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

2020
*
arXiv
*
pre-print

The proofs reveal an intriguing connection between

arXiv:2003.04834v1
fatcat:xv532a7arjaljbu6znckcsocpm
*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. ...##
###
Algebraic Branching Programs, Border Complexity, and Tangent Spaces

2020
*
Computational Complexity Conference
*

The proofs reveal an intriguing connection between

doi:10.4230/lipics.ccc.2020.21
dblp:conf/coco/BlaserIMPS20
fatcat:wtvmalfgdratvaq7cztsq7gqke
*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. ...##
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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 ...##
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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. ...##
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Noise-driven Topological Changes in Chaotic Dynamics
[article]

2021
*
arXiv
*
pre-print

Noise modifies the behavior of chaotic systems in both quantitative

arXiv:2010.09611v7
fatcat:oksdcqijsbgvhluaimbokglbka
*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 ...##
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Subdivision Methods for the Topology of 2d and 3d Implicit Curves
[chapter]

2008
*
Geometric Modeling and Algebraic Geometry
*

In this paper, we describe a subdivision method for handling

doi:10.1007/978-3-540-72185-7_11
fatcat:wf3goegqajgefcwjroi6xwtd7y
*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. ...##
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Algebraic method for constructing singular steady solitary waves: a case study

2016
*
Proceedings of the Royal Society A
*

localised solutions using the methods of the effective

doi:10.1098/rspa.2016.0194
fatcat:xacfzxjmlndazhnfzwxq2khlm4
*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. ...##
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Shape description using weighted symmetric axis features

1978
*
Pattern Recognition
*

These are based on both width

doi:10.1016/0031-3203(78)90025-0
fatcat:rp4sfedknvdgva7x7lhvlsswhy
*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. ...##
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Tiling spaces are inverse limits

2003
*
Journal of Mathematical Physics
*

Let M be an arbitrary Riemannian homogeneous

doi:10.1063/1.1613041
fatcat:aqojda5rd5ht7ox66a4vy2ic6u
*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 ...##
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Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension
[chapter]

2001
*
Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems
*

For these equations we develop a

doi:10.1007/978-3-642-56589-2_3
fatcat:cmqwoy4ewzalll6ikwovrwbu6e
*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 ...##
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Unifying Known Lower Bounds via Geometric Complexity Theory

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

We show that most

doi:10.1109/ccc.2014.35
dblp:conf/coco/Grochow14
fatcat:z5eytgp64jh2pclri4tlmky53e
*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 ...##
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Unifying Known Lower Bounds via Geometric Complexity Theory

2015
*
Computational Complexity
*

We show that most arithmetic circuit lower bounds

doi:10.1007/s00037-015-0103-x
fatcat:ifoaveduubhmxlokr4pygqzjqq
*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 ...##
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Logic and linear algebra: an introduction
[article]

2017
*
arXiv
*
pre-print

This is made explicit by showing how to represent proofs in linear logic as linear maps between vector

arXiv:1407.2650v3
fatcat:hjiw7vripnecdk5352eznu7nwe
*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*. ...##
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A Subdivision Approach to Planar Semi-algebraic Sets
[chapter]

2010
*
Lecture Notes in Computer Science
*

Semi-

doi:10.1007/978-3-642-13411-1_8
fatcat:ggqtdkjvfzccpgkrgukbdarlfm
*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] . ...##
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Calculus for the Liberal Arts: A Humanistic Approach

1999
*
Humanistic Mathematics Network Journal
*

I followed that plan of action in four classes in spring

doi:10.5642/hmnj.199901.21.13
fatcat:53ja4cml3ve63p4uwnfb3p3vem
*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. ...
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