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In [[mathematics]], an '''addition chain''' for computing a positive integer ''n'' can be given by a [[sequence]] of [[natural number]]s ''v'' and a sequence of index pairs ''w'' such that each term in ''v'' is the sum of two previous terms, the indices of those terms being specified by ''w'': | |||
: ''v'' =(''v''<sub>0</sub>,...,''v''<sub>''s''</sub>), with ''v''<sub>0</sub> = 1 and ''v''<sub>''s''</sub> = ''n'' | |||
:for each 0< ''i'' ≤ ''s'' holds: ''v''<sub>''i''</sub> = ''v''<sub>''j''</sub> + ''v''<sub>''k''</sub>, with ''w''<sub>''i''</sub>=(''j,k'') and 0 ≤ ''j,k'' ≤ ''i'' − 1 | |||
Often only ''v'' is given since it is easy to extract ''w'' from ''v'', but sometimes ''w'' is not uniquely reconstructible. An introduction is given in.<ref>D. E. Knuth, ''The Art of Computer Programming'', Vol 2, "Seminumerical Algorithms", Section 4.6.3, 3rd edition, 1997</ref> | |||
==Examples== | |||
As an example: ''v'' = (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since | |||
:2 = 1 + 1 | |||
:3 = 2 + 1 | |||
:6 = 3 + 3 | |||
:12 = 6 + 6 | |||
:24 = 12 + 12 | |||
:30 = 24 + 6 | |||
:31 = 30 + 1 | |||
Addition chains can be used for [[addition-chain exponentiation]]: so for example we only need 7 [[multiplication]]s to calculate 5<sup>31</sup>: | |||
:5<sup>2</sup> = 5<sup>1</sup> × 5<sup>1</sup> | |||
:5<sup>3</sup> = 5<sup>2</sup> × 5<sup>1</sup> | |||
:5<sup>6</sup> = 5<sup>3</sup> × 5<sup>3</sup> | |||
:5<sup>12</sup> = 5<sup>6</sup> × 5<sup>6</sup> | |||
:5<sup>24</sup> = 5<sup>12</sup> × 5<sup>12</sup> | |||
:5<sup>30</sup> = 5<sup>24</sup> × 5<sup>6</sup> | |||
:5<sup>31</sup> = 5<sup>30</sup> × 5<sup>1</sup> | |||
== | ==Methods for computing addition chains== | ||
Calculating an addition chain of minimal length is not easy; a generalized version of the problem, in which one must find a chain that simultaneously forms each of a sequence of values, is NP-complete.<ref>{{Cite journal|first1=Peter|last1=Downey|first2=Benton|last2=Leong|first3=Ravi|last3=Sethi|title=Computing sequences with addition chains|journal=SIAM Journal on Computing|volume=10|issue=3|year=1981|pages=638–646|doi=10.1137/0210047}}. A number of other papers state that finding a single addition chain is NP-complete, citing this paper, but it does not claim or prove such a result.</ref> There is no known algorithm which can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or small memory usage. However, several techniques to calculate relatively short chains exist. | |||
One very well known technique to calculate relatively short addition chains is the ''binary method'', similar to [[exponentiation by squaring]]. Other well-known methods are the ''factor method'' and ''window method''.{{Citation needed|date=December 2009}} | |||
:<math> | ==Chain length== | ||
Let <math>l(n)</math> denote the smallest ''s'' so that there exists an addition chain | |||
of length ''s'' which computes ''n''. | |||
It is known that <ref>A. Schonhage A lower bound on the length of addition chains, Theoret. Comput. Sci. 1 (1975), 1–12.</ref> | |||
:<math>\log(n)+ \log(\nu(n))-2.13\leq l(n) \leq \log(n) + \log(n)(1+o(1))/\log(\log(n))</math>, | |||
where <math>\nu(n)</math> is [[Hamming weight]] of binary expansion of ''n''. | |||
It is clear that ''l''(2''n'') ≤ ''l''(''n'')+1. Strict inequality is possible, as ''l''(382) = ''l''(191) = 11, observed by Knuth.<ref name=G169/> | |||
A | ==Brauer chain== | ||
A '''Brauer chain''' or '''star addition chain''' is an addition chain in which one of the summands is always the previous chain: that is, | |||
:< | :for each ''k''>0: ''a''<sub>''k''</sub> = ''a''<sub>''k-1''</sub> + ''a''<sub>''j''</sub> for some ''j'' < ''k''. | ||
A '''Brauer number''' is one for which the Brauer chain is minimal.<ref name=G169/> | |||
Brauer proved that | |||
:''l''*(2<sup>''n''</sup>−1) ≤ ''n'' − 1 + ''l''*(''n'') | |||
where ''l''* is the length of the shortest star chain. For many values of ''n'',and in particular for ''n'' ≤ 2500, they are equal: ''l''(''n'') = ''l''*(''n''). But Hansen showed that there are some values of ''n'' for which ''l''(''n'') ≠ ''l''*(''n''), such as ''n'' = 2<sup>6106</sup> + 2<sup>3048</sup> + 2<sup>2032</sup> + 2<sup>2016</sup> + 1 which has ''l''*(''n'') = 6110, ''l''(''n'') ≤ 6109. | |||
==Scholz conjecture== | |||
{{main|Scholz conjecture}} | |||
The [[Scholz conjecture]] (sometimes called the ''Scholz–Brauer'' or ''Brauer–Scholz conjecture''), named after [[A. Scholz]] and Alfred T. Brauer), is a [[conjecture]] from 1937 stating that | |||
:''l''(2<sup>''n''</sup> − 1) ≤ ''n'' − 1 + ''l''(''n'') . | |||
N. Clift checked this by computer for ''n'' ≤ 46. It is known to be true for Brauer numbers.<ref name=G169>Guy (2004) p.169</ref> | |||
==See also== | ==See also== | ||
*[[ | * [[Addition chain exponentiation]] | ||
*[[ | * [[Addition-subtraction chain]] | ||
*[[ | * [[Vectorial addition chain]] | ||
* [[Lucas chain]] | |||
==References== | ==References== | ||
{{reflist}} | {{reflist}} | ||
*{{cite journal | last1=Brauer | first1=Alfred | title=On addition chains | doi=10.1090/S0002-9904-1939-07068-7 | mr=0000245 | year=1939 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=45 | issue=10 | pages=736–739}} | |||
* {{cite book|author=Richard K. Guy|authorlink=Richard K. Guy|title=[[Unsolved Problems in Number Theory]]|publisher=[[Springer-Verlag]]|year=2004|isbn=0-387-20860-7|oclc=54611248 | zbl=1058.11001}} Section C6. | |||
== External links == | |||
* http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html | |||
* {{SloanesRef |sequencenumber=A003313|name=Length of shortest addition chain for n}} | |||
*[http://www.numdam.org/item?id=JTNB_1994__6_1_21_0 F. Bergeron, J. Berstel. S. Brlek "Efficient computation of addition chains"] | |||
{{DEFAULTSORT:Addition Chain}} | |||
[[Category:Addition chains|*]] | |||
* | |||
[[ | [[es:Suma encadenada]] | ||
Revision as of 09:55, 12 August 2014
In mathematics, an addition chain for computing a positive integer n can be given by a sequence of natural numbers v and a sequence of index pairs w such that each term in v is the sum of two previous terms, the indices of those terms being specified by w:
- v =(v0,...,vs), with v0 = 1 and vs = n
- for each 0< i ≤ s holds: vi = vj + vk, with wi=(j,k) and 0 ≤ j,k ≤ i − 1
Often only v is given since it is easy to extract w from v, but sometimes w is not uniquely reconstructible. An introduction is given in.[1]
Examples
As an example: v = (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since
- 2 = 1 + 1
- 3 = 2 + 1
- 6 = 3 + 3
- 12 = 6 + 6
- 24 = 12 + 12
- 30 = 24 + 6
- 31 = 30 + 1
Addition chains can be used for addition-chain exponentiation: so for example we only need 7 multiplications to calculate 531:
- 52 = 51 × 51
- 53 = 52 × 51
- 56 = 53 × 53
- 512 = 56 × 56
- 524 = 512 × 512
- 530 = 524 × 56
- 531 = 530 × 51
Methods for computing addition chains
Calculating an addition chain of minimal length is not easy; a generalized version of the problem, in which one must find a chain that simultaneously forms each of a sequence of values, is NP-complete.[2] There is no known algorithm which can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or small memory usage. However, several techniques to calculate relatively short chains exist. One very well known technique to calculate relatively short addition chains is the binary method, similar to exponentiation by squaring. Other well-known methods are the factor method and window method.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
Chain length
Let denote the smallest s so that there exists an addition chain of length s which computes n. It is known that [3]
- ,
where is Hamming weight of binary expansion of n.
It is clear that l(2n) ≤ l(n)+1. Strict inequality is possible, as l(382) = l(191) = 11, observed by Knuth.[4]
Brauer chain
A Brauer chain or star addition chain is an addition chain in which one of the summands is always the previous chain: that is,
- for each k>0: ak = ak-1 + aj for some j < k.
A Brauer number is one for which the Brauer chain is minimal.[4]
Brauer proved that
- l*(2n−1) ≤ n − 1 + l*(n)
where l* is the length of the shortest star chain. For many values of n,and in particular for n ≤ 2500, they are equal: l(n) = l*(n). But Hansen showed that there are some values of n for which l(n) ≠ l*(n), such as n = 26106 + 23048 + 22032 + 22016 + 1 which has l*(n) = 6110, l(n) ≤ 6109.
Scholz conjecture
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The Scholz conjecture (sometimes called the Scholz–Brauer or Brauer–Scholz conjecture), named after A. Scholz and Alfred T. Brauer), is a conjecture from 1937 stating that
- l(2n − 1) ≤ n − 1 + l(n) .
N. Clift checked this by computer for n ≤ 46. It is known to be true for Brauer numbers.[4]
See also
References
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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 - 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.
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External links
- http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html
- Template:SloanesRef
- F. Bergeron, J. Berstel. S. Brlek "Efficient computation of addition chains"
- ↑ D. E. Knuth, The Art of Computer Programming, Vol 2, "Seminumerical Algorithms", Section 4.6.3, 3rd edition, 1997
- ↑ 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. A number of other papers state that finding a single addition chain is NP-complete, citing this paper, but it does not claim or prove such a result. - ↑ A. Schonhage A lower bound on the length of addition chains, Theoret. Comput. Sci. 1 (1975), 1–12.
- ↑ 4.0 4.1 4.2 Guy (2004) p.169