Spherical Functions of Mathematical Geosciences

Two-dimensional crystals on curved manifolds exhibit nontrivial defect structures. Here we consider "active crystals" on a sphere, which are composed of self-propelled colloidal particles. Our work is based on a phase-field-crystal-type model that involves a density and a polarization field on the sphere. Depending on the strength of the self-propulsion, three different types of crystals are found: a static crystal, a self-spinning "vortex-vortex" crystal containing two vortical poles of the local velocity, and a self-translating "source-sink" crystal with a source pole where crystallization occurs and a sink pole where the active crystal melts. These different crystalline states as well as their defects are studied theoretically here and can in principle be confirmed in experiments.

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“…the spherical harmonics are dense in the function space L 2 (S) [37,38]. Therefore, the scalar field ψ can be represented as the series expansion…”

Section: A (Vector) Spherical Harmonics Expansionmentioning

confidence: 99%

Active crystals on a sphere

et al. 2018

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“…This is the reason why they do not show any space localization, but ideal frequency (momentum) localization (for a more detailed description see Freeden and Schreiner 2009). This property makes outer harmonics difficult to use for high resolution modeling at local scale.…”

Section: Tensorial Runge-walsh Approximation Propertymentioning

confidence: 95%

“…(for explicit representations see Freeden and Schreiner (2009)). Note that, for sufficiently smooth tensor fields f : → R 3 ⊗ R 3 of the form…”

Section: Tensor Spherical Harmonicsmentioning

confidence: 99%

“…Conventionally, the sequence { ∧ (n)} n∈N 0 is called the symbol of the scalar PDO . Tensorial PDO's can be defined in analogous way (see Freeden and Schreiner 2009): Let { ∧ (n)} n∈N 0 be a sequence of real numbers. The operator :…”

Section: Pseudodifferential Operatorsmentioning

confidence: 99%

“…For a more detailed introduction to the theory of scalar and tensor spherical harmonics including the development of associated addition theorems the reader is referred to Freeden and Schreiner (2009).…”

Section: Tensor Spherical Harmonicsmentioning

confidence: 99%

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In the context of inverse problems of mathematical geodesy, the calculation of the gravitational potential at the Earth's surface from its Hesse tensor at satellite's height turns out to be exponentially ill-posed. In fact, it requires specific tensorial procedures for its solution. This paper proposes a wavelet-based regularization method to overcome the calamities of the ill-posedness, thereby providing a "zooming-in" technique of modeling the gravitational potential from global to local scale. As a particularly remarkable ingredient the paper offers a new procedure of multiscale regularization by use of locally adapted regularization parameters.

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Three‐dimensional simulation of nonwoven fabrics using a greedy approximation of the distribution of fiber directions

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An elementary algorithm is used to simulate the industrial production of a fiber of a 3-dimensional nonwoven fabric. The algorithm simulates the fiber as a polyline where the direction of each segment is stochastically drawn based on a given probability density function (PDF) on the unit sphere. This PDF is obtained from data of directions of fiber fragments which originate from computer tomography scans of a real nonwoven fabric. However, the simulation algorithm requires numerous evaluations of the PDF. Since the established technique of a kernel density estimator leads to very high computational costs, a novel greedy algorithm for estimating a sparse representation of the PDF is introduced. Numerical tests for a synthetic and a real example are presented. In a realistic scenario, the introduced sparsity ansatz leads to a reduction of the computation time for 100 fibers from around 80 days to 2.5 hours

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Spherical Functions of Mathematical Geosciences

Cited by 182 publications

References 0 publications

Active crystals on a sphere

Active crystals on a sphere

Satellite gravity gradiometry as tensorial inverse problem

Three‐dimensional simulation of nonwoven fabrics using a greedy approximation of the distribution of fiber directions

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