*Zhen-Qing Chen and Masatoshi Fukushima*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691136059
- eISBN:
- 9781400840564
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691136059.003.0003
- Subject:
- Mathematics, Probability / Statistics

This chapter studies a symmetric Hunt process associated with a regular Dirichlet form. Without loss of generality, the majority of the chapter assumes that E is a locally compact separable metric ...
More

This chapter studies a symmetric Hunt process associated with a regular Dirichlet form. Without loss of generality, the majority of the chapter assumes that E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, and X = (Xₜ, Pₓ) is an m-symmetric Hunt process on (E,B(E)) whose Dirichlet form (E,F) is regular on L²(E; m). It adopts without any specific notices those potential theoretic terminologies and notations that are formulated in the previous chapter for the regular Dirichlet form (E,F). Furthermore, throughout this chapter, the convention that any numerical function on E is extended to the one-point compactification E∂ = E ∪ {∂} by setting its value at δ to be zero is adopted.Less

This chapter studies a symmetric Hunt process associated with a regular Dirichlet form. Without loss of generality, the majority of the chapter assumes that *E* is a locally compact separable metric space, *m* is a positive Radon measure on *E* with supp[*m*] = *E*, and *X* = (*X*ₜ, **P**ₓ) is an *m*-symmetric Hunt process on (*E*,*B*(*E*)) whose Dirichlet form (*E*,*F*) is regular on *L*²(*E*; *m*). It adopts without any specific notices those potential theoretic terminologies and notations that are formulated in the previous chapter for the regular Dirichlet form (*E*,*F*). Furthermore, throughout this chapter, the convention that any numerical function on *E* is extended to the one-point compactification *E*_{∂} = *E* ∪ {∂} by setting its value at δ to be zero is adopted.

*Christophe Reutenauer*

- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.003.0013
- Subject:
- Mathematics, Pure Mathematics

The geometrical definition of Christoffel words shows that such a word is of the form amb for some palindrome m. These palindromes are special palindromes on a two-letter alphabet,withmany surprising ...
More

The geometrical definition of Christoffel words shows that such a word is of the form amb for some palindrome m. These palindromes are special palindromes on a two-letter alphabet,withmany surprising characterizations; they are called centralwords. In particular, they may be computed by iterated palindromization, as follows from a theorem of Aldo de Luca. This operation may be simulated by a formula of Justin. Central words are characterized also by the existence of two relatively prime periods, p and q, and length equal to p + q − 2 (Mignosi and de Luca). Moreover, Christoffel words are products of two palindromes (Chuan).Less

The geometrical definition of Christoffel words shows that such a word is of the form amb for some palindrome m. These palindromes are special palindromes on a two-letter alphabet,withmany surprising characterizations; they are called centralwords. In particular, they may be computed by iterated palindromization, as follows from a theorem of Aldo de Luca. This operation may be simulated by a formula of Justin. Central words are characterized also by the existence of two relatively prime periods, p and q, and length equal to p + q − 2 (Mignosi and de Luca). Moreover, Christoffel words are products of two palindromes (Chuan).