Selfcoincidences and roots in Nielsen theory

Koschorke, Ulrich

doi:10.1007/s11784-007-0046-1

“…Versions very close to that stated below can be found in [21] Corollary 3.3 and [9]; see, also, [24]. Proof.…”

Section: Given By the Inclusion Of Null(s) In H-null(s) The Homotopysupporting

confidence: 66%

“…In a series of recent papers [16][17][18][19][20][21][22] Koschorke has studied, by differential-topological methods, when B is a closed manifold, rather than a general ENR, what we shall call the homotopy coincidence index. This index is an element of a certain stable homotopy group and in a range of dimensions (dim B < 2(dim N − 1)) it is the precise obstruction to the existence of a deformation of e and f to maps with empty coincidence set.…”

Section: Introductionmentioning

confidence: 99%

The homotopy coincidence index

Crabb

¹

2010

J. Fixed Point Theory Appl.

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“…Like the topological degree, it is a homotopy invariant and does not change under small perturbations of f . For pairs of maps f 1 , f 2 : M m → N n between compact manifolds of different dimensions m, n, Nielsen theory has been applied to compute a lower bound for the number of connected components of the coincidence set f 1 = f 2 , where f i f i are homotopic for i = 1, 2 [32,33]. Surprisingly, the author of [32] shows that under the condition m ≤ 2n − 3-the same as our stable dimension range-a computable number N (f 1 , f 2 ) coincides with the minimal number of connected components of…”

Section: Related Workmentioning

confidence: 99%

Robust Satisfiability of Systems of Equations

Franek

¹

,

Krčál

²

2015

J. ACM

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We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f : K → R n on a finite simplicial complex K and α > 0, it holds that each function g : K → R n such that g − f ∞ ≤ α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dim K ≤ 2n−3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis.Via a reverse reduction we prove that the problem is undecidable when dim K ≥ 2n − 2, where the threshold comes from the stable range in homotopy theory.For the lucidity of our exposition, we focus on the setting when f is piecewise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.

show abstract

“…Like the topological degree, it is a homotopy invariant and does not change under small perturbations of f . For pairs of maps f 1 , f 2 : M m → N n between compact manifolds of different dimensions m, n, Nielsen theory has been applied to compute a lower bound for the number of connected components of the coincidence set f 1 = f 2 , where f i f i are homotopic for i = 1, 2 [32,33]. Surprisingly, the author of [32] shows that under the condition m ≤ 2n − 3-the same as our stable dimension range-a computable number N (f 1 , f 2 ) coincides with the minimal number of connected components of {f 1 = f 2 } for f 1 resp.…”

mentioning

confidence: 99%

Robust Satisfiability of Systems of Equations

Franek

¹

,

Krčál

²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f : K → R n on a finite simplicial complex K and α > 0, it holds that each function g : K → R n such that g − f ∞ ≤ α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dim K ≤ 2n−3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis.Via a reverse reduction we prove that the problem is undecidable when dim K ≥ 2n − 2, where the threshold comes from the stable range in homotopy theory.For the lucidity of our exposition, we focus on the setting when f is piecewise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.

show abstract

Selfcoincidences and roots in Nielsen theory

Cited by 10 publications

References 16 publications

The homotopy coincidence index

The homotopy coincidence index

Robust Satisfiability of Systems of Equations

Robust Satisfiability of Systems of Equations

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