The $n$ Body Matrix and Its Determinant

Grinberg, Darij; Olver, Peter J.

doi:10.1137/18m1175410

Cited by 6 publications

(4 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In either case, the quotient (usually a much more complicated polynomial 3 ) remains mysterious; our proofs are indirect and reveal little about it. Our second result generalizes a curious property of n 2 × n 2 -determinants [6,Theorem 10] that arose from the study of the n-body problem (see Example 2.4 for details).…”

Section: Introductionsupporting

confidence: 60%

“…In a polynomial ring, each indeterminate is regular; hence, each monomial (without coefficient) is regular (since any product of two regular elements is regular). The following fact is easy to see: 6 6 We recall a few standard concepts from commutative algebra: Let K be a commutative ring. A multiplicative subset of K means a subset S of K that contains the unity 1 K of K and has the property that every a, b ∈ S satisfy ab ∈ S.…”

Section: The Proofsmentioning

confidence: 99%

See 1 more Smart Citation

A double Sylvester determinant

Grinberg

2021

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

Section: Introductionsupporting

confidence: 60%

Section: The Proofsmentioning

confidence: 99%

A double Sylvester determinant

Grinberg

2021

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

“…This section reviews the past attempts at Problem 1.1 starting from the simplest case when points are ordered. The case of labelled clouds C ⊂ R n is easy for isometry classification because the matrix of distances d ij between indexed points p i , p j allows us to reconstruct C by using the known distances to the previously constructed points [34,Theorem 9]. For any clouds of the same number m of labelled points, the difference between m × m matrices of distances (or Gram matrices of p i • p j ) can be converted into a continuous metric by taking a matrix norm.…”

Section: Related Work On Point Cloud Classificationsmentioning

confidence: 99%

The strength of a simplex is the key to a continuous isometry classification of Euclidean clouds of unlabelled points

Kurlin¹

2023

Preprint

View full text Add to dashboard Cite

This paper solves the continuous classification problem for finite clouds of unlabelled points under Euclidean isometry. The Lipschitz continuity of required invariants in a suitable metric under perturbations of points is motivated by the inevitable noise in measurements of real objects.The best solved case of this isometry classification is known as the SSS theorem in school geometry saying that any triangle up to congruence (isometry in the plane) has a continuous complete invariant of three side lengths.However, there is no easy extension of the SSS theorem even to four points in the plane partially due to a 4parameter family of 4-point clouds that have the same six pairwise distances. The computational time of most past metrics that are invariant under isometry was exponential in the size of the input. The final obstacle was the discontinuity of previous invariants at singular configurations, for example, when a triangle degenerates to a straight line.All the challenges above are now resolved by the Simplexwise Centred Distributions that combine inter-point distances of a given cloud with the new strength of a simplex that finally guarantees the Lipschitz continuity. The computational times of new invariants and metrics are polynomial in the number of points for a fixed Euclidean dimension.

show abstract

“…The case of labeled point sets in R n is easy for isometry classification because the matrix of distances d ij between indexed points p i , p j allows us to build a set by using the known distances to the previously constructed points [28,Theorem 9]. For any sets A, B ⊂ R n of the same number m of points, the difference between m × m matrices of distances (or Gram matrices of p i • p j ) can be converted into a continuous metric by taking a matrix norm.…”

Section: A Review Of Past Work On Isometry Invariants Of Finite Point...mentioning

confidence: 99%

Computable complete invariants for finite clouds of unlabeled points under Euclidean isometry

Kurlin¹

2022

Preprint

View full text Add to dashboard Cite

A finite cloud of unlabeled points is the simplest representation of many real objects such as rigid shapes considered modulo rigid motion or isometry preserving inter-point distances. The distance matrix uniquely determines any finite cloud of labeled (ordered) points under Euclidean isometry but is intractable for comparing clouds of unlabeled points due to a huge number of permutations.The past work developed approximate algorithms for Hausdorff-like distances minimized over translations, rotations, and general isometries. One big success is the exact algorithm by Paul Chew et al for matching sets in the plane under Euclidean motion with a polynomial complexity of degree five in the number of points. We introduce continuous and complete isometry invariants on the spaces of finite clouds of unlabeled points considered under isometry in any Euclidean space. The continuity under perturbations of points in the bottleneck distance is proved in terms of new metrics that are exactly computable in polynomial time in the number of points for a fixed dimension.

show abstract

The $n$ Body Matrix and Its Determinant

Cited by 6 publications

References 19 publications

A double Sylvester determinant

A double Sylvester determinant

The strength of a simplex is the key to a continuous isometry classification of Euclidean clouds of unlabelled points

Computable complete invariants for finite clouds of unlabeled points under Euclidean isometry

Contact Info

Product

Resources

About