On the Morse number of embedded and non-embedded minimal immersions spanning wires on the boundary of special bodies in ?3

1985 ◽  
Vol 188 (2) ◽  
pp. 149-170 ◽  
Author(s):  
A. J. Tromba
Keyword(s):  
Morse Number ◽  
Minimal Immersions

Minimal immersions of spheres into spheres

2012 ◽  
pp. 93-112
Author(s):  
Manfredo P. do Carmo ◽  
Nolan R. Wallach
Keyword(s):  
Minimal Immersions

scholarly journals ON GENERALIZED THUE–MORSE FUNCTIONS AND THEIR VALUES

2019 ◽  
Vol 108 (2) ◽  
pp. 177-201
Author(s):  
DZMITRY BADZIAHIN ◽  
EVGENIY ZORIN
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Laurent Series ◽  
Regular Structure ◽  
Continued Fraction Expansion ◽  
Infinite Products ◽  
Morse Functions ◽  
Morse Number ◽  
Badly Approximable

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products $f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$, $d\in \mathbb{N}$, $d\geq 2$, which generalize the generating function $f_{2}(x)$ of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_{d}(x)$ have a regular structure. We also address the question of whether the corresponding Mahler numbers $f_{d}(a)\in \mathbb{R}$, $a,d\in \mathbb{N}$, $a,d\geq 2$, are badly approximable.

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