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Computing critical points for invariant algebraic systems
[article]

2020
*
arXiv
*
pre-print

We consider the problem of

arXiv:2009.00847v1
fatcat:icnflywcivas5gxm3fqbktg5am
*computing*the*points*at which 𝐟 vanish and the Jacobian matrix associated to 𝐟, ϕ is rank deficient provided that this set is finite. ... We exploit the*invariance*properties of the input to split the solution space according to the orbits of 𝒮_n. ... Algorithms*for**computing**critical**points*We can now turn to the main question in this article. ...##
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Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials
[article]

2022
*
arXiv
*
pre-print

., x_n] with s<n, we consider the problem of

arXiv:2203.16094v2
fatcat:cnbp5vlqxzdlpnv4u2j6g3unk4
*computing*the set W(ϕ, 𝐟) of*points*at which 𝐟 vanishes and the Jacobian matrix of 𝐟, ϕ with respect to x_1, ..., x_n does not have full rank. ... The runtime of our algorithm is polynomial in the total number of*points*described by the output. ...*critical**points**for**algebraic**systems*. ...##
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Computing Critical Points for Algebraic Systems Defined by Hyperoctahedral Invariant Polynomials

2022
*
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation
*

CCS CONCEPTS •

doi:10.1145/3476446.3536181
fatcat:lafepkefoffuvaa4g7gb7htjmi
*Computing*methodologies →*Algebraic*algorithms; • Theory of*computation*→ Design and analysis of algorithms. ... . , x n ] with s < n, we consider the problem of*computing*the set W (ϕ, f ) of*points*at which f vanishes and the Jacobian matrix of f , ϕ with respect to x 1 , . . . , x n does not have full rank. ... in*computing**critical**points**for**algebraic**systems*. ...##
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Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems

2005
*
Bulletin des Sciences Mathématiques
*

This work deals with planar polynomial differential

doi:10.1016/j.bulsci.2004.09.002
fatcat:hvl3g5ydcberfpax4pxmbxm56u
*systems*{ẋ} = P (x, y), {ẏ} = Q(x, y). We give a set of necessary conditions*for*a*system*to have an*invariant**algebraic*curve. ... These conditions are determined from the value of the cofactor at the singular*points*of the*system*, once considered in a compact space. ...*Critical**points*at infinity We first define a polynomial differential equation in CP(2) and the notion of*invariant**algebraic*curve and*critical**point**for*the equation. ...##
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Synthesizing Switching Controllers for Hybrid Systems by Generating Invariants
[chapter]

2013
*
Lecture Notes in Computer Science
*

We extend a template-based approach

doi:10.1007/978-3-642-39698-4_22
fatcat:na6ge7pjandallod5rdqksmyd4
*for*synthesizing switching controllers*for*semi-*algebraic*hybrid*systems*, in which all expressions are polynomials. ... This is achieved by combining a QE (quantifier elimination)-based method*for*generating*invariants*with a qualitative approach*for*predefining templates. ... Jiang Liu*for*his great contribution to our previous joint work on*invariant*generation. We also thank Dr. Matthias Horbach, Mr. ...##
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Homogeneous mechanical systems with symmetry, and geometry of the coadjoint action

1976
*
Bulletin of the American Mathematical Society
*

In this note we announce further results on the global behaviour of the so called transitive mechanical

doi:10.1090/s0002-9904-1976-14002-5
fatcat:us6ofb6eanhk3iywitirpcelje
*systems*[4], i.e., mechanical*systems*with symmetry on homogeneous spaces with an*invariant*riemannian ... The germ of this idea appeared in Arnold [1, especially §6], linked to the Euler equations*for*Lie groups. Detailed proofs will appear in [5]. ... If ad(G) C GL(G) is*algebraic*, then lm(J) and lm(I) are semialgebraic sets*for*any transitive mechanical*system*with G as its group of symmetries. ...##
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Conformal algebras of two-dimensional disordered systems

2002
*
Journal of Physics A: Mathematical and General
*

Remarkably, we also find this

doi:10.1088/0305-4470/35/27/101
fatcat:6zkgxjcu4ncefm5ysq6wf7unr4
*algebra*, and thereby an ensuing hidden supersymmetry, realized at general replica-averaged*critical**points*,*for*which we derive an explicit formula*for*b. ... We discuss the structure of 2D conformal field theories (CFT) at central charge c=0 describing*critical*disordered*systems*, polymers and percolation. ... Cardy,*for*useful discussions. One of us (V.G.) is supported by the NSF grant PHY-94-07194. ...##
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A cubic Kolmogorov system with six limit cycles

2002
*
Computers and Mathematics with Applications
*

We show in particular that a maximum of six small amplitude limit cycles can bifurcate from a

doi:10.1016/s0898-1221(02)00161-x
fatcat:qwsfpnmlunbbdfqi372yzxszay
*critical**point*in the first quadrant, and we discuss the number of*invariant*lines. ... we consider a class of cubic Kolmogorov*systems*. ... The derivation of necessary conditions often involves extensive*computing*. We use the*computer**algebra**system*Reduce predominantly, but occasionally also Maple. ...##
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Distinct small-distance scaling behavior of on-off intermittency in chaotic dynamical systems

1996
*
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
*

Although the statistics of their laminar phase obeys the

doi:10.1103/physreve.54.321
pmid:9965075
fatcat:5dmx23babrgzjc3pcvjfz5k3da
*algebraic*scaling, quantities such as the average transient time*for*trajectories to fall in a small neighborhood of the asymptotic off state exhibit ... On-off intermittency in chaotic dynamical*systems*refers to the situation where some dynamical variables exhibit two distinct states in their course of time evolution. ... These singularities come from the successive iterations of the*critical**point*of the map x c ϭ0.5 ͓9͔. ...##
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Real Root Finding for Equivariant Semi-algebraic Systems

2018
*
Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18
*

Combining this geometric result with efficient algorithms

doi:10.1145/3208976.3209023
dblp:conf/issac/RienerD18
fatcat:2xpe5foybvgxfaw7mat3gays6i
*for*real root finding (based on the*critical**point*method), one can decide the emptiness of basic semi-*algebraic*sets defined by s polynomials ... Such a semi-*algebraic*set is*invariant*by the action of the symmetric group. We show that such a set is either empty or it contains a*point*with at most 2d − 1 distinct coordinates. ... With a more*algebraic*flavour,*computer**algebra*has been developed to solve polynomial*systems*which are*invariant*under the action of some groups. ...##
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Some criteria for the existence of limit cycles for quadratic vector fields

2003
*
Journal of Mathematical Analysis and Applications
*

In this paper we consider real quadratic

doi:10.1016/s0022-247x(03)00149-5
fatcat:3h7fciuyora7lhze6gvoh3rc4y
*systems*. We present new criteria*for*the existence and uniqueness of limit cycles*for*such*systems*by using Darbouxian particular solutions. ... Some results are based on the study of such*systems*in CP 2 . We also generalize the well-know result of Bautin on the nonexistence of limit cycles*for*quadratic Lotka-Volterra*systems*. (J. ... It is well known (see,*for*instance, [16] ) that a limit cycle of a quadratic*system*surrounds an unique*critical**point*(x 0 , y 0 ) of the*system*. ...##
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Non-nested configuration of algebraic limit cycles in quadratic systems

2006
*
Journal of Differential Equations
*

This work deals with

doi:10.1016/j.jde.2006.01.009
fatcat:viqe3xhzyzbnlf57as3adjakfa
*algebraic*limit cycles of planar polynomial differential*systems*of degree two. ... More concretely, we show among other facts that a quadratic vector field cannot possess two non-nested*algebraic*limit cycles contained in different irreducible*invariant**algebraic*curves. ... He has also*pointed*out us Ref. [15] . ...##
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Real root finding for equivariant semi-algebraic systems
[article]

2018
*
arXiv
*
pre-print

Combining this geometric result with efficient algorithms

arXiv:1806.08121v1
fatcat:3ba7dehbuffezkyndhud34lsme
*for*real root finding (based on the*critical**point*method), one can decide the emptiness of basic semi-*algebraic*sets defined by s polynomials ... Such a semi-*algebraic*set is*invariant*by the action of the symmetric group. We show that such a set is either empty or it contains a*point*with at most 2d-1 distinct coordinates. ... With a more*algebraic*flavour,*computer**algebra*has been developed to solve polynomial*systems*which are*invariant*under the action of some groups. ...##
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Page 8530 of Mathematical Reviews Vol. , Issue 2002K
[page]

2002
*
Mathematical Reviews
*

The

*critical**point*at infinity occurs only in spatial problems. We*compute*the homology of the integral manifold*for*each regular value of the Jacobi constant. ... “There are five positive*critical*values of the Jacobi constant: one is due to a*critical**point*at infinity, another is due to the Lagrangian*critical**points*and three are due to the Eulerian*critical*...##
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Noncommutative topological dynamics

2006
*
Chaos, Solitons & Fractals
*

We study noncommutative dynamical

doi:10.1016/j.chaos.2005.03.016
fatcat:4viewhmd3jd6xkcifi7kmkwjo4
*systems*associated to unimodal and bimodal maps of the interval. To these maps we associate subshifts and the correspondent AF-*algebras*and Cuntz-Krieger*algebras*. ... As an example we consider*systems*having equal topological entropy log(1 + /), where / is the golden number, but distinct chaotic behavior and we show how a new numerical*invariant*allows to distinguish ... The K 0 -group of F s;r is not a complete*invariant**for*the bimodal dynamical*system*. Proof. Set s fixed. Let r 1 and r 2 be two distinct values in Z½s; s À1 . ...
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