scholarly journals On the Stanley ring of cubical complex

1995 ◽  
Vol 14 (3) ◽  
pp. 305-330 ◽  
Author(s):  
G. Hetyei
Keyword(s):  
Cubical Complex

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals The configuration space of a robotic arm in a tunnel of width 2

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Federico Ardila ◽  
Hanner Bastidas ◽  
Cesar Ceballos ◽  
John Guo
Keyword(s):  
Configuration Space ◽  
Robotic Arm ◽  
Configuration Spaces ◽  
Cubical Complex ◽  
International Audience ◽  
Rectangular Tunnel

International audience We study the motion of a robotic arm inside a rectangular tunnel of width 2. We prove that the configuration space S of all possible positions of the robot is a CAT(0) cubical complex. Before this work, very few families of robots were known to have CAT(0) configuration spaces. This property allows us to move the arm optimally from one position to another.

scholarly journals Dipaths and dihomotopies in a cubical complex

2005 ◽  
Vol 35 (2) ◽  
pp. 188-206 ◽  
Author(s):  
Lisbeth Fajstrup
Keyword(s):  
Cubical Complex

scholarly journals The combinatorics of CAT(0) cubical complexes

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Tia Baker ◽  
Rika Yatchak
Keyword(s):  
Remote Control ◽  
Shortest Path ◽  
Robotic Arm ◽  
General Strategy ◽  
Cubical Complex ◽  
Cubical Complexes ◽  
Square Grid ◽  
Reconfigurable System ◽  
International Audience ◽  
Simultaneous Moves

International audience Given a reconfigurable system $X$, such as a robot moving on a grid or a set of particles traversing a graph without colliding, the possible positions of $X$ naturally form a cubical complex $\mathcal{S}(X)$. When $\mathcal{S}(X)$ is a CAT(0) space, we can explicitly construct the shortest path between any two points, for any of the four most natural metrics: distance, time, number of moves, and number of steps of simultaneous moves. CAT(0) cubical complexes are in correspondence with posets with inconsistent pairs (PIPs), so we can prove that a state complex $\mathcal{S}(X)$ is CAT(0) by identifying the corresponding PIP. We illustrate this very general strategy with one known and one new example: Abrams and Ghrist's ``positive robotic arm" on a square grid, and the robotic arm in a strip. We then use the PIP as a combinatorial ``remote control" to move these robots efficiently from one position to another.

scholarly journals A Second Look at the Toric h-Polynomial of a Cubical Complex

2012 ◽  
Vol 16 (3) ◽  
pp. 517-541 ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Cubical Complex ◽  
Second Look

scholarly journals Applications to discrete Morse theory: The collapsibility of CAT(0) cubical complexes of dimension 2 and 3

2011 ◽  
Vol 27 (2) ◽  
pp. 225-237
Author(s):  
IOANA-CLAUDIA LAZAR ◽  
Keyword(s):  
Morse Theory ◽  
Discrete Morse Theory ◽  
Cubical Complex ◽  
Cubical Complexes ◽  
Euclidean Metric ◽  
The Face ◽  
Dimension 2 ◽  
Positively Curved

We investigate, using discrete Morse theory, the collapsibility of finite cubical complexes of dimension 2 and 3. Our main result states that any finite cubical complex of dimension 3 or less endowed with the standard piecewise Euclidean metric that is non-positively curved, and that satisfies the property that every 2-cell is the face of at most two 3-cells in the complex, collapses to a point.

scholarly journals Trace spaces in a pre-cubical complex

2009 ◽  
Vol 156 (9) ◽  
pp. 1718-1728 ◽  
Author(s):  
Martin Raussen
Keyword(s):  
Cubical Complex ◽  
Trace Spaces

scholarly journals The Toric h-Vector of a Cubical Complex in Terms of Noncrossing Partition Statistics

2014 ◽  
Vol 18 (4) ◽  
pp. 585-606
Author(s):  
Sarah Birdsong ◽  
Gábor Hetyei
Keyword(s):  
Cubical Complex ◽  
Noncrossing Partition ◽  
Partition Statistics
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