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.

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 Unitary Representations of Some Linear Groups

1952 ◽  
Vol 4 ◽  
pp. 1-13 ◽  
Author(s):  
Seizô Itô
Keyword(s):  
Euclidean Space ◽  
Analogous Result ◽  
Unitary Representations ◽  
Dimensional Euclidean Space ◽  
Linear Groups ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Linear Transformations ◽  
Straight Line ◽  
Irreducible Unitary Representations

Recently I. Gelfand and M. Neumark [2] have determined the types of irreducible unitary representations of the group G1 of linear transformations of the straight line. The analogous result is obtained for the group G2 of transformations z → az + b in the complex-number plane , where a and b run over all complex numbers with the exception of a = 0, which may be considered as the group of all sense-preserving similar transformations in the two-dimensional euclidean space E2.

An asymptotic analysis for Hamilton–Jacobi equations with large Hamiltonian drift terms

2017 ◽  
Vol 0 (0) ◽  
Author(s):  
Taiga Kumagai
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Asymptotic Analysis ◽  
Open Subset ◽  
Dimensional Euclidean Space ◽  
Asymptotic Behavior Of Solutions ◽  
Two Dimensional ◽  
Jacobi Equations ◽  
Drift Terms

AbstractWe investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by

scholarly journals Trading spaces: building three-dimensional nets from two-dimensional tilings

2012 ◽  
Vol 2 (5) ◽  
pp. 555-566 ◽  
Author(s):  
Toen Castle ◽  
Myfanwy E. Evans ◽  
Stephen T. Hyde ◽  
Stuart Ramsden ◽  
Vanessa Robins
Keyword(s):  
Euclidean Space ◽  
Ab Initio ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Complex Patterns ◽  
Geometrically Complex

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic ( S 2 ), Euclidean ( E 2 ) and hyperbolic ( H 2 ) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

Results on the intersection of randomly located sets

1975 ◽  
Vol 12 (4) ◽  
pp. 817-823 ◽  
Author(s):  
Franz Streit
Keyword(s):  
Euclidean Space ◽  
Surface Area ◽  
Arbitrary Dimension ◽  
Expected Value ◽  
Dimensional Euclidean Space ◽  
Common Intersection ◽  
Point Set ◽  
The Common

Randomly generated subsets of a point-set A0 in the k-dimensional Euclidean space Rk are investigated. Under suitable restrictions the probability is determined that a randomly located set which hits A0. is a subset of A0. Some results on the expected value of the measure and the surface area of the common intersection-set formed by n randomly located objects and A0 are generalized and derived for arbitrary dimension k.

scholarly journals On the spectrum of the distributional kernel related to the residue

2001 ◽  
Vol 27 (12) ◽  
pp. 715-723
Author(s):  
Amnuay Kananthai
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Nonnegative Integer ◽  
Dimensional Euclidean Space ◽  
Complex Numbers ◽  
Alpha Beta ◽  
The Fourier Transform

We study the spectrum of the distributional kernelKα,β(x), whereαandβare complex numbers andxis a point in the spaceℝnof then-dimensional Euclidean space. We found that for any nonzero pointξthat belongs to such a spectrum, there exists the residue of the Fourier transform(−1)kK2k,2k(ξ)ˆ, whereα=β=2k,kis a nonnegative integer andξ∈ℝn.

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