (mathematics)\|field]] ''F'' and <math>P(x_1,\ldots,x_n)</math> is a polynomial over ''F''.

			When <math>P(x_1,\ldots,x_n)=1</math>, ''S'' is the usual [[sumset]] <math>A_1+\cdots+A_n</math> which is denoted by ''nA'' if <math>A_1=\cdots=A_n=A</math>; when

			:<math>P(x_1,\ldots,x_n)=\prod_{1\le i<j\le n}(x_j-x_i),</math>

			''S'' is written as <math>A_1\dotplus\cdots\dotplus A_n</math> which is denoted by <math>n^{\wedge} A</math> if <math>A_1=\cdots=A_n=A</math>. Note that \|''S''\| > 0 if and only if there exist <math>a_1\in A_1,\ldots,a_n\in A_n</math> with <math>P(a_1,\ldots,a_n)\not=0</math>.

			== Cauchy–Davenport theorem ==
			The '''Cauchy–Davenport theorem''' named after [[Augustin Louis Cauchy]] and [[Harold Davenport]] asserts that for any prime ''p'' and nonempty subsets ''A'' and ''B'' of the prime order cyclic group '''Z'''/''p'''''Z''' we have the inequality<ref>Nathanson (1996) p.44</ref><ref name=GR1412>Geroldinger & Ruzsa (2009) pp.141–142</ref>

			:<math>\|A+B\|\ge\min\{p,\ \|A\|+\|B\|-1\}.\,</math>

			We may use this to deduce the [[Erdős-Ginzburg-Ziv theorem]]: given any 2''n''−1 elements of '''Z'''/''n'', there is a non-trivial subset that sums to zero modulo ''n''. (Here ''n'' does not need to be prime.)<ref>Nathanson (1996) p.48</ref><ref name=GR53>Geroldinger & Ruzsa (2009) p.53</ref>

			A direct consequence of the Cauchy-Davenport theorem is: Given any set ''S'' of ''p''−1 or more elements, not necessarily distinct, of '''Z'''/''p'''''Z''', every element of '''Z'''/''p'''''Z''' can be written as the sum of the elements of some subset (possibly empty) of ''S''.<ref>Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.</ref>

			[[Kneser's theorem (combinatorics)\|Kneser's theorem]] generalises this to finite abelian groups.<ref name=GR143>Geroldinger & Ruzsa (2009) p.143</ref>

			== Erdős–Heilbronn conjecture ==
			The '''Erdős–Heilbronn conjecture''' posed by [[Paul Erdős]] and [[Hans Heilbronn]] in 1964 states that <math>\|2^\wedge A\|\ge\min\{p,2\|A\|-3\}</math> if ''p'' is a prime and ''A'' is a nonempty subset of the field '''Z'''/''p'''''Z'''.<ref>Nathanson (1996) p.77</ref> This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994<ref>{{cite journal
			\| author = Dias da Silva, J. A.; Hamidoune, Y. O.
			\| title = Cyclic spaces for Grassman derivatives and additive theory
			\| journal = [[London Mathematical Society\|Bulletin of the London Mathematical Society]]
			\| volume = 26
			\| year = 1994
			\| pages = 140–146
			\| doi = 10.1112/blms/26.2.140
			\| issue = 2}}</ref>
			who showed that

			:<math>\|n^\wedge A\|\ge\min\{p(F),\ n\|A\|-n^2+1\},</math>

			where ''A'' is a finite nonempty subset of a field ''F'', and ''p''(''F'') is a prime ''p'' if ''F'' is of characteristic ''p'', and ''p''(''F'') = ∞ if ''F'' is of characteristic 0. Various extensions of this result were given by [[Noga Alon]], M. B. Nathanson and [[Imre Z. Ruzsa\|I. Ruzsa]] in 1996,<ref name="Alon1996"/> Q. H. Hou and [[Zhi-Wei Sun]] in 2002,<ref>{{cite journal
			\| author = Hou, Qing-Hu; [[Zhi-Wei Sun\|Sun, Zhi-Wei]]
			\| title = Restricted sums in a field
			\| journal = [[Acta Arithmetica]]
			\| volume = 102
			\| year = 2002
			\| issue = 3
			\| pages = 239–249
			\| mr = 1884717
			\| doi = 10.4064/aa102-3-3}}</ref>
			and G. Karolyi in 2004.<ref>{{cite journal
			\| author = Károlyi, Gyula
			\| title = The Erdős–Heilbronn problem in abelian groups
			\| journal = [[Israel Journal of Mathematics]]
			\| volume = 139
			\| year = 2004
			\| pages = 349–359
			\| mr = 2041798
			\| doi = 10.1007/BF02787556}}</ref>

			== Combinatorial Nullstellensatz ==
			A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz.<ref name="Alon1999">{{cite journal
			\| author = [[Noga Alon\|Alon, Noga]]
			\| url = http://www.math.tau.ac.il/~nogaa/PDFS/null2.pdf
			\| title = Combinatorial Nullstellensatz
			\| journal = [[Combinatorics, Probability and Computing]]
			\| volume = 8
			\| issue = 1–2
			\| year = 1999
			\| pages = 7–29
			\| mr = 1684621
			\| doi = 10.1017/S0963548398003411}}</ref> Let <math>f(x_1,\ldots,x_n)</math> be a polynomial over a field ''F''. Suppose that the coefficient of the monomial <math>x_1^{k_1}\cdots x_n^{k_n}</math> in <math>f(x_1,\ldots,x_n)</math> is nonzero and <math>k_1+\cdots+k_n</math> is the [[total degree]] of <math>f(x_1,\ldots,x_n)</math>. If <math>A_1,\ldots,A_n</math> are finite subsets of ''F'' with <math>\|A_i\|>k_i</math> for <math>i=1,\ldots,n</math>, then there are <math>a_1\in A_1,\ldots,a_n\in A_n</math> such that <math>f(a_1,\ldots,a_n)\not = 0 </math>.

			The method using the combinatorial Nullstellensatz is also called the polynomial method. This tool was rooted in a paper of N. Alon and M. Tarsi in 1989,<ref>{{cite journal
			\| author = [[Noga Alon\|Alon, Noga]]; Tarsi, Michael
			\| title = A nowhere-zero point in linear mappings
			\| journal = [[Combinatorica]]
			\| volume = 9
			\| year = 1989
			\| pages = 393–395
			\| mr = 1054015
			\| doi = 10.1007/BF02125351
			\| issue = 4}}</ref>
			and developed by Alon, Nathanson and Ruzsa in 1995-1996,<ref name="Alon1996">{{cite journal
			\| author = [[Noga Alon\|Alon, Noga]]; Nathanson, Melvyn B.; Ruzsa, Imre
			\| url = http://www.math.tau.ac.il/~nogaa/PDFS/anrf3.pdf
			\| title = The polynomial method and restricted sums of congruence classes
			\| journal = [[Journal of Number Theory]]
			\| volume = 56
			\| issue = 2
			\| year = 1996
			\| pages = 404–417
			\| mr = 1373563
			\| doi = 10.1006/jnth.1996.0029}}</ref>
			and reformulated by Alon in 1999.<ref name="Alon1999"/>

			==References==
			{{reflist\|2}}
			* {{cite book \| editor1-last=Geroldinger \| editor1-first=Alfred \| editor2-last=Ruzsa \| editor2-first=Imre Z. \| others=Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse) \| title=Combinatorial number theory and additive group theory \| series=Advanced Courses in Mathematics CRM Barcelona \| location=Basel \| publisher=Birkhäuser \| year=2009 \| isbn=978-3-7643-8961-1 \| zbl=1177.11005 }}
			* {{cite book \| first=Melvyn B. \| last=Nathanson \| title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets \| volume=165 \| series=[[Graduate Texts in Mathematics]] \| publisher=[[Springer-Verlag]] \| year=1996 \| isbn=0-387-94655-1 \| zbl=0859.11003 }}

			== External links ==
			*{{cite journal \| author = [[Zhi-Wei Sun\|Sun, Zhi-Wei]] \| title = An additive theorem and restricted sumsets \| year = 2006 \| pages = 1263–1276 \| volume = 15 \| issue = 6 \| journal = Math. Res. Lett. , no. \| arxiv = math.CO/0610981 }}
			* [[Zhi-Wei Sun]]: [http://math.nju.edu.cn/~zwsun/EHLS.pdf On some conjectures of Erdős-Heilbronn, Lev and Snevily] ([[PDF]]), a survey talk.
			*{{mathworld \| urlname = Erdos-HeilbronnConjecture \| title = Erdos-Heilbronn Conjecture}}

			[[Category:Sumsets]]
			[[Category:Additive combinatorics]]
			[[Category:Additive number theory]]

en>Bogus.Koszalka: Undid revision 339123585 by 125.60.173.71 (talk)

2010-01-23T10:30:29Z

Undid revision 339123585 by 125.60.173.71 (talk)

New page

Hello. Allow me introduce the writer. Her name is Refugia Shryock. My day job is a meter reader. North Dakota is her beginning place but she will have to move 1 working day or an additional. Doing ceramics is what her family and her enjoy.<br><br>Look at my weblog - [http://www.blaze16.com/blog/257588 http://www.blaze16.com]

← Older revision		Revision as of 00:58, 4 October 2014
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Generating function (physics) - Revision history

en>Archelon: /* See also */ eliminate redirect

en>B-80: /* In Canonical Transformation */

en>Red Act: /* Example */ Substantial copy editing.

en>Bogus.Koszalka: Undid revision 339123585 by 125.60.173.71 (talk)