2.4 The Loop Space and the Space of Closed Curves

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.

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

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-2-7 ◽

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.

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Functional Band and Loop Space Maintainers in Children

Case Reports in Dentistry ◽

10.1155/2019/4312049 ◽

2019 ◽

Vol 2019 ◽

pp. 1-4

Author(s):

V. Vinothini ◽

A. Sanguida ◽

A. Selvabalaji ◽

G. S. Prathima ◽

M. Kavitha

Keyword(s):

Clinical Application ◽

Loop Space ◽

Single Tooth ◽

Space Maintainer ◽

Space Maintenance ◽

Band And Loop

Premature loss of teeth in children leads to space loss and affects arch integrity. The band and loop space maintainer is used in majority of patients requiring single tooth space maintenance in both primary and mixed dentitions. It preserves the proximal dimensions, but it is nonfunctional. This paper describes a method to modify the conventional band and loop space maintainer into a functional one and reports its clinical application and follow-up in five children.

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