Isoelastic function
In computer science, the complexity function of a string, a finite or infinite sequence u of letters from some alphabet, is the function pu(n) of a positive integer n that counts the number of different factors (consecutive substrings) of length n from the string u.[1][2][3][4][5]
For a string u of length at least n over an alphabet of size k we clearly have
the bounds being achieved by the constant word and a disjunctive word,[6] for example, the Champernowne word respectively.[7] For infinite words u, we have pu(n) bounded if u is ultimately periodic (a finite, possibly empty, sequence followed by a finite cycle). Conversely, if pu(n) ≤ n for some n, then u is ultimately periodic.[3][8]
An aperiodic sequence is one which is not ultimately periodic. An aperiodic sequence has strictly increasing complexity function (this is the Morse–Hedlund theorem),[9] so p(n) is at least n+1.[10]
A set S of finite binary words is balanced if for each n the subset Sn of words of length n has the property that the Hamming weight of the words in Sn takes at most two distinct values. A balanced sequence is one for which the set of factors is balanced.[11] A balanced sequence has complexity sequence at most n+1.[12]
A Sturmian word over a binary alphabet is one with complexity function n + 1.[13] A sequence is Sturmian if and only if it is balanced and aperiodic.[2][14] An example is the Fibonacci word.[13][15] More generally, a Sturmian word over an alphaber of size k is one with complexity n+k−1. An Arnoux-Rauzy word over a ternary alphabet has complexity 2n + 1:[13] an example is the Tribonacci word.[16]
For recurrent words, those in which each factor appears infinitely often, the complexity function almosr characterises the set of factors: if s is a recurrent word with the same complexity function as t are then s has the same set of factors as t or δt where δ denotes the letter doubling morphism a → aa.[17]
The topological entropy of an infinite sequence u is defined by
The limit exists as the logarithm of the complexity function is subadditive.[18][19] Every real number between 0 and 1 occurs as the topological entry of some sequence.[20]
For x a real number and b an integer ≥ 2 then the complexity function of x in base b is the complexity function p(x,b,n) of the sequence of digits of x written in base b. If x is an irrational number then p(x,b,n) ≥ n+1; if x is rational then p(x,b,n) ≤ C for some constant C depending on x and b.[6]
References
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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.
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- ↑ Lothaire (2011) p.7
- ↑ 2.0 2.1 Lothaire (2011) p.46
- ↑ 3.0 3.1 Pytheas Fogg (2002) p.3
- ↑ Berstel et al (2009) p.82
- ↑ Allouche & Shallit (2003) p.298
- ↑ 6.0 6.1 Bugeaud (2012) p.91
- ↑ Cassaigne & Nicolas (2010) p.165
- ↑ Allouche & Shallit (2003) p.302
- ↑ Cassaigne & Nicolas (2010) p.166
- ↑ Lothaire (2011) p.22
- ↑ Allouche & Shallit (2003) p.313
- ↑ Lothaire (2011) p.48
- ↑ 13.0 13.1 13.2 Pytheas Fogg (2002) p.6
- ↑ Allouche & Shallit (2003) p.318
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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 - ↑ Pytheas Fogg (2002) p.368
- ↑ Berstel et al (2009) p.84
- ↑ Pytheas Fogg (2002) p.4
- ↑ Allouche & Shallit (2003) p.303
- ↑ Cassaigne & Nicolas (2010) p.169