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Solving schubert problems with Littlewood-Richardson homotopies

Frank Sottile, Ravi Vakil, Jan Verschelde
2010 Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10  
For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.  ...  Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem.  ...  Solving Schubert Problems The global structure of the Littlewood-Richardson homotopy is encoded by a graded poset.  ... 
doi:10.1145/1837934.1837971 dblp:conf/issac/SottileVV10 fatcat:dudusfcgy5hmlgnipvnwpdhyii

Solving Schubert Problems with Littlewood-Richardson Homotopies [article]

Frank Sottile, Ravi Vakil, Jan Verschelde
2010 arXiv   pre-print
For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.  ...  Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem.  ...  Solving Schubert Problems The global structure of the Littlewood-Richardson homotopy is encoded by a graded poset.  ... 
arXiv:1001.4125v1 fatcat:lvj3op2udnbsnkzawzo33ldkxq

Cooperative Path Planning for Aerial Recovery of a UAV Swarm Using Genetic Algorithm and Homotopic Approach

Yongbei Liu, Naiming Qi, Weiran Yao, Jun Zhao, Song Xu
2020 Applied Sciences  
In particular, the operation mode of "launch-recovery-relaunch" will greatly improve the efficiency of a UAV swarm.  ...  In this study, the recovery problem of a UAV swarm by a mother aircraft has been investigated.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/app10124154 fatcat:sxcwb4humrcylgqa2eq32paxgq

Transforming Curves on Surfaces Redux [chapter]

Jeff Erickson, Kim Whittlesey
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
We describe a simple variant of Dehn's algorithm that runs in linear time, with no hidden dependence on the genus of the surface.  ...  Our algorithm combines components of these earlier algorithms, classical results in small cancellation theory by Gersten and Short [Inventiones 1990], and simple run-length encoding.  ...  We can easily remove any spur or bracket from the run-length-encoded turn sequence in constant time, without changing the homotopy type of the underlying cycle.  ... 
doi:10.1137/1.9781611973105.118 dblp:conf/soda/EricksonW13 fatcat:wowewdjslbg4janu7b6kqwffdm

Numerical Schubert Calculus in Macaulay2 [article]

Anton Leykin, Abraham Martín del Campo, Frank Sottile, Ravi Vakil, Jan Verschelde
2021 arXiv   pre-print
One is in the scripting language of Macaulay2 using the package NumericalAlgebraicGeometry, and the other is in the compiled code of PHCpack.  ...  It implements both the Pieri homotopy algorithm and the Littlewood-Richardson homotopy algorithm. Each algorithm has two independent implementations in this package.  ...  The authors thank the American Institute of Mathematics for supporting the project through their SQuaREs program. The package NumericalSchubertCalculus is included in Macaulay2 version 1.18.  ... 
arXiv:2105.04494v1 fatcat:i4ioapn5rnfoflvovg35k22c5e

Numerical Evidence for a Conjecture in Real Algebraic Geometry

Jan Verschelde
2000 Experimental Mathematics  
The implementation of SAGBI homotopies involves polyhedral continuation, at deformation and cheater's homotopy.  ...  The optimality of the solving procedure is a promising rst step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory.  ...  I thank Marie-Fran coise Roy for pointing to the property of random real polynomials in the Chebychev basis. The discussions with T.Y. Li and Xiaoshen Wang on cheater's homotopy were most fruitful.  ... 
doi:10.1080/10586458.2000.10504645 fatcat:oxkgobdtobdbjhln2rjm4xwqqm

Robust multirobot coordination using priority encoded homotopic constraints [article]

Jean Gregoire, Silvère Bonnabel, Arnaud de La Fortelle
2015 arXiv   pre-print
Then we prove that two paths in the coordination space are continuously deformable into each other if and only if they induce the same priority graph, that is, the priority graph uniquely encodes homotopy  ...  Finally, we give a simple control law allowing to safely navigate into homotopy classes under kinodynamic constraints even in the presence of unexpected events, such as a sudden robot deceleration without  ...  In the following, we present the main result of this section: priorities uniquely encode homotopy classes of feasible paths in the coordination space.  ... 
arXiv:1306.0785v3 fatcat:cwhuh74mb5hbfd5nd2wedhpd4u

Beyond the Tenfold Way: Non-Abelian Topology in Noninteracting Metals [article]

QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek
2019 arXiv   pre-print
Starting from two-band considerations, we develop the complete many-band description of nodes in the presence of PT and mirror symmetries.  ...  This theory allows us to investigate the topological stability of nodal chains in metals.  ...  The blue path γ is closed and represents a non-trivial element of the homotopy group in Eq. (25a). The particular choice of γ in (a) encodes an order-parameter field similar to that ofFig. S-2(d).  ... 
arXiv:1808.07469v2 fatcat:lkjsyoid4raolc2itzv7rjqkv4

Planning under topological constraints using beam-graphs

Venkatraman Narayanan, Paul Vernaza, Maxim Likhachev, Steven M. LaValle
2013 2013 IEEE International Conference on Robotics and Automation  
We present a framework based on graph search for navigation in the plane with a variety of topological constraints.  ...  The method is based on modifying a standard graph-based navigation approach to keep an additional state variable that encodes topological information about the path.  ...  The other authors were partially supported by MURI/ONR grant N00014-09-1-1052.  ... 
doi:10.1109/icra.2013.6630611 dblp:conf/icra/NarayananVLL13 fatcat:lyacldygvrhr7dtjwtncbpfdry

A Systematic Framework for Solving Geometric Constraints Analytically

Cassiano Durand, Christoph M. Hoffmann
2000 Journal of symbolic computation  
The framework has been used successfully to solve a family of spatial geometric constraint problems. The approach combines geometric reasoning, symbolic reduction, and homotopy continuation.  ...  A systematic framework is presented for solving algebraic equations arising in geometric constraint solving.  ...  The term homotopy continuation refers to a set of techniques for numerically approximating the homotopy path.  ... 
doi:10.1006/jsco.2000.0392 fatcat:qwvat2n7pfcuzoukrqwxemxhfm

Efficiently finding optimal winding-constrained loops in the plane

Paul Vernaza, Venkatraman Narayanan, Maxim Likhachev
2012 Robotics: Science and Systems VIII  
Our approach is similar to a typical graph-based search for an optimal path in the plane, but with an additional state variable that encodes information about path homotopy.  ...  We present a method to efficiently find windingconstrained loops in the plane that are optimal with respect to a minimum-cost objective and in the presence of obstacles.  ...  Furthermore, we employ a real-valued encoding of homotopy information, as opposed to the complex-valued encoding derived in [3] .  ... 
doi:10.15607/rss.2012.viii.053 dblp:conf/rss/VernazaNL12 fatcat:xamde6wryrgrvmuhf7eyoxcly4

Bounded stretch geographic homotopic routing in sensor networks

Kan Huang, Chien-Chun Ni, Rjk Sarkar, Jie Gao, Joseph S. B. Mitchell
2014 IEEE INFOCOM 2014 - IEEE Conference on Computer Communications  
We propose the first lightweight homotopic routing scheme that generates constant bounded stretch compared to the shortest path of the same homotopy type.  ...  We also present simulation analysis in a variety of settings and show that the paths indeed have small stretch in practice, considerably shorter than the bounds guaranteed by the theory. 978-1-4799-3360  ...  Mitchell acknowledges support from the National Science Foundation (CCF-1018388). Jie Gao and Chien-Chun Ni acknowledge support from NSF grants DMS-1221339, CNS-1217823 and CNS-1116881.  ... 
doi:10.1109/infocom.2014.6848027 dblp:conf/infocom/HuangNSGM14 fatcat:2lmw32aisngxxipdpsim3ya7lm

On the Calculation of Fundamental Groups in Homotopy Type Theory by Means of Computational Paths [article]

Tiago Mendonça Lucena de Veras, Arthur F. Ramos, Ruy J. G. B. de Queiroz, Anjolina G. de Oliveira
2018 arXiv   pre-print
One of the most interesting entities of homotopy type theory is the identity type.  ...  We review the concept of computational paths and the LND_EQ-TRS, which is a term rewriting system proposed by De Oliveira in 1994 to map redundancies between computational paths.  ...  Each of these lines is a point in the projective plane.  ... 
arXiv:1804.01413v2 fatcat:s7dg5htbcvaifmy5zmtvs767iu

Identification and Representation of Homotopy Classes of Trajectories for Search-based Path Planning in 3D

Subhrajit Bhattacharya, Maxim Likhachev, Vijay Kumar
2011 Robotics: Science and Systems VII  
The Biot-Savart law lets us design an appropriate vector field, the line integral of which, using the integral form of Ampere's Law, encodes information about homotopy classes in three dimensions.  ...  The Residue Theorem allows the use of integration along the path to distinguish between trajectories in different homotopy classes. However, this idea is fundamentally limited to two dimensions.  ...  Skeleton of a SHIO In [1] , each obstacle in a 2-dimensional plane that induces the notion of multiple homotopy classes is assigned a representative point.  ... 
doi:10.15607/rss.2011.vii.002 dblp:conf/rss/BhattacharyaLK11 fatcat:5smr4xka6nhkhgsclivjawylai

Topology and Routing Problems: The Circular Frame [article]

Rak-Kyeong Seong, Chanho Min, Sang-Hoon Han, Jaeho Yang, Seungwoo Nam, Kyusam Oh
2021 arXiv   pre-print
In this work, we solve the problem of finding non-intersecting paths between points on a plane with a new approach by borrowing ideas from geometric topology, in particular, from the study of polygonal  ...  do not intersect in the interior of the circle.  ...  Acknowledgements The authors would like to thank Seungjai Min and Minsoo Kim at Samsung SDS for helpful discussions and for suggesting the problem.  ... 
arXiv:2105.03386v1 fatcat:l34ojqojyfaixpuyuesd2n5m7q
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