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# Geography 4110/5100 Remote Sensing of the Environment Lecture: Tues.

Description

• cours magistral
Geography 4110/5100 Advance Remote Sensing Lecture: Tues. & Thurs. 9:30 – 10:45 Ekeley, W240 Office Hours: Tuesday 2:00 - 3:30 and by appointment Teaching Assistant: Pakorn Petchprayoon Office Hours: Tues. 10:45-12:00, Thurs. 1:00-2:00, and by appointment (W240)
• extent of vegetation cover
• ice sheet topography
• relative resolution for various sensors
• types of orbits
• spectral bands
• remote sensing

Subjects

##### Végétation

Informations

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Order and quasiperiodicity in episturmian words
Amy Glen LaCIM,Universit´eduQu´ebec`aMontre´al, C.P.8888,succursaleCentreville,Montre´al,Qu´ebec,CANADA,H3C3P8 Extended Abstract
Introduction
July 13, 2007
In this paper, we build upon previous work concerning inequalities characterizing Sturmian and episturmian words – see [19, 29, 30, 14, 16]. First let us recall from [29] the following notion relating tolexicographic order. LetAbe atotally orderedﬁnite alphabet consisting of at least two letters. To any inﬁnite wordxover A, we can associate two inﬁnite words min(x) and max(x) such that any preﬁx of min(x) (resp. max(x)) is thelexicographicallysmallest (resp. greatest) amongst the factors ofxMore precisely, ifof the same length. we denote by min(x|k) (resp. max(x|k)) the lexicographically smallest (resp. greatest) factor ofxof length kfor the given order, then min(x|k) and max(x|k) are clearly preﬁxes of the respective words min(x|k+ 1) and max(x|kSo we can deﬁne, by taking limits, the following two inﬁnite words+ 1).
min(xlim min() = x|k) k→∞
and
max(xlim max() = x|k). k→∞
An important point here is that, for any aperiodic inﬁnite wordxwhich is uniformly recurrent, min(x) is aninﬁnite Lyndon word, i.e., it is (strictly) lexicographically smaller than all of its proper suﬃxes for the given order onA. In 2003, Pirillo [28] (also see [29]) proved that, for inﬁnite wordsson a 2letter alphabet{a, b}with a < b, the inequalityasmin(s)max(s)bscharacterizesstandard Sturmian words(aperiodic and periodic). Equivalently, an inﬁnite sequences= (sn)n0over{a, b}is standard Sturmian if and only if
k asT (s)bs,
for allk0,
(1.1)
k k where T is thekth iterate of theshift map: T ((sn)n0) = (sn+k)n0(cf.our analogue for episturmian sequences – Corollary 3.2). Actually, this result was known much earlier, dating back to the work of P. Veerman [38, 39] in the mid 80’s. Since that time, these ‘Sturmian inequalities’ have been rediscovered numerous times under diﬀerent guises, as discussed in our survey paper [4]. In the case of an arbitrary ﬁnite alphabetA, Pirillo [29] generalized the above inequalities by proving that an inﬁnite wordsoverAisstandard episturmian(orepistandardfor short) if and only if, for any lexicographic order, we have asmin(s) wherea= min(A).(1.2) Moreover,sis astrictepistandard word (i.e., astandard ArnouxRauzy sequence[7, 37]) if and only if (1.2) holds with strict equality for any order [19]. In a similar spirit, Glen, Justin, and Pirillo [16] recently proved the following characterization of allepisturmian words.
Proposition 1.1.[16]A recurrent inﬁnite wordtoverAis episturmian if and only if there exists an inﬁnite wordssuch that, for any lexicographic order, we haveasmin(t)wherea= min(A).
From the proof of the above result, it is not immediately clear what form is taken by the inﬁnite words, if it exists. We now prove further (in Section 3) thatsis in fact the ‘unique’ epistandard word with the same set of factors ast, i.e., the corresponding epistandard word in theshift orbit closureoft(see Section 2). As the title of this paper suggests, these results have a connection withquasiperiodicityexactly?. What Well, roughly speaking, an inﬁnite wordxisquasiperiodicif there exists a ﬁnite wordusuch that the Email:amy.glen@gmail.com(with the support of CRMISMLaCIM) Same as the full version, but all proofs (and some lemmas) have been omitted.
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