Betti Numbers of Semialgebraic and Sub-Pfaffian Sets

Gabrielov, Andrei; Vorobjov, Nicolai; Zell, Thierry

doi:10.1112/s0024610703004939

Cited by 36 publications

(67 citation statements)

References 23 publications

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“…The case where Π| X is closed is treated in [14], and the proof is identical in the locally split case.…”

Section: Betti Numbers Of Sub-pfaffian Sets and Of Hausdorff Limitsmentioning

confidence: 99%

“…The first upper-bounds for the Betti numbers of Pfaffian sets which were not defined by quantifier-free formulas were obtained in [14] (for sub-Pfaffian sets) and in [25] (for Hausdorff limits). These results will be the key to our estimates for relative closures.…”

Section: Betti Numbers Of Sub-pfaffian Sets and Of Hausdorff Limitsmentioning

confidence: 99%

“…It seems it first appeared in [5] in the case of proper maps, and it has been rediscovered many times since. The reader can find proofs in [14] for the closed case and in [2] or [6] for the locally split case. Example 1.19.…”

Section: Betti Numbers Of Sub-pfaffian Sets and Of Hausdorff Limitsmentioning

confidence: 99%

“…It also presents all the spectral sequence machinery and its corollaries that appeared first in [14] and [25], and which will be used in our proofs. Section 2 is devoted to the Borel-Moore estimates, and Section 3 to the singular case.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Topological Complexity of the Relative Closure of a Semi-Pfaffian Couple

Zell

2008

Discrete Comput Geom

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“…The case where Π| X is closed is treated in [14], and the proof is identical in the locally split case.…”

Section: Betti Numbers Of Sub-pfaffian Sets and Of Hausdorff Limitsmentioning

confidence: 99%

Section: Betti Numbers Of Sub-pfaffian Sets and Of Hausdorff Limitsmentioning

confidence: 99%

Section: Betti Numbers Of Sub-pfaffian Sets and Of Hausdorff Limitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Topological Complexity of the Relative Closure of a Semi-Pfaffian Couple

Zell

2008

Discrete Comput Geom

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“…Gabrielov, Vorobjov, and Zell (2003) showed how to compute bounds on the number of connected components of sets of the form (1). However, in their general approach it is often rather difficult to obtain good bounds because they, in fact, bound the sum of all Betti numbers Φ, while the number of connected components equals the zeroth Betti number.…”

Section: Pfaffian Functionsmentioning

confidence: 99%

Approximate Generalizations and Computational Experiments

Kübler¹

2007

Econometrica

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Abstract:In this paper I demonstrate how one can generalize finitely many examples to statements about (infinite) classes of economic models. If there exist upper bounds on the number of connected components of one-dimensional linear subsets of the set of parameters for which a conjecture is true, one can conclude that it is correct for all parameter values in the class considered, except for a small residual set, once one has verified the conjecture for a predetermined finite set of points. I show how to apply this insight to computational experiments and spell out assumptions on the economic fundamentals that ensure that the necessary bounds on the number of connected components exist. I argue that these methods can be fruitfully utilized in applied general equilibrium analysis. I provide general assumptions on preferences and production sets that ensure that economic conjectures define sets with a bounded number of connected components. Using the theoretical results, I give an example of how one can explore qualitative and quantitative implications of general equilibrium models using computational experiments. Finally, I show how random algorithms can be used for generalizing examples in high-dimensional problems. In this paper I demonstrate how one can generalize finitely many examples to statements about (infinite) classes of economic models. If there exist upper bounds on the number of connected components of one-dimensional linear subsets of the set of parameters for which a conjecture is true, one can conclude that it is correct for all parameter values in the class considered, except for a small residual set, once one has verified the conjecture for a predetermined finite set of points. I show how to apply this insight to computational experiments and spell out assumptions on the economic fundamentals that ensure that the necessary bounds on the number of connected components exist.I argue that these methods can be fruitfully utilized in applied general equilibrium analysis. I provide general assumptions on preferences and production sets that ensure that economic conjectures define sets with a bounded number of connected components. Using the theoretical results, I give an example of how one can explore qualitative and quantitative implications of general equilibrium models using computational experiments. Finally, I show how random algorithms can be used for generalizing examples in high-dimensional problems.

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Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems

Korovina

Vorobjov

2006

Logical Approaches to Computational Barriers

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Abstract. In this paper we study a class of dynamical systems defined by Pfaffian maps. It is a sub-class of o-minimal dynamical systems which capture rich continuous dynamics and yet can be studied using finite bisimulations. The existence of finite bisimulations for o-minimal dynamical and hybrid systems has been shown by several authors (see e.g. [2,3,11]). The next natural question to investigate is how the sizes of such bisimulations can be bounded. The first step in this direction was done in [9] where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems. In the present paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of systems on which the exponential bound is attained. The bounds provide a basis for designing efficient algorithms for computing bisimulations, solving reachability and motion planning problems.

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Betti Numbers of Semialgebraic and Sub-Pfaffian Sets

Cited by 36 publications

References 23 publications

Topological Complexity of the Relative Closure of a Semi-Pfaffian Couple

Topological Complexity of the Relative Closure of a Semi-Pfaffian Couple

Approximate Generalizations and Computational Experiments

Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems

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