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The geometry and topology of reconfiguration
2007
Advances in Applied Mathematics
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A number of reconfiguration problems in robotics, biology, computer science, combinatorics, and group theory coordinate local rules to effect global changes in system states. ...
We define for any such reconfigurable system a cubical complex-the state complex-which coordinates independent local moves. ...
Acknowledgments Many of the details in the definition of reconfigurable systems and state complexes were developed in conversations with A. Abrams. ...
doi:10.1016/j.aam.2005.08.009
fatcat:6qpx7oxsbzct3nec2r6r2jzb7y
Geodesics in CAT(0) Cubical Complexes
[article]
2011
arXiv
pre-print
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This correspondence also gives an explicit realization of such a complex as the state complex of a reconfigurable system, and a way to embed any interval in the integer lattice cubing of its dimension. ...
We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. ...
We would like to acknowledge Rick Scott for valuable discussions on CAT(0) cubical complexes, and for an excellent talk at the Bay Area Discrete Mathematics Day in 2007 which first inspired Theorems 2.5 ...
arXiv:1101.2428v1
fatcat:hwbi4exsdrcjlga4ohpeabedva
Rank Rigidity for Cat(0) Cube Complexes
2011
Geometric and Functional Analysis
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2011 pre-print arXiv:1005.5687v2 fatcat:oc7i2ypagbctveikk64qyvgw44Other Versions
We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits Alternative, an existence result for regular elements in (possibly non-uniform) lattices ...
acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology. ...
We express our gratitude to Frà c dà c ric Haglund and the referee for long lists of detailed comments on an earlier version of this manuscript, which helped much in improving the presentation. ...
doi:10.1007/s00039-011-0126-7
fatcat:q6auv767zvcplo52lhjmwu2ta4
Infima of length functions and dual cube complexes
2017
Algebraic and Geometric Topology
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2015 pre-print arXiv:1505.07944v1 fatcat:s6htui6ltzh6xolhkm2spzcmi4Other Versions
In the presence of certain topological conditions, we provide lower bounds for the infimum of the length function associated to a collection of curves on Teichmüller space that depend on the dual cube ...
As an application of our bounds, we obtain estimates for the 'longest' curve with k self-intersections, complementing work of Basmajian basmajian. ...
A construction of Sageev [Sag1] associates to a curve system Γ an isometric action of π 1 Σ on a finite-dimensional cube complex, or the dual cube complex C(Γ) of Γ. ...
doi:10.2140/agt.2017.17.1041
fatcat:c2l4fcawybf73nzwn3yytppyny
CAT(0) geometry, robots, and society
[article]
2020
arXiv
pre-print
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In this case, using ideas from combinatorics and geometric group theory, we can construct a 'remote control' to navigate these complicated maps, and move the robots optimally. ...
Fortunately, for some discrete robots they are CAT(0) cubical complexes, a family of spaces with favorable properties. ...
Every dcube in X has d hyperplanes that bisect its edges. Whenever two cubes share an edge, let us glue the two hyperplanes bisecting it. The result is a system of hyperplanes associated to X [17]. ...
arXiv:1912.10007v2
fatcat:66ggi4hcyvaxllknaubnlkga74
Geodesics in CAT(0) cubical complexes
2012
Advances in Applied Mathematics
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This correspondence also gives an explicit realization of such a complex as the state complex of a reconfigurable system, and a way to embed any interval in the integer lattice cubing of its dimension. ...
We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. ...
Acknowledgment We would like to acknowledge Rick Scott for valuable discussions on CAT(0) cubical complexes, and for an excellent talk at the Bay Area Discrete Mathematics Day in 2007 which first inspired ...
doi:10.1016/j.aam.2011.06.004
fatcat:vvsxt2lb5jegvnizyn3mxabhjy
Multiscale structural optimization with concurrent coupling between scales
2021
Structural And Multidisciplinary Optimization
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accessibility to a greater range of extremal point properties during optimization. ...
Only a small fraction of the microstructure database is required to derive the optimized multiscale solutions, which demonstrates a significant reduction in the computational expense of optimization in ...
Acknowledgments RM gratefully acknowledges funding through the EPSRC Industrial Case Award with Airbus Central R&T. ...
doi:10.1007/s00158-020-02773-3
fatcat:v3ikjsx54rbwha257tt43yrrmi
Numerical Algorithms and Visualization in Medical Treatment Planning
[chapter]
1997
Visualization and Mathematics
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Then we continue with a description of our medical planning system HyperPlan. ...
After a short summary on therapy planning and the underlying technologies we discuss quantitative medicine by giving a short overview on medical image data, summarizing some applications of computer based ...
For discussions on the electro-technical questions and the provision of a finite-difference Maxwell-solver we are grateful to Jacek Nadobny. ...
doi:10.1007/978-3-642-59195-2_20
dblp:conf/vismath/BeckDHSS95
fatcat:sjkfyggxu5fitmyjm37wpyil74
A Geometric Approach to Sample Compression
[article]
2009
arXiv
pre-print
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This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. ...
The construction of the PL sweeping involves Pachner moves on the one-inclusion graph, corresponding to moves of a hyperplane across the intersection of d other hyperplanes. ...
Acknowledgments We thank Peter Bartlett for his very helpful feedback, and gratefully acknowledge the support of the NSF through grants DMS-0434383 and DMS-0707060, and the support of the Siebel Scholars ...
arXiv:0911.3633v1
fatcat:3eyqdnf76vfp3plzowwu257oua
Cubical-like geometry of quasi-median graphs and applications to geometric group theory
[article]
2017
arXiv
pre-print
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In the first part of our work, we extend the definition of hyperplanes from CAT(0) cube complexes, and we show that the geometry of a quasi-median graph essentially reduces to the combinatorics of its ...
The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. ...
I am grateful to Jérémie Chalopin and Victor Chepoï, for a useful ...
arXiv:1712.01618v1
fatcat:ab2w2scic5d63onzhqfyvcmhie
Multiple-Output Quantile Regression
[chapter]
2017
Handbook of Quantile Regression
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All those approaches crucially depend on the choice of a coordinate system (hence a unit cube). ...
A map T P1;P2 that attains this infimum is called an "optimal transport map", in short, an "optimal transport" mapping P 1 to P 2 . ...
doi:10.1201/9781315120256-12
fatcat:inz5vngtznazji4ze6htxemhki
Page 214 of Mathematical Reviews Vol. , Issue 80A
[page]
1980
Mathematical Reviews
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An affine diameter of a convex body K in Euclidean n-space is a chord whose endpoints lie in a parallel pair of support hyperplanes to K. B. Griinbaum [Proc. Sympos. Pure Math., Vol. ...
The results of an example in an 8-dimensional space are dealt
80a:52008
with. ...
A quantitative Doignon-Bell-Scarf theorem
2016
Combinatorica
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In this case c(n, 0) = 2 n as in the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. ...
The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. ...
Acknowledgements The research of the third author was partially supported by a UCMEXUS project grant and by the Institute for Mathematics and its Applications with funds provided by the National Science ...
doi:10.1007/s00493-015-3266-9
fatcat:jnt62cfuzfb5nj42744b35bzgy
A Quantitative Doignon-Bell-Scarf Theorem
[article]
2015
arXiv
pre-print
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In this case c(n,0) = 2^n is the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. ...
The famous Doignon-Bell-Scarf Theorem is a Helly-type result about the existence of integer solutions on systems of linear inequalities. ...
Acknowledgements The research of the third author was partially supported by a UCMEXUS project grant and by the Institute for Mathematics and its Applications with funds provided by the National Science ...
arXiv:1405.2480v3
fatcat:4c5tazkxvvegpnrsimk7hpnlqq
Page 3353 of Mathematical Reviews Vol. , Issue 97E
[page]
1997
Mathematical Reviews
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Optim. 6 (1996), no. 3, 638-652.
Let I’ be a convex subset of the unit cube in R” having non-empty interior. ...
The result is of a theoretical nature in the sense that in order to ensure that the limit point is an ¢-optimal solution, one may have to restrict the steplength to the order of O(c). ...
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