Bifurcations of Steiner minimal trees and minimal fillings for non-convex four-point boundaries and Steiner subratio for the Euclidean plane

2016 ◽  
Vol 71 (2) ◽  
pp. 79-81
Author(s):  
E. I. Stepanova
Keyword(s):  
Euclidean Plane ◽  
Steiner Minimal Trees

GENERALIZED MELZAK'S CONSTRUCTION IN THE STEINER TREE PROBLEM

2002 ◽  
Vol 12 (06) ◽  
pp. 481-488 ◽  
Author(s):  
JIA F. WENG
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Steiner Tree Problem ◽  
Minimal Tree ◽  
Steiner Minimal Tree ◽  
The Core ◽  
Steiner Minimal Trees ◽  
Minimum Networks ◽  
Set Of Points ◽  
The Given

For a given set of points in the Euclidean plane, a minimum network (a Steiner minimal tree) can be constructed using a geometric method, called Melzak's construction. The core of the Melzak construction is to replace a pair of terminals adjacent to the same Steiner point with a new terminal. In this paper we prove that the Melzak construction can be generalized to constructing Steiner minimal trees for circles so that either the given points (terminals) are constrained on the circles or the terminal edges are tangent to the circles. Then we show that the generalized Melzak construction can be used to find minimum networks separating and surrounding circular objects or to find minimum networks connecting convex and smoothly bounded objects and avoiding convex and smoothly bounded obstacles.

IDENTIFYING STEINER MINIMAL TREES ON FOUR POINTS IN SPACE

2009 ◽  
Vol 01 (03) ◽  
pp. 401-411 ◽  
Author(s):  
J. F. WENG ◽  
D. A. THOMAS ◽  
I. MAREELS
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Sufficient Conditions ◽  
Steiner Tree Problem ◽  
Minimum Length ◽  
Minimal Tree ◽  
Steiner Minimal Tree ◽  
Steiner Minimal Trees ◽  
Necessary And Sufficient ◽  
Set Of Points

A Steiner minimal tree is a network with minimum length spanning a given set of points in space. There are several criteria for identifying the Steiner minimal tree on four points in the Euclidean plane. However, it has been proved that the length of the Steiner minimal tree on four points cannot be computed using radicals if the four points lie in Euclidean space. This unsolvability implies that it is unlikely that similar necessary and sufficient conditions exist in the spatial case. Hence, a problem arises: Is it possible to generalize the known planar criteria to space in the sense that they are sufficient to identify Steiner minimal trees on four points in space? This problem is investigated and some sufficient conditions are proved in this paper. These sufficient conditions can help us to solve the general Steiner tree problem on n(> 4) points in Euclidean space.

scholarly journals Cutoff AdS3 versus $$ T\overline{T} $$ CFT2 in the large central charge sector: correlators of energy-momentum tensor

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yi Li ◽  
Yang Zhou
Keyword(s):  
Field Theory ◽  
Euclidean Plane ◽  
Energy Momentum Tensor ◽  
Central Charge ◽  
Momentum Tensor ◽  
Two Dimensional ◽  
Conformal Field ◽  
Operator Identity ◽  
Pure Gravity ◽  
Charge Sector

Abstract In this article we probe the proposed holographic duality between $$ T\overline{T} $$ T T ¯ deformed two dimensional conformal field theory and the gravity theory of AdS3 with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $$ T\overline{T} $$ T T ¯ CFT in a Euclidean plane and a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in Euclidean AdS3 with the corresponding cutoff surface, given a holographic dictionary which identifies gravity parameters with $$ T\overline{T} $$ T T ¯ CFT parameters.

scholarly journals On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane

2004 ◽  
Vol 11 (3) ◽  
pp. 479-487
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Plane ◽  
Identity Transformation ◽  
Extension Problem ◽  
Measure Extension ◽  
Nonmeasurable Set

Abstract The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑2 which are 𝑇2-negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇2 denotes the group of all translations of 𝐑2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇2, where 𝑔 is an arbitrary rotation of 𝐑2 distinct from the identity transformation and all central symmetries of 𝐑2.

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Algorithmica ◽  
2001 ◽  
Vol 31 (4) ◽  
pp. 479-500 ◽  
Author(s):  
Xue ◽  
-H. Lin ◽  
-Z. Du
Keyword(s):  
Euclidean Plane ◽  
Grade Of Service

scholarly journals Steiner minimal trees on sets of four points

1987 ◽  
Vol 2 (4) ◽  
pp. 401-414 ◽  
Author(s):  
D. Z. Du ◽  
F. K. Hwang ◽  
G. D. Song ◽  
G. Y. Ting
Keyword(s):  
Steiner Minimal Trees

scholarly journals How to Collect Balls Moving in the Euclidean Plane

2004 ◽  
Vol 91 ◽  
pp. 229-245 ◽  
Author(s):  
Yuichi Asahiro ◽  
Takashi Horiyama ◽  
Kazuhisa Makino ◽  
Hirotaka Ono ◽  
Toshinori Sakuma ◽  
...  
Keyword(s):  
Euclidean Plane

Generalized evolutes of planar curves

2021 ◽  
Author(s):  
Yongqiao Wang ◽  
Yuan Chang ◽  
Haiming Liu
Keyword(s):  
Euclidean Plane ◽  
Singular Points ◽  
Moving Frames ◽  
Planar Curves ◽  
Fixed Direction ◽  
Spatial Curves

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

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