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In mathematics, '''arithmetic combinatorics''' arose out of the interplay between [[number theory]], [[combinatorics]], [[ergodic theory]] and [[harmonic analysis]]. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). '''Additive combinatorics''' is the special case when only the operations of addition and subtraction are involved.


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the difference set
:<math>A-A := \{x-y: x,y \in A\},</math>


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:<math>A\times A := \{xy: x,y \in A\}</math>
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be, and how are the sizes of these sets related?  (Not to be confused: the terms [[difference set]] and [[product set]] can have other meanings.)


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The sets being studied may also be subsets of algebraic structures other than the integers, for example, [[group (mathematics)|groups]], [[ring (mathematics)|rings]] and [[field (mathematics)|fields]].<ref>{{cite journal |title=A sum-product estimate in finite fields, and applications |first=Jean |last=Bourgain |first2=Nets |last2=Katz |first3=Terence |last3=Tao |year=2004 |journal=Geometric And Functional Analysis |volume=14 |issue=1 |pages=27–57 |doi=10.1007/s00039-004-0451-1 }}</ref>
 
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Arithmetic combinatorics is explained in [[Ben J. Green|Green's]] [http://www.ams.org/bull/2009-46-03/S0273-0979-09-01231-2/S0273-0979-09-01231-2.pdf review] of "Additive Combinatorics" by [[Terence Tao|Tao]] and [[Van H. Vu|Vu]].


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==See also==
 
*[[Additive number theory]]
  <li>[http://222.243.160.155/forum.php?mod=viewthread&tid=12889812&fromuid=322012 http://222.243.160.155/forum.php?mod=viewthread&tid=12889812&fromuid=322012]</li>
*[[Corners theorem]]
 
*[[Ergodic Ramsey theory]]
  <li>[http://elec.newroomschat.com/spip.php?article30/ http://elec.newroomschat.com/spip.php?article30/]</li>
*[[Green–Tao theorem]]
 
*[[Problems involving arithmetic progressions]]
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24013326 http://enseignement-lsf.com/spip.php?article64#forum24013326]</li>
<!-- *[[Restricted sum set]] -->
 
*[[Schnirelmann density]]
  <li>[http://www.francochilenos.com/spip.php?article2750/ http://www.francochilenos.com/spip.php?article2750/]</li>
*[[Shapley–Folkman lemma]]
 
*[[Sidon set]]
  <li>[http://lmusicradio.altervista.org/osclass/index.php?page=item&id=74610 http://lmusicradio.altervista.org/osclass/index.php?page=item&id=74610]</li>
*[[Sum-free set]]
 
*[[Szemerédi's theorem]]
</ul>


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==Notes==
{{Reflist}}


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==References==
 
* {{cite journal | first= Izabella | last = Łaba | authorlink = Izabella Łaba | title=From harmonic analysis to arithmetic combinatorics | journal=Bull. Amer. Math. Soc. | volume=45 | year=2008 | issue=01 | pages=77–115 | doi=10.1090/S0273-0979-07-01189-5 }}
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24015270 http://enseignement-lsf.com/spip.php?article64#forum24015270]</li>
*[http://www.cs.berkeley.edu/~luca/pubs/addcomb-sigact.pdf Additive Combinatorics and Theoretical Computer Science], Luca Trevisan, SIGACT News, June 2009
 
*{{cite book |last=Bibak|first=Khodakhast |editor-last1=Borwein |editor-first1=Jonathan M. |editor-last2=Shparlinski |editor-first2=Igor E. |editor-last3=Zudilin |editor-first3=Wadim |title=Number Theory and Related Fields: In Memory of Alf van der Poorten |publisher= Springer Proceedings in Mathematics & Statistics, Vol. 43, Springer, New York |date=2013 |pages= 99-128|chapter=Additive combinatorics with a view towards computer science and cryptography |doi=10.1007/978-1-4614-6642-0_4 |isbn=978-1-4614-6642-0}}
  <li>[http://elec.newroomschat.com/spip.php?article30/ http://elec.newroomschat.com/spip.php?article30/]</li>
*[http://people.math.gatech.edu/~ecroot/E2S-01-11.pdf Open problems in additive combinatorics], E Croot, V Lev
 
*[http://www.ams.org/notices/200103/fea-tao.pdf From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE], [[Terence Tao]], AMS Notices March 2001
  <li>[http://www.achicourtautrement.fr/spip.php?article451/ http://www.achicourtautrement.fr/spip.php?article451/]</li>
* {{cite book | last1=Tao | first1=Terence | author1-link=Terence Tao | last2=Vu | first2=Van H. | author2-link=Van H. Vu | title=Additive combinatorics | series=Cambridge Studies in Advanced Mathematics | volume=105 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-85386-9 | zbl=1127.11002 }}
 
* {{cite book | editor1-first=Andrew | editor1-last=Granville | editor1-link=Andrew Granville | editor2-first=Melvyn B. | editor2-last=Nathanson| editor3-first=József | editor3-last=Solymosi | title=Additive Combinatorics | series= CRM Proceedings & Lecture Notes | volume=43 | publisher=[[American Mathematical Society]] | year=2007 | isbn=978-0-8218-4351-2 | zbl=1124.11003 }}
  <li>[http://ai33228.com/forum.php?mod=viewthread&tid=445130&fromuid=79508 http://ai33228.com/forum.php?mod=viewthread&tid=445130&fromuid=79508]</li>
*{{cite book | first=Henry | last=Mann |authorlink=Henry Mann
 
|title=Addition Theorems: The Addition Theorems of Group Theory and Number Theory
  <li>[http://lab.nqnwebs.com/lavoz_bak/spip.php?article5818/ http://lab.nqnwebs.com/lavoz_bak/spip.php?article5818/]</li>
|publisher=[http://www.krieger-publishing.com/subcats/MathematicsandStatistics/mathematicsandstatistics.html Robert E. Krieger Publishing Company]
 
|location=Huntington, New York
  </ul>
|year=1976
|edition=Corrected reprint of 1965 Wiley
|isbn=0-88275-418-1
}}
* {{cite book | title=Additive Number Theory: the Classical Bases | volume=164 | series=Graduate Texts in Mathematics | author=Melvyn B. Nathanson | publisher=Springer-Verlag | year=1996 | isbn=0-387-94656-X }}
* {{cite book | title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets | volume=165 | series=Graduate Texts in Mathematics | author=Melvyn B. Nathanson | publisher=Springer-Verlag | year=1996 | isbn=0-387-94655-1 }}


== Nike Air Max Suisse Alors ==
==Further reading==
*[http://www.math.ucla.edu/~tao/254a.1.03w/ Some Highlights of Arithmetic Combinatorics], resources by [[Terence Tao]]
*[http://math.stanford.edu/~ksound/Notes.pdf Additive Combinatorics: Winter 2007], K Soundararajan
*[http://lucatrevisan.wordpress.com/2009/04/17/earliest-connections-of-additive-combinatorics-and-computer-science/ Earliest Connections of Additive Combinatorics and Computer Science], Luca Trevisan


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[[Category:Additive number theory]]
 
[[Category:Sumsets]]
  <li>[http://www.loocai.cn/forum.php?mod=viewthread&tid=236692 http://www.loocai.cn/forum.php?mod=viewthread&tid=236692]</li>
[[Category:Harmonic analysis]]
 
[[Category:Ergodic theory]]
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Latest revision as of 23:09, 21 August 2013

In mathematics, arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved.

For example: if A is a set of N integers, how large or small can the sumset

A+A:={x+y:x,yA},

the difference set

AA:={xy:x,yA},

and the product set

A×A:={xy:x,yA}

be, and how are the sizes of these sets related? (Not to be confused: the terms difference set and product set can have other meanings.)

The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.[1]

Arithmetic combinatorics is explained in Green's review of "Additive Combinatorics" by Tao and Vu.

See also

Notes

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.

References

  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • Additive Combinatorics and Theoretical Computer Science, Luca Trevisan, SIGACT News, June 2009
  • 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
  • Open problems in additive combinatorics, E Croot, V Lev
  • From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE, Terence Tao, AMS Notices March 2001
  • 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
  • 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
  • 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
  • 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
  • 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

Further reading

Template:Numtheory-stub

  1. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang