A discrete Gauss-Bonnet type theorem

Knill, Oliver

doi:10.4171/em/188

Cited by 11 publications

(11 citation statements)

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“…the graph obtained by adding extra edges until all non-triangular faces have been decomposed into triangles. A recent discrete Gauss-Bonnet theorem by Knill [18] says that this curvature integrates to a multiple of the Euler characteristic of the bond graph (see also Theorem 2.1 below, which extends this result to irregular boundaries as needed here). This leads to the following exact expression for the energy (1.1), (1.3) of an arbitrary configuration X = (x 1 , .., x N ) (see Theorem 3.1 below):…”

Section: Introductionmentioning

confidence: 94%

“…What turns out to work is a discrete version of Puiseux curvature, applied not to the bond graph but the triangulated bond graph. Discrete Puiseux curvature was recently introduced and studied in the context of subdomains of the triangular lattice by Knill [18], and can be thought of as a boundary-corrected version of Euler curvature, corrected in such a way that flat boundaries have curvature zero. First we define the triangulated bond graph.…”

Section: 4mentioning

confidence: 99%

“…We endow them with additional mathematical structure (graph; metric; discrete curvature) and derive a discrete Gauss-Bonnet theorem by adapting recent work of Knill [18] to general triangular graphs.…”

Section: Discrete Differential Geometry Of Particle Configurationsmentioning

confidence: 99%

“…As turns out, we will work with the combinatorial Puiseux curvature (see [18] or (2.7) below) of the triangulated bond graph, i.e. the graph obtained by adding extra edges until all non-triangular faces have been decomposed into triangles.…”

Section: Introductionmentioning

confidence: 99%

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Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Luca

Friesecke

2017

J Nonlinear Sci

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Abstract. We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem [16], which concerns a system of N identical atoms in two dimensions interacting via the idealized pair potential V (r) = +∞ if r < 1, −1 if r = 1, 0 if r > 1. This is done by endowing the bond graph of a general particle configuration with a suitable notion of discrete curvature, and appealing to a discrete GaussBonnet theorem [18] which, as its continuous cousins, relates the sum/integral of the curvature to topological invariants. This leads to an exact geometric decomposition of the Heitmann-Radin energy into (i) a combinatorial bulk term, (ii) a combinatorial perimeter, (iii) a multiple of the Euler characteristic, and (iv) a natural topological energy contribution due to defects. An analogous exact geometric decomposition is also established for soft potentials such as the Lennard-Jones potential V (r) = r −6 − 2r −12 , where two additional contributions arise, (v) elastic energy and (vi) energy due to non-bonded interactions.

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Section: Introductionmentioning

confidence: 94%

Section: 4mentioning

confidence: 99%

Section: Discrete Differential Geometry Of Particle Configurationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Luca

Friesecke

2017

J Nonlinear Sci

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“…is called the discrete curvature on the graph G (Knill 2012). The definition of discrete curvature is relevant for planar graphs only.…”

Section: Gastrulation: Discrete Curvaturementioning

confidence: 99%

Topological Invariance of Biological Development

2013

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A topological inevitability of early developmental events through the use of classical topological concepts is discussed. Topological dynamics of forms and maps in embryo development are presented. Forms of a developing organism such as cell sets and closed surfaces are topological objects. Maps (or mathematical functions) are additional topological constructions in these objects and include polarization, singularities and curvature. Topological visualization allows us to analyze relationships that link local morphogenetic processes and integral developmental structures and also to find stable spatio-temporal topological characteristics that are invariant for a taxonomic group. The application of topological principles reveals a topological imperative: certain topological rules define and direct embryogenesis. A topological stability of embryonic morphogenesis is proposed and a topological scenario of developmental and evolutionary transformations is presented.

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Other Network Dimensions

Rosenberg

2020

Fractal Dimensions of Networks

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A discrete Gauss-Bonnet type theorem

Abstract: Oliver Knill hat an der ETH Zürich in Mathematik promoviert. Seit zehn Jahren wirkt er an der Harvard Universität im Preceptor Team, das dort für das Calculus Programm verantwortlich ist.In memory of Ernst Specker.

Cited by 11 publications

References 25 publications

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Topological Invariance of Biological Development

Other Network Dimensions

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