Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

Kurlin, Vitaliy

doi:10.48550/arxiv.2205.04388

Cited by 2 publications

(2 citation statements)

References 24 publications

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“…This paper focused on foundations, so experiments are postponed to future work. The exactly computable metric on WMIs can be adapted to the complete isometry invariants of periodic crystals [61], which has been done only for 1-periodic sequences [62][63][64]. The earlier invariants [1,46,48] detected geometric duplicates, which had wrong atomic types but were deposited in the well-curated (mostly by experienced eyes) world's largest collection of real materials (Cambridge Structural Database).…”

Section: Exactly Computable Metrics All Cloudsmentioning

confidence: 99%

Polynomial-Time Algorithms for Continuous Metrics on Atomic Clouds of Unordered Points

Kurlin

2023

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The most fundamental model of a molecule is a cloud of unordered atoms, even without chemical bonds that can depend on thresholds for distances and angles. The strongest equivalence between clouds of atoms is rigid motion, which is a composition of translations and rotations. The existing datasets of experimental and simulated molecules require a continuous quantification of similarity in terms of a distance metric. While clouds of m ordered points were continuously classified by Lagrange’s quadratic forms (distance matrices or Gram matrices), their extensions to m unordered points are impractical due to the exponential number of m! permutations. We propose new metrics that are continuous in general position and are computable in a polynomial time in the number m of unordered points in any Euclidean space of a fixed dimension n.

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Section: Exactly Computable Metrics All Cloudsmentioning

confidence: 99%

Polynomial-Time Algorithms for Continuous Metrics on Atomic Clouds of Unordered Points

Kurlin

2023

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“…For 1-periodic sequences of points in R n , complete isometry invariants with continuous and computable metrics appeared in [12], see related results for finite clouds of unlabeled points [11,17].…”

Section: Review Of Related Past Workmentioning

confidence: 99%

Density functions of periodic sequences of continuous events

Anosova¹,

Kurlin²

2023

Preprint

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Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations. The motivations come from periodic crystals whose structures are determined in a rigid form but any minimal cells can discontinuously change due to small noise in measurements. For any integer k ≥ 0, the density function of a periodic set S was previously defined as the fractional volume of all k-fold intersections (within a minimal cell) of balls that have a variable radius t and centers at all points of S. This paper introduces the density functions for periodic sets of points with different initial radii motivated by atomic radii of chemical elements and by continuous events occupying disjoint intervals in time series. The contributions are explicit descriptions of the densities for periodic sequences of intervals. The new densities are strictly stronger and distinguish periodic sequences that have identical densities in the case of zero radii.

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Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

Cited by 2 publications

References 24 publications

Polynomial-Time Algorithms for Continuous Metrics on Atomic Clouds of Unordered Points

Polynomial-Time Algorithms for Continuous Metrics on Atomic Clouds of Unordered Points

Density functions of periodic sequences of continuous events

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