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The geometry and topology of reconfiguration

R. Ghrist, V. Peterson
2007 Advances in Applied Mathematics  
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]

Federico Ardila, Megan Owen, Seth Sullivant
2011 arXiv   pre-print
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

Pierre-Emmanuel Caprace, Michah Sageev
2011 Geometric and Functional Analysis  
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

Jonah Gaster
2017 Algebraic and Geometric Topology  
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]

Federico Ardila
2020 arXiv   pre-print
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

Federico Ardila, Megan Owen, Seth Sullivant
2012 Advances in Applied Mathematics  
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

Ryan Murphy, Chikwesiri Imediegwu, Robert Hewson, Matthew Santer
2021 Structural And Multidisciplinary Optimization  
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]

Rudolf Beck, Peter Deuflhard, Hans-Christian Hege, Martin Seebaß, Detlev Stalling
1997 Visualization and Mathematics  
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]

Benjamin I. P. Rubinstein, J. Hyam Rubinstein
2009 arXiv   pre-print
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]

Anthony Genevois
2017 arXiv   pre-print
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]

Marc Hallin, Miroslav Šiman
2017 Handbook of Quantile Regression  
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  
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

Iskander Aliev, Robert Bassett, Jesús A. De Loera, Quentin Louveaux
2016 Combinatorica  
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]

Iskander Aliev, Robert Bassett, Jesus A. De Loera, Quentin Louveaux
2015 arXiv   pre-print
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  
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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