arXiv subjects

Olga Anosova

Publications and source records attributed to Olga Anosova.

10 recordsLinked to original sources

arXiv2026

The strength of a geometric simplex

The basic input for many real objects is a finite cloud of unordered points. The strongest equivalence between objects in practice is rigid motion in a Euclidean space. A recent polynomial-time classification of point clouds required a Lipschitz continuous function that vanishes on degenerate simplices, while the usual volume is not Lipschitz. We define the strength of any geometric simplex and prove its continuity under perturbations with explicit bounds for Lipschitz constants.

math.MG↗

arXiv2026

Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry

Ideal symmetry is known to break down under almost any noise. One measure of asymmetry in a periodic crystal is the relative multiplicity Z' of geometrically non-equivalent units. However, Z' discontinuously changes under almost any displacement of atoms, which can arbitrarily scale up a primitive cell. This discontinuity was recently resolved by a hierarchy of invariant descriptors that continuously change under all small perturbations. We introduce a Continuous Invariant-based Asymmetry (CIA) to quantify (in physically meaningful Angstroms) the deviation of a periodic crystal from a higher symmetry form. Our experiments on several Crystal Structure Prediction datasets show that about a half of simulated crystals have high values of CIA, while all experimental structures in these datasets have CIA=0. On another hand, many crystals with high values Z' in the Cambridge Structural Database (CSD) turned out to be close to more symmetric forms with Z'<=1 due to low values of CIAs.

math.MG↗

arXiv2025

Complete and bi-continuous invariant of protein backbones under rigid motion

Proteins are large biomolecules that regulate all living organisms and consist of one or several chains. The primary structure of a protein chain is a sequence of amino acid residues whose three main atoms (alpha-carbon, nitrogen, and carbonyl carbon) form a protein backbone. The tertiary structure is the rigid shape of a protein chain represented by atomic positions in 3-dimensional space. Because different geometric structures often have distinct functional properties, it is important to continuously quantify differences in rigid shapes of protein backbones. Unfortunately, many widely used similarities of proteins fail axioms of a distance metric and discontinuously change under tiny perturbations of atoms. This paper develops a complete invariant that identifies any protein backbone in 3-dimensional space, uniquely under rigid motion. This invariant is Lipschitz bi-continuous in the sense that it changes up to a constant multiple of a maximum perturbation of atoms, and vice versa. The new invariant has been used to detect thousands of (near-)duplicates in the Protein Data Bank, whose presence inevitably skews machine learning predictions. The resulting invariant space allows low-dimensional maps with analytically defined coordinates that reveal substantial variability in the protein universe.

cs.CG↗

arXiv2025

Recognition of near-duplicate periodic patterns by continuous metrics with approximation guarantees

This paper rigorously solves the challenging problem of recognizing periodic patterns under rigid motion in Euclidean geometry. The 3-dimensional case is practically important for justifying the novelty of solid crystalline materials (periodic crystals) and for patenting medical drugs in a solid tablet form. Past descriptors based on finite subsets fail when a unit cell of a periodic pattern discontinuously changes under almost any perturbation of atoms, which is inevitable due to noise and atomic vibrations. The major problem is not only to find complete invariants (descriptors with no false negatives and no false positives for all periodic patterns) but to design efficient algorithms for distance metrics on these invariants that should continuously behave under noise. The proposed continuous metrics solve this problem in any Euclidean dimension and are algorithmically approximated with small error factors in times that are explicitly bounded in the size and complexity of a given pattern. The proved Lipschitz continuity allows us to confirm all near-duplicates filtered by simpler invariants in major databases of experimental and simulated crystals. This practical detection of noisy duplicates will stop the artificial generation of `new' materials from slight perturbations of known crystals. Several such duplicates are under investigation by five journals for data integrity.

math.MG↗

arXiv2024

The importance of definitions in crystallography

This paper was motivated by the articles "Same or different - that is the question" in CrystEngComm (July 2020) and "Change to the definition of a crystal" in the IUCr newsletter (June 2021). Experimental approaches to crystal comparisons require rigorously defined classifications in crystallography and beyond. Since crystal structures are determined in a rigid form, their strongest equivalence in practice is rigid motion, which is a composition of translations and rotations in 3-dimensional space. Conventional representations based on reduced cells and standardizations theoretically distinguish all periodic crystals. However, all cell-based representations are inherently discontinuous under almost any atomic displacement that can arbitrarily scale up a reduced cell. Hence comparing millions of known structures in materials databases needs continuous distance metrics.

cond-mat.mtrl-sci↗

arXiv2023

Density functions of periodic sequences of continuous events

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.

cs.CG↗

arXiv2022

Density functions of periodic sequences

Periodic point sets model all solid crystalline materials whose structures are determined in a rigid form and should be studied up to rigid motion or isometry preserving inter-point distances. In 2021 H.Edelsbrunner et al. introduced an infinite sequence of density functions that are continuous isometry invariants of periodic point sets. These density functions turned out to be highly non-trivial even in dimension 1 for periodic sequences of points in the line. This paper fully describes the density functions of any periodic sequence and their symmetry properties. The explicit description theoretically confirms coincidences of density functions that were previously computed only through finite samples.

cs.CG↗

arXiv2021

Fast predictions of lattice energies by continuous isometry invariants of crystal structures

Crystal Structure Prediction (CSP) aims to discover solid crystalline materials by optimizing periodic arrangements of atoms, ions or molecules. CSP takes weeks of supercomputer time because of slow energy minimizations for millions of simulated crystals. The lattice energy is a key physical property, which determines thermodynamic stability of a crystal but has no simple analytic expression. Past machine learning approaches to predict the lattice energy used slow crystal descriptors depending on manually chosen parameters. The new area of Periodic Geometry offers much faster isometry invariants that are also continuous under perturbations of atoms. Our experiments on simulated crystals confirm that a small distance between the new invariants guarantees a small difference of energies. We compare several kernel methods for invariant-based predictions of energy and achieve the mean absolute error of less than 5kJ/mole or 0.05eV/atom on a dataset of 5679 crystals.

cond-mat.mtrl-sci↗

arXiv2021

Introduction to Periodic Geometry and Topology

This monograph introduces key concepts and problems in the new research area of Periodic Geometry and Topology for materials applications.Periodic structures such as solid crystalline materials or textiles were previously classified in discrete and coarse ways that depend on manual choices or are unstable under perturbations. Since crystal structures are determined in a rigid form, their finest natural equivalence is defined by rigid motion or isometry, which preserves inter-point distances. Due to atomic vibrations, isometry classes of periodic point sets form a continuous space whose geometry and topology were unknown. The key new problem in Periodic Geometry is to unambiguously parameterize this space of isometry classes by continuous coordinates that allow a complete reconstruction of any crystal. The major part of this manuscript reviews the recently developed isometry invariants to resolve the above problem: (1) density functions computed from higher order Voronoi zones, (2) distance-based invariants that allow ultra-fast visualizations of huge crystal datasets, and (3) the complete invariant isoset (a DNA-type code) with a first continuous metric on all periodic crystals. The main goal of Periodic Topology is to classify textiles up to periodic isotopy, which is a continuous deformation of a thickened plane without a fixed lattice basis. This practical problem substantially differs from past research focused on links in a fixed thickened torus.

cs.CG↗

arXiv2021

A Proof of the Invariant Based Formula for the Linking Number and its Asymptotic Behaviour

In 1833 Gauss defined the linking number of two disjoint curves in 3-space. For open curves this double integral over the parameterised curves is real-valued and invariant modulo rigid motions or isometries that preserve distances between points, and has been recently used in the elucidation of molecular structures. In 1976 Banchoff geometrically interpreted the linking number between two line segments. An explicit analytic formula based on this interpretation was given in 2000 without proof in terms of 6 isometry invariants: the distance and angle between the segments and 4 coordinates specifying their relative positions. We give a detailed proof of this formula and describe its asymptotic behaviour that wasn't previously studied.

math.GT↗