The generalized uniqueness wavelet descriptor for planar closed curves

2000 ◽  
Vol 9 (5) ◽  
pp. 834-845 ◽  
Author(s):  
King-Chu Hung
Keyword(s):  
Closed Curves ◽  
Wavelet Descriptor

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.

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

Periodic wavelet descriptor of plant leaf and its application in botany

2015 ◽  
Vol 13 (06) ◽  
pp. 1550043 ◽  
Author(s):  
Qing-Mao Zeng ◽  
Tong-Lin Zhu ◽  
Xue-Ying Zhuang ◽  
Ming-Xuan Zheng
Keyword(s):  
Species Identification ◽  
Plant Species ◽  
Shape Descriptor ◽  
Plant Leaf ◽  
Practical Applications ◽  
Wide Range ◽  
Periodic Wavelet ◽  
Plant Species Identification ◽  
Botanical Research ◽  
Wavelet Descriptor

Leaf is one of the most important organs of plant. Leaf contour or outline, usually a closed curve, is a fundamental morphological feature of leaf in botanical research. In this paper, a novel shape descriptor based on periodic wavelet series and leaf contour is presented, which we name as Periodic Wavelet Descriptor (PWD). The PWD of a leaf actually expresses the leaf contour in a vector form. Consequently, the PWD of a leaf has a wide range in practical applications, such as leaf modeling, plant species identification and classification, etc. In this work, the plant species identification and the leaf contour reconstruction, as two practical applications, are discussed to elaborate how to employ the PWD of a plant leaf in botanical research.

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