Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

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Configuration spaces of n lines in affine (n + k)-space

Journal of Mathematical Sciences ◽

10.1007/s10958-007-0359-z ◽

2007 ◽

Vol 146 (1) ◽

pp. 5474-5482

Author(s):

Margareta Boege ◽

Luis Montejano

Keyword(s):

Configuration Spaces

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Standard Deep Generative Models for Density Estimation in Configuration Spaces: A Study of Benefits, Limits and Challenges

2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) ◽

10.1109/iros45743.2020.9340994 ◽

2020 ◽

Author(s):

Robert Gieselmann ◽

Florian T. Pokorny

Keyword(s):

Density Estimation ◽

Generative Models ◽

Configuration Spaces

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Collision-Free Motion Planning for an Aligned Multiple-turret System Operating in Extreme Environment

Robotica ◽

10.1017/s026357472100076x ◽

2021 ◽

pp. 1-30

Author(s):

Ümit Yerlikaya ◽

R.Tuna Balkan

Keyword(s):

Path Planning ◽

Extreme Environment ◽

Configuration Spaces ◽

Planning Problem ◽

Different Types ◽

Planning Algorithm ◽

Line Path ◽

Path Planning Algorithm ◽

Path Planning Problem ◽

System Operating

Abstract Instead of using the tedious process of manual positioning, an off-line path planning algorithm has been developed for military turrets to improve their accuracy and efficiency. In the scope of this research, an algorithm is proposed to search a path in three different types of configuration spaces which are rectangular-, circular-, and torus-shaped by providing three converging options named as fast, medium, and optimum depending on the application. With the help of the proposed algorithm, 4-dimensional (D) path planning problem was realized as 2-D + 2-D by using six sequences and their options. The results obtained were simulated and no collision was observed between any bodies in these three options.

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