Clique problem - Revision history

en>David Eppstein: /* References */ url => chapter-url

2014-12-02T22:14:53Z

References: url => chapter-url

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46.121.232.249: /* Definitions */

2014-02-28T06:48:09Z

Definitions

← Older revision		Revision as of 08:48, 28 February 2014
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	~~{{Unreferenced\|date=August 2008}}~~		Name: Saul Pinkerton<br>My age: 24 years old<br>Country: France<br>Home town: Bois-Colombes <br>Postal code: 92270<br>Street: 70 Avenue De L'amandier<br><br>Here is my blog :: [http://www.staymarlborough.co.nz/_admin/cheap-ghd-hair-straighteners.asp ghd straighteners nz]
	~~{{numeral systems}}~~

	'''Mixed radix''' [[numeral system]]s are [[non-standard positional numeral systems]] in which the numerical [[radix\|base]] varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same factor. Such units are common for instance in measuring time; a time of 32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds might be expressed as a number of minutes in mixed-radix notation as:

	~~... 32, 5, 7, 45; 15, 500~~
	~~... ∞, 7, 24, 60; 60, 1000~~

	~~or as~~

	~~:32~~<~~sub>∞</sub>5<sub>7</sub>7<sub~~>24<~~/sub~~>~~45<sub>60~~<~~/sub~~>~~.15<sub>60</sub>500<sub>1000</sub>~~

	In the tabular format, the digits are written above their base, and a [[semicolon]] is used to indicate the [[radix point]]. In numeral format, each digit has its associated base attached as a subscript, and the position of the radix point is indicated by a [[full stop]] or [[full stop\|period]]. The base for each digit is the number of corresponding units that make up the next larger unit. As a consequence there is no base (written as ∞) for the first (most significant) digit, since here the "next larger unit" does not exist (and note that one could not add a larger unit of "month" or "year" to the sequence of units, as they are not integer multiples of "week").

	~~==Examples==~~
	The most familiar example of mixed radix systems is in timekeeping and calendars. Western time radicies vary from [[decimal]] centuries, decades and years to [[duodecimal]] months, [[trigesimal]] (and untrigesimal) days, overlapped with base 52 weeks, and [[septenary]] days. One variant uses [[base 13\|tridecimal]] months, [[Quaternary numeral system\|quaternary]] weeks, and septenary days. Time is further divided by [[base 24\|quadrivigesimal]] hours, [[sexagesimal]] minutes and seconds, then decimal fractions thereof.

	~~A mixed radix numeral system can often benefit from a tabular summary. The system for describing the 604800 seconds of a week starting from midnight on Sunday runs as follows:~~

	~~{\| class="wikitable"~~
	~~\| '''Radix~~:~~''' \|\| 7 \|\| 2 \|\| 12 \|\| 60 \|\| 60~~
	\|-
	~~\| '''Denomination:''' \|\| day \|\| half-day \|\| hour \|\| minute \|\| second~~
	\|-
	~~\| '''Place value (seconds):''' \|\| 86400 \|\| 43200 \|\| 3600 \|\| 60 \|\| 1~~
	\|-
	\|-
	~~\| colspan=6\| '''Digit translations …'''~~
	\|-
	~~\| '''day:'''~~
	~~\| colspan=5 \| 0=Sunday, 1=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday, 5=Friday, 6=Saturday~~
	\|-
	~~\| '''half-day:'''~~
	~~\| colspan=5 \| 0=AM, 1=PM~~
	\|-
	~~\| '''hour:'''~~
	~~\| colspan=5 \| 0 is written as "12"~~
	\|}

	~~In this numeral system, the mixed radix numeral 3~~<~~sub~~>~~7</sub>1<sub>2</sub>5<sub>12</sub>51<sub>60</sub>57<sub>60</sub> seconds would be interpreted as 05:51~~:~~57 p.m. on Wednesday, and 0<sub>7</sub>0<sub>2</sub>0<sub>12</sub>02<sub>60</sub>24~~<~~sub~~>~~60</sub> would be 12:02:24 a.m. on Sunday. ''Ad hoc'' notations for mixed radix numeral systems are commonplace.~~

	The [[maya calendar]] consists of several overlapping cycles of different radicies. A short count ''[[tzolk'in]]'' overlaps [[vigesimal]] named days with [[base 13\|tridecimal]] numbered days. A ''[[haab']]'' consists of vigesimal days, [[octodecimal]] ''months'', and base-52 years forming a ''round''. In addition, a ''long count'' of vigesimal days, octodecimal ''winal'', then vigesimal ''tun'', ''k'atun'', ''b'ak'tun'', etc. tracks historical dates.

	A second example of a mixed radix [[numeral system]] in current use is in the design and use of [[currency]], where a limited set of denominations are printed or minted with the objective of being able to represent any monetary quantity; the amount of money is then represented by the number of [[coins]] or [[banknotes]] of each denomination. When deciding which denominations to create (and hence which radices to mix), a compromise is aimed for between a minimal number of different denominations, and a minimal number of individual pieces of coinage required to represent typical quantities. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, and coins are minted for £2, £1, 50p, 20p, 10p, 5p, 2p and 1p—these follow the [[Preferred values#1-2-5 series\| 1-2-5 series of preferred values]].

	Mixed-radix representation is also relevant to mixed-radix versions of the [[Cooley-Tukey FFT algorithm]], in which the indices of the input values are expanded in a mixed-radix representation, the indices of the output values are expanded in a corresponding mixed-radix representation with the order of the bases and digits reversed, and each subtransform can be regarded as a Fourier transform in one digit for all values of the remaining digits.

	~~==Manipulation==~~
	Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. Conversion of values from one mixed base to another is easily accomplished by first converting the place values of the one system into the other, and then applying the digits from the one system against these.

	~~[[APL programming language\|APL]] includes operators to convert to and from mixed-radix systems.~~

	~~== Factorial number system ==~~
	~~{{main\|Factorial number system}}~~

	~~Another proposal is the so-called [[factorial]] number system~~:

	~~{\| class="wikitable"~~
	\|-
	~~\| '''Radix''' \|\| 8 \|\| 7 \|\| 6 \|\| 5 \|\| 4 \|\| 3 \|\| 2 \|\| 1~~
	\|-
	~~\| '''Place value'~~'~~' \|\| 7! \|\| 6! \|\| 5! \|\| 4! \|\| 3! \|\| 2! \|\| 1! \|\| 0!~~
	\|-
	~~\| '''Place value in decimal''' \|\| 5040 \|\| 720 \|\| 120 \|\| 24 \|\| 6 \|\| 2 \|\| 1 \|\| 1~~
	\|-
	~~\| '''Highest digit allowed''' \|\| 7 \|\| 6 \|\| 5 \|\| 4 \|\| 3 \|\| 2 \|\| 1 \|\| 0~~
	\|}

	For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in [[decimal]]: 5×5! + 4×4! + 3×3! + 2×2! + 1×1! It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:

	: <~~math~~> ~~\sum_{i=0}^{n} (([i+1]+1)-1) \cdot ([i]+1)! = ([n+1]+1)! - 1~~ <~~/math~~>

	~~There~~ is a natural mapping between the integers 0, ..., ''n''! − 1 and [[permutation]]s of ''n'' elements in lexicographic order, which uses the factorial representation of the integer, followed by an interpretation as a [[Permutations#Numbering permutations\|Lehmer code]].

	The above equation is a particular case of the following general rule for any radix (either standard or mixed) base representation which expresses the fact that any radix (either standard or mixed) base representation is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective weights multiplied by the index is always the next weight minus one:

	: ~~<math> \sum_{i=0}^{n} (m_{i+1} - 1) \cdot M_i = M_{n+1} - 1 </math>, where <math>M_i = \prod_{j=1}^{i} m_j, m_j > 1, M_0 = 1 </math>,~~
	~~which can be easily proved with [~~[~~mathematical induction]].~~

	~~== Primorial number system ==~~
	~~Another proposal is the number system with successive prime numbers as radix, whose place values are [[primorial]] numbers~~:

	~~{\| class="wikitable"~~
	\|-
	~~\| '''radix:''' \|\| 17 \|\| 13 \|\| 11 \|\| 7 \|\| 5 \|\| 3 \|\| 2~~
	\|-
	~~\| '''place value:''' \|\| (p<sub>6</sub>=13)# \|\| (p<sub>5</sub>=11)# \|\| (p<sub>4</sub>=7)# \|\| (p<sub>3</sub>=5)# \|\| (p<sub>2</sub>=3)# \|\| (p<sub>1</sub>=2)# \|\| (p<sub>0</sub>=1)#~~
	\|-
	~~\| '''decimal:''' \|\| 30030 \|\| 2310 \|\| 210 \|\| 30 \|\| 6 \|\| 2 \|\| 1~~
	\|}

	~~: <math> \sum_{i=0}^{n} (p_{i+1} - 1) \cdot p_i\# = p_{n+1}\# - 1 <~~/~~math> where <math>p_i\# = \prod_{j=1}^{i} p_j<~~/~~math>, and p<sub>j</sub> = j<sup>th</sup> prime, p<sub>0</sub># = p<sub>0</sub> = 1~~.

	~~==References==~~

	* [[Donald Knuth]]. ~~''[[The Art of Computer Programming]]'', Volume 2: ''Seminumerical Algorithms'', Third Edition~~. ~~Addison-Wesley, 1997. ISBN 0~~-~~201~~-~~89684~~-~~2. Pages 65–66, 208–209, and 290.~~
	* [[Georg Cantor]]. ''Über einfache Zahlensysteme'', Zeitschrift für Math. und Physik '''14'''(1869), 121–128.


	~~[[Category:Non-standard positional numeral systems]~~]

en>Quebec99: fixed references

2013-11-27T18:21:13Z

fixed references

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en>David Eppstein: /* Maximal versus maximum */ co-well-covered

2012-08-23T21:53:44Z

Maximal versus maximum: co-well-covered

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