2,215 research outputs found
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media
This paper proposes to address the issue of complexity reduction for the
numerical simulation of multiscale media in a quasi-periodic setting. We
consider a stationary elliptic diffusion equation defined on a domain such
that is the union of cells and we
introduce a two-scale representation by identifying any function defined
on with a bi-variate function , where relates to the
index of the cell containing the point and relates to a local
coordinate in a reference cell . We introduce a weak formulation of the
problem in a broken Sobolev space using a discontinuous Galerkin
framework. The problem is then interpreted as a tensor-structured equation by
identifying with a tensor product space of
functions defined over the product set . Tensor numerical methods
are then used in order to exploit approximability properties of quasi-periodic
solutions by low-rank tensors.Comment: Changed the choice of test spaces V(D) and X (with regard to
regularity) and the argumentation thereof. Corrected proof of proposition 3.
Corrected wrong multiplicative factor in proposition 4 and its proof (was 2
instead of 1). Added remark 6 at the end of section 2. Extended remark 7.
Added references. Some minor improvements (typos, typesetting
Non-geometric Kaluza-Klein monopoles and magnetic duals of M-theory R-flux backgrounds
We introduce a magnetic analogue of the seven-dimensional nonassociative
octonionic R-flux algebra that describes the phase space of M2-branes in
four-dimensional locally non-geometric M-theory backgrounds. We show that these
two algebras are related by a Spin(7) automorphism of the 3-algebra that
provides a covariant description of the eight-dimensional M-theory phase space.
We argue that this algebra also underlies the phase space of electrons probing
a smeared magnetic monopole in quantum gravity by showing that upon appropriate
contractions, the algebra reduces to the noncommutative algebra of a spin foam
model of three-dimensional quantum gravity, or to the nonassociative algebra of
electrons in a background of uniform magnetic charge. We realise this set-up in
M-theory as M-waves probing a delocalised Kaluza-Klein monopole, and show that
this system also has a seven-dimensional phase space. We suggest that the
smeared Kaluza-Klein monopole is non-geometric because it cannot be described
by a local metric. This is the magnetic analogue of the local non-geometry of
the R-flux background and arises because the smeared Kaluza-Klein monopole is
described by a U(1)-gerbe rather than a U(1)-fibration.Comment: 19 pages, 2 figures; v2: dimensionful factors corrected throughout,
exposition improved; Final version to be published in JHE
A micromechanics-enhanced finite element formulation for modelling heterogeneous materials
In the analysis of composite materials with heterogeneous microstructures,
full resolution of the heterogeneities using classical numerical approaches can
be computationally prohibitive. This paper presents a micromechanics-enhanced
finite element formulation that accurately captures the mechanical behaviour of
heterogeneous materials in a computationally efficient manner. The strategy
exploits analytical solutions derived by Eshelby for ellipsoidal inclusions in
order to determine the mechanical perturbation fields as a result of the
underlying heterogeneities. Approximation functions for these perturbation
fields are then incorporated into a finite element formulation to augment those
of the macroscopic fields. A significant feature of this approach is that the
finite element mesh does not explicitly resolve the heterogeneities and that no
additional degrees of freedom are introduced. In this paper, hybrid-Trefftz
stress finite elements are utilised and performance of the proposed formulation
is demonstrated with numerical examples. The method is restricted here to
elastic particulate composites with ellipsoidal inclusions but it has been
designed to be extensible to a wider class of materials comprising arbitrary
shaped inclusions.Comment: 28 pages, 12 figures, 2 table
Fibre-like cylinders, their packings and coverings in space
In this paper we define the notion of infinite or bounded fibre-like geodesic
cylinder in space, develop a
method to determine its volume and total surface area. We prove that the common
part of the above congruent fibre-like cylinders with the base plane are
Euclidean circles and determine their radii.
Using the former classified infinite or bounded congruent regular prism
tilings with generating groups we introduce the notions of
cylinder packings, coverings and their densities. Moreover, we determine the
densest packing, the thinnest covering cylinder arrangements in
space, their densities, their
connections with the extremal hyperbolic circle arrangements and with the
extremal fibre-like cylinder arrangements in space
In our work we use the projective model of
introduced by E. Moln\'ar in
\cite{M97}.Comment: 23 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:1403.3192, arXiv:1304.054
Matter from Space
General Relativity offers the possibility to model attributes of matter, like
mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed
only with a single field that represents its Riemannian geometry. I review this
picture of `Geometrodynamics' and comment on various developments after
Einstein.Comment: 37 Pages, 17 figures. Based on a talk delivered at the conference
"Beyond Einstein: Historical Perspectives on Geometry, Gravitation, and
Cosmology in the Twentieth Century", September 2008 at the University of
Mainz in Germany. To appear in the Einstein-Studies Series, Birkhaeuser,
Boston. v2: Reference [7] added and typo in formula [42] correcte
Spectral curves and the mass of hyperbolic monopoles
The moduli spaces of hyperbolic monopoles are naturally fibred by the
monopole mass, and this leads to a nontrivial mass dependence of the
holomorphic data (spectral curves, rational maps, holomorphic spheres)
associated to hyperbolic multi-monopoles. In this paper, we obtain an explicit
description of this dependence for general hyperbolic monopoles of magnetic
charge two. In addition, we show how to compute the monopole mass of higher
charge spectral curves with tetrahedral and octahedral symmetries. Spectral
curves of euclidean monopoles are recovered from our results via an
infinite-mass limit.Comment: 43 pages, LaTeX, 3 figure
Symetric Monopoles
We discuss Bogomolny monopoles of arbitrary charge invariant
under various symmetry groups. The analysis is largely in terms of the spectral
curves, the rational maps, and the Nahm equations associated with monopoles. We
consider monopoles invariant under inversion in a plane, monopoles with cyclic
symmetry, and monopoles having the symmetry of a regular solid. We introduce
the notion of a strongly centred monopole and show that the space of such
monopoles is a geodesic submanifold of the monopole moduli space.
By solving Nahm's equations we prove the existence of a tetrahedrally
symmetric monopole of charge and an octahedrally symmetric monopole of
charge , and determine their spectral curves. Using the geodesic
approximation to analyse the scattering of monopoles with cyclic symmetry, we
discover a novel type of non-planar -monopole scattering process
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