Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space

2018 ◽  
Vol 15 (06) ◽  
pp. 1850092 ◽  
Author(s):  
Djavvat Khadjiev ◽  
İdri̇s Ören ◽  
Ömer Pekşen
Keyword(s):  
Euclidean Space ◽  
Existence And Uniqueness ◽  
Dimensional Euclidean Space ◽  
Uniqueness Theorems ◽  
Two Dimensional ◽  
Existence And Uniqueness Theorems ◽  
Global Invariants ◽  
The Given

Let [Formula: see text] be the [Formula: see text]-dimensional Euclidean space, [Formula: see text] be the group of all linear similarities of [Formula: see text] and [Formula: see text] be the group of all orientation-preserving linear similarities of [Formula: see text]. The present paper is devoted to solutions of problems of global [Formula: see text]-equivalence of paths and curves in [Formula: see text] for the groups [Formula: see text]. Complete systems of global [Formula: see text]-invariants of a path and a curve in [Formula: see text] are obtained. Existence and uniqueness theorems are given. Evident forms of a path and a curve with the given global invariants are obtained.

scholarly journals Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space

2019 ◽  
Vol 27 (2) ◽  
pp. 37-65 ◽  
Author(s):  
Djavvat Khadjiev ◽  
İdris Ören
Keyword(s):  
Euclidean Space ◽  
Orthogonal Group ◽  
Dimensional Euclidean Space ◽  
Plane Curves ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Special Orthogonal Group ◽  
The Common ◽  
Global Invariants ◽  
The Given

AbstractIn this paper, for the orthogonal group G = O(2) and special orthogonal group G = O+(2) global G-invariants of plane paths and plane curves in two-dimensional Euclidean space E2 are studied. Using complex numbers, a method to detect G-equivalences of plane paths in terms of the global G-invariants of a plane path is presented. General evident form of a plane path with the given G-invariants are obtained. For given two plane paths x(t) and y(t) with the common G-invariants, evident forms of all transformations g ∈ G, carrying x(t) to y(t), are obtained. Similar results have obtained for plane curves.

scholarly journals Existence and Uniqueness Theorems for the Two-Dimensional Ericksen–Leslie System

2016 ◽  
Vol 18 (3) ◽  
pp. 571-589 ◽  
Author(s):  
Gregory A. Chechkin ◽  
Tudor S. Ratiu ◽  
Maxim S. Romanov ◽  
Vyacheslav N. Samokhin
Keyword(s):  
Existence And Uniqueness ◽  
Uniqueness Theorems ◽  
Two Dimensional ◽  
Existence And Uniqueness Theorems ◽  
Leslie System
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