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In [[number theory]], the '''minimum overlap problem''' is a problem proposed by [[Hungary|Hungarian]] [[mathematician]] [[Paul Erdős]] in 1955.<ref name="rguy">{{cite book |last1=Guy |first1= Richard K. |author1-link=Richard Guy |title=Unsolved Problems in Number Theory |url= |accessdate=29 January 2011 |issue=3rd edition|year=2004 |editor1-last= Bencsáth |editor1-first=Katalin A. |editor1-link= |editor2-last=Halmos |editor2-first= Paul R. |publisher=Springer Science+Business Media Inc.|location=New York |isbn=0-387-20860-7 |chapter=C17 |pages=199–200}}</ref><ref name="finch">{{cite web |url=http://www.people.fas.harvard.edu/~sfinch/csolve/ovr.pdf |title=Erdös' minimum overlap problem |accessdate=15 December 2013 |last=Finch |first=Steven |authorlink1= |date=2 July 2004 |format=PDF}}</ref>
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== Formal statement of the problem ==
 
Let ''A'' = {''a''<sub>''i''</sub>} and ''B'' = {''b''<sub>''j''</sub>} be two [[Complement (set theory)|complementary subsets]], a splitting of the set of [[natural number]]s {1,2,…,2''n''}, such that both have the same [[cardinal number|cardinality]], namely ''n''. Denote by ''M''<sub>''k''</sub> the number of solutions of the equation ''a''<sub>''i''</sub>&nbsp;−&nbsp;''b''<sub>''j''</sub>&nbsp;=&nbsp;''k'', where ''k'' is an [[integer number|integer]] varying between −2''n'' and 2''n''. ''M''(''n'') is defined as:
:<math>M(n)=\min_{A,B}\max_k M_k \,\!</math>
 
The problem is to estimate ''M''(''n'') when ''n'' is sufficiently large.<ref name="finch" />
 
== History ==
 
This problem can be found amongst the problems proposed by [[Paul Erdős]] in [[combinatorial number theory]], known by English speakers as the ''Minimum overlap problem''. It was first formulated in 1955 in the article ''Some remark on number theory'' of Riveon Lematematica, and has become one of the classical problems described by [[Richard Guy]] in his book ''Unsolved problems in number theory''.<ref name="rguy" />
 
== Partial results ==
 
Since it was first formulated, there has been continuous progress made in the calculation of [[Inferior limit|lower bounds]] and [[Superior limit|upper bounds]] of ''M''(''n''), with the following results:<ref name="rguy" /><ref name="finch" />
 
=== Lower ===
 
{| class="wikitable"
! Limit inferior
! Author(s)
! Year
|-
|<math>M(n) > n/4</math>|| P. Erdős ||1955
|-
|<math>M(n) > (1 - 2^{-1/2} )n </math>||P. Erdős, Scherk||1955
|-
|<math>M(n) > (4 - 6^{-1/2})n/5</math>||S. Swierczkowski||1958
|-
|<math>M(n) > (4 - 15^{1/2})^{1/2}(n -1)</math>||L. Moser||1966
|-
|<math>M(n) > (4 - 15^{1/2})^{1/2} n</math>||J. K. Haugland ||1996
|}
 
=== Upper ===
 
{| class="wikitable"
! Limit superior
! Author(s)
! Year
|-
|<math>M(n)/n = 1/2</math>|| P. Erdős ||1955
|-
|<math>M(n)/n < 2/5</math>||T. S. Motzkin, K. E. Ralston and J. L. Selfridge,||1956
|-
|<math>M(n)/n < 0.38201</math>||J. K. Haugland ||1996
|}
 
J. K. Haugland showed that the [[Limit (mathematics)|limit]] of ''M''(''n'')/''n'' < 0.38569401 unconditionally. For his research, he was awarded a prize in the competition for young scientists in 1993.<ref>{{cite web |url=http://www.neutreeko.net/mop.htm |title= The minimum overlap problem |accessdate=15 December 2013 |last=Haugland |first= Jan Kristian}}</ref> In 1996 he showed that the superior and inferior limits of ''M''(''n'')/''n'' are same, thus proving that the [[Limit (mathematics)|limit]] of ''M''(''n'')/''n'' exists.<ref name="finch" /><ref>{{cite journal
| last=Haugland
| first=Jan Kristian
| year=1996
| title=Advances in the Minimum Overlap Problem
| journal=The Journal of Number Theory
| volume=58
| issue=1
| location= Ohio (USA)
| pages= 71–78
| issn= 0022-314X
| doi=10.1006/jnth.1996.0064
}}</ref>
 
=== The first known values of ''M''(''n'') ===
 
The values of ''M''(''n'') for the first 15 positive integers are the following:<ref name="rguy" />
 
{|style="border: 1px solid #000000;" cellpadding="5" cellspacing="0"
|style="background: LightGrey" | '''''n'''''
|1 ||2||3||4||5||6||7||8||9||10||11||12||13||14||15||...
|-align="center"
|style="background: LightGrey" | '''''M''(''n'')'''
|1||1||2||2||3||3||3||4||4||5||5||5||6||6||6||...
|}
 
== References ==
 
{{reflist}}
 
[[Category:Additive number theory]]
 
 
{{Mathematics-stub}}

Latest revision as of 00:59, 24 November 2014

Lille has been very environment friendly in every aspect of their work. Initially, she did very detailed calculations concerning the buying and selling course of. She obviously knows her work very properly as she also highlighted and smoothed out quite a lot of issues that I am not even aware of as a seller.

of Buy Value Upon the grant of Option to Buy 5% – 10% (Reserving/Possibility charge) Typically 5% Upon signing of the Sale & Buy Agreement or inside eight weeks from the Possibility date 20% much less reserving charge 1. Completion of foundation work 10% 2. Completion of bolstered concrete framework of unit 10% 3. Completion of brick walls of unit 5% 4. Completion of roofing/ceiling of unit 5% 5. Completion of electrical wiring, plumbing and set up of door and window

Mar 31, 2013 The Midtown Hougang New Launch Mixed deveopment nearHougang MRT. Apr 02, 2013 NeWest - Clementi/ West Coast New Launch Mixeddeveopment nearHougang MRT. Glorious location,Premium condominium with Bayfront resort life-style theme and views ofwaterscape. Close to EC pricing - Value for cash! Apr 02, 2013 Alexandra New Launch - Echelon is just 3 minutes walkfrom Redhil MRTstation. Apr 02, 2013 Sengkang New Condominium Launch, La Fiesta- Sengkang MRTstation at your gate. La Fiesta is the last apartment residential development that is withinwalking distance from Sengkang MRT station. First thing first, there are no restrictions on the acquisition of properties by the expats. Expats can easily purchase a property on this city nation. Stratum Condo @ Elias Declare VVIP Discount!

Tuesday's measures – which tightened mortgage phrases for state-constructed homes and delayed public-housing eligibility for certain non-residents – comply with a series of presidency curbs that stretch again over 4 years. It also prompted some investors to unload their stakes in Singaporean property builders on Wednesday, amid a broader market droop. In the midst of this ‘uniquely singapore property search' Property GSS, our most wished foreign customers are nowhere to be seen. With the most recent spherical of government intervention policies geared toward cooling off the property market, there's some confusion of what LTV you're eligible for. Listed below are some clarifications. Globalisation of business real property funding gathers tempo in 2014 as transaction volumes up 23% in Q1 2014

To steer, help consumers to search out their dream homes, their investments of selection in new launch property in Singapore and the regions." Roger Seah 谢富帆 (sixty five) 9620 1223 Builders could conduct property launches in Singapore by way of an in-house advertising and marketing crew, or extra doubtless (to minimise staffing), by appointing external real property businesses to do the marketing. For all new Singapore property gross sales, authorized completion, ie. when authorized titles are conveyed to consumers buying property in Singapore, must happen inside three years of supply of vacant possession. Typically talking, as communities develop, the worth of property also grows. The longer a property in held, the extra potential of getting a high return. This factor works at its finest in Singapore. New Rental SKIES MILTONIA

Presently, the SSD rates are levied at the same charge as buyer's stamp responsibility, i.e. 1 per cent for the primary $180,000, 2 per cent for the next $a hundred and eighty,000 and 3% on the steadiness. The SSD rates are tiered in response to the duration of the holding period, i.e. the seller pays the total SSD charge if the residential property is bought in the first 12 months of purchase; 2/three the full SSD fee if the sale is in the second yr; 1/three the complete SSD rate if within the third year.

Few properties are listed on the Web, although, so discovering a rental lengthy-distance is tough. As an alternative, come in person and guide into a hotel for a couple of days when you scout an area. Most flats for lease have indicators exterior with costs and phone numbers, and most supply long-time period rental choices. I remember a case of a rented condominium in finest location right here in Tokyo, about 120 m2, and the company paid rent for over 20 years. - For this cash you might BUY simply a reasonably large house+land+automobile+parking-lot or condominium, about 7 to 10 km away from the prime location and instead of paying rent give the money to the financial institution for paying off the mortgage. Posted by Edison Foo November 24, 2013 Open for booking NOW !!! Posted by Edison Foo September 12, 2013