The Lie bracket of undirected closed curves on a surface

2021 ◽  
Author(s):  
Moira Chas ◽  
Arpan Kabiraj
Keyword(s):  
Closed Curves ◽  
Lie Bracket

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

scholarly journals Stochastic Lie bracket (derivation, derivation) in MB-algebras

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Masoumeh Madadi ◽  
Reza Saadati ◽  
Choonkil Park ◽  
John Michael Rassias
Keyword(s):  
Lie Bracket

Nontransversal curves of T-points: a source of closed curves of global bifurcations

2002 ◽  
Vol 303 (2-3) ◽  
pp. 204-211 ◽  
Author(s):  
A. Algaba ◽  
F. Fernández-Sánchez ◽  
E. Freire ◽  
M. Merino ◽  
A.J. Rodrı́guez-Luis
Keyword(s):  
Global Bifurcations ◽  
Closed Curves
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