scholarly journals Quasilinear Approximation Scheme for Steiner Multi Cycle in the Euclidean plane

2019 ◽  
Vol 346 ◽  
pp. 589-601
Author(s):  
Carla N. Lintzmayer ◽  
Flávio K. Miyazawa ◽  
Phablo F.S. Moura ◽  
Eduardo C. Xavier
Keyword(s):  
Euclidean Plane ◽  
Approximation Scheme ◽  
Quasilinear Approximation

scholarly journals The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Haiming Liu ◽  
Jiajing Miao ◽  
Wanzhen Li ◽  
Jianyun Guan
Keyword(s):  
Lie Group ◽  
Smooth Surface ◽  
Euclidean Plane ◽  
Approximation Scheme ◽  
Geodesic Curvature ◽  
Smooth Curves ◽  
3 Dimensional ◽  
Characteristic Points ◽  
Curves On Surfaces ◽  
Bonnet Theorem

The rototranslation group ℛ T is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.

Randomized approximation scheme for Steiner Multi Cycle in the Euclidean plane

2020 ◽  
Vol 835 ◽  
pp. 134-155
Author(s):  
Carla N. Lintzmayer ◽  
Flávio K. Miyazawa ◽  
Phablo F.S. Moura ◽  
Eduardo C. Xavier
Keyword(s):  
Euclidean Plane ◽  
Approximation Scheme

scholarly journals PTAS FOR k-TOUR COVER PROBLEM ON THE PLANE FOR MODERATELY LARGE VALUES OF k

2010 ◽  
Vol 21 (06) ◽  
pp. 893-904 ◽  
Author(s):  
ANNA ADAMASZEK ◽  
ARTUR CZUMAJ ◽  
ANDRZEJ LINGAS
Keyword(s):  
Vehicle Routing Problem ◽  
Euclidean Plane ◽  
Approximation Scheme ◽  
Constant Factor ◽  
Polynomial Time Approximation Scheme ◽  
Routing Problem ◽  
Small Set ◽  
Cover Problem ◽  
Capacitated Vehicle ◽  
Origin Point

Let P be a set of n points in the Euclidean plane and let O be the origin point in the plane. In the k-tour cover problem (called frequently the capacitated vehicle routing problem), the goal is to minimize the total length of tours that cover all points in P, such that each tour starts and ends in O and covers at most k points from P. The k-tour cover problem is known to be [Formula: see text]-hard. It is also known to admit constant factor approximation algorithms for all values of k and even a polynomial-time approximation scheme (PTAS) for small values of k, k = Ø( log n/ log log n). In this paper, we significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of k ≤ 2 log δ n, where δ = δ(ε). The main technical result proved in the paper is a novel reduction of the k-tour cover problem with a set of n points to a small set of instances of the problem, each with Ø((k/ε)Ø(1)) points.

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.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

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.

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