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Algebraic Branching Programs, Border Complexity, and Tangent Spaces [article]

Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, Nitin Saurabh
2020 arXiv   pre-print
The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP.  ...  Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices  ...  Acknowledgements We thank Michael Forbes for illuminating discussions and for telling us about his (correct) intuition concerning Nisan's result.  ... 
arXiv:2003.04834v1 fatcat:xv532a7arjaljbu6znckcsocpm

Algebraic Branching Programs, Border Complexity, and Tangent Spaces

Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, Nitin Saurabh, Shubhangi Saraf
2020 Computational Complexity Conference  
The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP.  ...  Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices  ...  Acknowledgements We thank Michael Forbes for illuminating discussions and for telling us about his (correct) intuition concerning Nisan's result.  ... 
doi:10.4230/lipics.ccc.2020.21 dblp:conf/coco/BlaserIMPS20 fatcat:wtvmalfgdratvaq7cztsq7gqke

Page 3110 of Mathematical Reviews Vol. , Issue 2001E [page]

2001 Mathematical Reviews  
As some parameters in the system are varied the orbit of a state can become tangent to a boundary of a region. This leads to a border collision or grazing bifurcation.  ...  A semi-algebraic criterion (anal- ogous to the classical stability criterion of Routh and Hurwitz) for a polynomial to have one symmetric pair of purely imaginary roots and no other root with vanishing  ... 

Page 7303 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
Maps of n-dimensional spaces ex- hibit the same complexity as vector fields in n + 1 dimensions. A generic orientation-preserving germ of a diffeomorphism is em- beddable.  ...  Let H be a Hilbert space, and consider the differential equation @ = F(0,A) defined by the analytic vector field F: H x R" > H.  ... 

Noise-driven Topological Changes in Chaotic Dynamics [article]

Gisela D. Charó, Mickaël D. Chekroun, Denisse Sciamarella, Michael Ghil
2021 arXiv   pre-print
Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways.  ...  Algebraic topology sheds light on the most striking effects involved in such an evolution.  ...  BraMAH builds a cell complex that serves as an algebraic representation of the branched manifold underlying a point cloud in phase space, and it computes the full set of topological properties that describe  ... 
arXiv:2010.09611v7 fatcat:oksdcqijsbgvhluaimbokglbka

Subdivision Methods for the Topology of 2d and 3d Implicit Curves [chapter]

Chen Liang, Bernard Mourrain, Jean-Pascal Pavone
2008 Geometric Modeling and Algebraic Geometry  
In this paper, we describe a subdivision method for handling algebraic implicit curves in 2d and 3d.  ...  We use the representation of polynomials in the Bernstein basis associated with a given box, to check if the topology of the curve is determined inside this box, from its points on the border of the box  ...  Wintz, for his very nice software axel 3 for the visualisation and manipulation of algebraic objects, that we used to produce the pictures of the curves.  ... 
doi:10.1007/978-3-540-72185-7_11 fatcat:wf3goegqajgefcwjroi6xwtd7y

Algebraic method for constructing singular steady solitary waves: a case study

Didier Clamond, Denys Dutykh, André Galligo
2016 Proceedings of the Royal Society A  
localised solutions using the methods of the effective Algebraic Geometry.  ...  This article describes the use of algebraic methods in a phase plane analysis of ordinary differential equations.  ...  Dieudonné and of the University of Nice -Sophia Antipolis during his visits to Nice.  ... 
doi:10.1098/rspa.2016.0194 fatcat:xacfzxjmlndazhnfzwxq2khlm4

Shape description using weighted symmetric axis features

Harry Blum, Roger N. Nagel
1978 Pattern Recognition  
These are based on both width and axis properties, and two linear combinations of them, as well as their derivatives. These are felt to have intuitively and geometrically simple meanings.  ...  We present here an application of symmetric axis geometry to shape classification and description. Shapes are segmented into simplified segments, in which a sequential string of features is derived.  ...  An indication of the complexity of the shape in the vicinity of a branch point is quantified as a busy measure, the number of branch and end points inside the branch disc.  ... 
doi:10.1016/0031-3203(78)90025-0 fatcat:rp4sfedknvdgva7x7lhvlsswhy

Tiling spaces are inverse limits

Lorenzo Sadun
2003 Journal of Mathematical Physics  
Let M be an arbitrary Riemannian homogeneous space, and let Omega be a space of tilings of M, with finite local complexity (relative to some symmetry group Gamma) and closed in the natural topology.  ...  The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Gamma.  ...  Acknowledgements Many thanks to the organizers and participants in the August 2002 conference on Aperiodic Order, Dynamical Systems, Operator Algebras and Topology at the University of Victoria, British  ... 
doi:10.1063/1.1613041 fatcat:aqojda5rd5ht7ox66a4vy2ic6u

Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension [chapter]

Wolf-Jürgen Beyn, Winfried Kleß, Vera Thümmler
2001 Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems  
For these equations we develop a bordered version of the Bartels-Stewart algorithm which allows to reduce the linear algebra to solving a sequence of bordered linear systems.  ...  We show that the predictor and the corrector step during continuation lead to bordered matrix equations of Sylvester type.  ...  In fact, if we pass a turning point of the (u, λ)-branch we expectu T 1u0 > 0 while (49) indicates that the tangent of this branch is reversed due to a turning point with respect to the parameter s as  ... 
doi:10.1007/978-3-642-56589-2_3 fatcat:cmqwoy4ewzalll6ikwovrwbu6e

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representationtheoretic framework suggested by geometric complexity theory (GCT), including: the  ...  ), the connected components technique (Ben-Or, Steele-Yao), depth 3 algebraic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre  ...  Landsberg, Ketan Mulmuley, Toni Pitassi, Peter Scheiblechner, Chris Umans, Alasdair Urquhart, Ryan Williams, and Yiwei She for useful discussions. In particular, Williams suggested the  ... 
doi:10.1109/ccc.2014.35 dblp:conf/coco/Grochow14 fatcat:z5eytgp64jh2pclri4tlmky53e

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2015 Computational Complexity  
We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including:  ...  the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC^0[p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen  ...  Landsberg, Ketan Mulmuley, Toni Pitassi, Peter Scheiblechner, Chris Umans, Alasdair Urquhart, Ryan Williams, and Yiwei She for useful discussions. In particular, Williams suggested the  ... 
doi:10.1007/s00037-015-0103-x fatcat:ifoaveduubhmxlokr4pygqzjqq

Logic and linear algebra: an introduction [article]

Daniel Murfet
2017 arXiv   pre-print
This is made explicit by showing how to represent proofs in linear logic as linear maps between vector spaces.  ...  The interesting part of this vector space semantics is based on the cofree cocommutative coalgebra of Sweedler.  ...  be compared with an actual map of tangent spaces.  ... 
arXiv:1407.2650v3 fatcat:hjiw7vripnecdk5352eznu7nwe

A Subdivision Approach to Planar Semi-algebraic Sets [chapter]

Angelos Mantzaflaris, Bernard Mourrain
2010 Lecture Notes in Computer Science  
Semi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them.  ...  The idea is to localize the boundary curves by subdividing the space and then deduce their shape within small enough cells using only boundary information.  ...  These constraints define a semi-algebraic set as the solution space, in which the optimal points will be searched [15] , [12] .  ... 
doi:10.1007/978-3-642-13411-1_8 fatcat:ggqtdkjvfzccpgkrgukbdarlfm

Calculus for the Liberal Arts: A Humanistic Approach

Kathleen Shannon
1999 Humanistic Mathematics Network Journal  
I followed that plan of action in four classes in spring and summer 1996.  ...  more expensive and even the space shuttle taking off.  ...  The problem that Calculus confronts is how to find the area of a region with curved borders. The third problem is the Tangent or Velocity Problem.  ... 
doi:10.5642/hmnj.199901.21.13 fatcat:53ja4cml3ve63p4uwnfb3p3vem
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