Algebra

Lang, Serge

doi:10.1007/978-1-4613-0041-0

Cited by 1,154 publications

(471 citation statements)

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“…Clearly Equations (12) and (14) always hold (see Proposition 3), and Equations (13) and (15) are the same equation (see again Proposition 3).…”

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confidence: 87%

Central Configurations of the Planar Coorbitalsatellite Problem

Cors¹,

Llibre²,

Ollé³

2004

Celestial Mechanics and Dynamical Astronomy

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We study the planar central configurations of the 1 + n body problem where one mass is large and the other n masses are infinitesimal and equal. We find analytically all these central configurations when 2 ≤ n ≤ 4. Numerically, first we provide evidence that when n ≥ 9 the only central configuration is the regular n-gon with the large mass in its barycenter, and second we provide also evidence of the existence of an axis of symmetry for every central configuration.

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“…Clearly Equations (12) and (14) always hold (see Proposition 3), and Equations (13) and (15) are the same equation (see again Proposition 3).…”

mentioning

confidence: 87%

Central Configurations of the Planar Coorbitalsatellite Problem

Cors¹,

Llibre²,

Ollé³

2004

Celestial Mechanics and Dynamical Astronomy

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“…Hence, the endomorphism ring of an indecomposable object is local and the Krull-Schmidt theorem is an easy consequence by the classical proof as in [4] or in [1].…”

Section: S ⊗ R (R/rad(r)) S/(s ⊗ R Rad(r))mentioning

confidence: 99%

A Noether–deuring Theorem for Derived Categories

Zimmermann¹

2012

Glasgow Math. J.

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Abstract. We prove a Noether-Deuring theorem for the derived category of bounded complexes of modules over a Noetherian algebra.2010 Mathematics Subject Classification. Primary 16E35; Secondary 11S36, 13J10, 18E30, 16G30. Introduction.The classical Noether-Deuring theorem states that given an algebra A over a field K and a finite extension field L of K, two A-modules M andIn 1972, Roggenkamp gave a nice extension of this result to extensions S of local commutative Noetherian rings R and modules over Noetherian R-algebras.For the derived category of A-modules no such generalisation was documented before. The purpose of this note is to give a version of the Noether-Deuring theorem, in the generalised version given by Roggenkamp, for right bounded derived categories of A-modules. If there is a morphism α ∈ Hom D( ) (X, Y ), then it is fairly easy to show that for a faithfully flat ring extension S over R the fact that id S ⊗ α is an isomorphism implies that α is an isomorphism. This is done in proposition (1). More delicate is the question if only an isomorphism in Hom D(S⊗ R ) (S ⊗ R X, S ⊗ R Y ) is given. Then, we need further finiteness conditions on and on R and proceed by completion of R and then a classical going-down argument. This is done in theorem (4) and corollary (8).For the notation concerning derived categories, we refer to Verdier [6]. In particular, D(A) (resp D − (A), resp D b (A)) denotes the derived category of complexes (resp. right bounded complexes, resp. bounded complexes) of finitely generated A-modules,is the homotopy category of right bounded complexes (resp. bounded complexes, resp. right bounded complexes with bounded homology) of finitely generated projective A-modules. For a complex Z, we denote by H i (Z) the homology of Z in degree i, and by H(Z) the graded module given by the homology of Z.

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“…Obviously, when y 1 is at the intersection of more than one manifold, then in (36) Proof Please refer to Lang (2002).…”

Section: Theorem 4 (Derivative Of the Fitting Polynomials) Letmentioning

confidence: 99%

“…To show the other direction, by Hilbert's Nullstellensatz theorem, we only need to prove that I F and I H are prime ideals (Lang 2002).…”

Section: Appendixmentioning

confidence: 99%

Robust Algebraic Segmentation of Mixed Rigid-Body and Planar Motions from Two Views

et al. 2010

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This paper studies segmentation of multiple rigidbody motions in a 3-D dynamic scene under perspective camera projection. We consider dynamic scenes that contain both 3-D rigid-body structures and 2-D planar structures. Based on the well-known epipolar and homography constraints between two views, we propose a hybrid perspective constraint (HPC) to unify the representation of rigidbody and planar motions. Given a mixture of K hybrid perspective constraints, we propose an algebraic process to partition image correspondences to the individual 3-D motions, called Robust Algebraic Segmentation (RAS). Particularly, we prove that the joint distribution of image correspondences is uniquely determined by a set of (2K)-th degree polynomials, a global signature for the union of K motions of possibly mixed type. The first and second deriva- tives of these polynomials provide a means to recover the association of the individual image samples to their respective motions. Finally, using robust statistics, we show that the polynomials can be robustly estimated in the presence of moderate image noise and outliers. We conduct extensive simulations and real experiments to validate the performance of the new algorithm. The results demonstrate that RAS achieves notably higher accuracy than most existing robust motion-segmentation methods, including random sample consensus (RANSAC) and its variations. The implementation of the algorithm is also two to three times faster than the existing methods. The implementation of the algorithm and the benchmark scripts are available at http:// perception.csl.illinois.edu/ras/.

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Algebra

Cited by 1,154 publications

References 0 publications

Central Configurations of the Planar Coorbitalsatellite Problem

Central Configurations of the Planar Coorbitalsatellite Problem

A Noether–deuring Theorem for Derived Categories

Robust Algebraic Segmentation of Mixed Rigid-Body and Planar Motions from Two Views

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