Spherical Harmonics having Polyhedral Symmetry

Poole, E. G. C.

doi:10.1112/plms/s2-33.1.435

Cited by 10 publications

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“…We present details for some that seem to be quite relevant to the experimentally obtained patterns presented in [4]. We mention that perhaps a more direct approach than (6.32) comes from the classical method in [34], restricted here to the null spaces . ℓ N We refer to [9] for details, which will also appear elsewhere.…”

Section: ∈ ℓ ℕmentioning

confidence: 96%

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

Healey¹,

Dharmavaram²

2017

SIAM J. Math. Anal.

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We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly underdetermined in-plane deformation. The resulting governing equations comprise a quasi-linear elliptic system with nonlinear constraints. We show the equivalence of the problem to that of finding the zeros of compact vector field. The latter is not routine. Using spectral and a-priori estimates together with Fredholm properties, we demonstrate that the principal part of the quasi-linear mapping defines an operator with compact resolvent. We then combine well known group-theoretic ideas for symmetry breaking with global bifurcation theory to obtain our results.

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Section: ∈ ℓ ℕmentioning

confidence: 96%

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

Healey¹,

Dharmavaram²

2017

SIAM J. Math. Anal.

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“…we can construct spherical harmonics that are invariant under the action of G. For axi-symmetric spherical harmonics, the obvious choice for directional derivatives is along the axis of symmetry. For polyhedral subgroups of O(3) the vectors may be chosen to lie along symmetry directions of the corresponding regular polyhedron [10], [11].…”

Section: Spherical Harmonics As Directional Derivativesmentioning

confidence: 99%

“…More systematically, the average of T over the subgroup G yields a projection onto a 1-dimensional subspace of R 2 +1 , any non-zero element of which serves as c. However, this requires assembling the representation T explicitly, which is an inconvenient if not formidable task for even moderate values of ∈ N. In this work we present a direct and elegant alternative to the right side of (14), based on an old observation of Maxwell, i.e., that any spherical harmonic can be realized as a sequence of directional derivatives acting on on the fundamental solution of Laplace's equation in R 3 , viz., 1/r, where r 2 = x 2 + y 2 + z 2 . It was first recognized by Poole [11], and later by Hodgkinson [7], that complete families of spherical harmonics invariant with respect to the symmetry of a Platonic solid can be constructed by such a procedure. Later Meyer 1672 SANJAY DHARMAVARAM AND TIMOTHY J. HEALEY [10] presented a rigorous, systematic approach to that construction accounting for all possible subgroups of O(3).…”

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confidence: 99%

Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry

Dharmavaram¹,

Healey²

2019

Discrete &Amp; Continuous Dynamical Systems - S

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We consider bifurcation problems in the presence of O(3) symmetry. Well known group-theoretic techniques enable the classification of all maximal isotropy subgroups of O(3), with associated mode numbers ∈ N, leading to 1-dimensional fixed-point subspaces of the (2 +1)-dimensional space of spherical harmonics. In each case the so-called equivariant branching lemma can then be used to establish the existence of a local branch of bifurcating solutions having the symmetry of the respective subgroup. To first-order, such a branch is a precise linear combination of the 2 + 1 spherical harmonics, which we call the bifurcation direction. Our work here is focused on the direct construction of these bifurcation directions, complementing the above-mentioned classification. The approach is an application of a general method for constructing families of symmetric spherical harmonics, based on differentiating the fundamental solution of Laplace's equation in R 3 .

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“…It was first considered in connection with solutions to the Laplace and the Helmholtz equation with polyhedral boundary conditions (Klein [46], Goursat [47], Pockels [48], Poole [49], Hodgkinson [50], Laporte [51]), in connection with the problem of term splitting when passing from spherical symmetry to another point symmetry (Bethe [52], see also the whole literature on ligand-field Unauthenticated Download Date | 5/11/18 1:52 PM theory), and in connection with the rotational-vibrational spectrum of the methane molecule (Ehlert [53], Jahn [54,55], Hecht [56], Moret-Bailly [57], Fox and Ozier [58]). Systematic group-theoretical studies for a larger number of point groups are due to Laporte [51], Meyer [59], Melvin [60], Altmann [61,62], Döring [63], Bradley and Cracknell [64], and Kurki-Suonio et al [65,66,43].…”

Section: Symmetry Adaptationmentioning

confidence: 99%

“…"Symmetry-adapted functions" [60] is too general, and we therefore use the precise and pictorial name "polyhedral (surface spherical) harmonics" = "polyedrische Kugelflächenfunktio-nen" [49][50][51]. They are functions on the surface of the unit sphere and possess the symmetry of a polyhedron.…”

Section: Symmetry Adaptationmentioning

confidence: 99%

Three-Dimensional Reciprocal Form Factors and Momentum Densities of Electrons from Compton Experiments I. Symmetry-Adapted Series Expansion of the Electron Momentum Density

Heuser-Hofmann

Weyrich

1985

Zeitschrift Für Naturforschung A

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All methods of reconstruction of the electron momentum density from Compton scattering data are based on a series expansion of this density in terms of symmetry-adapted surface spherical harmonics (polyhedral harmonics). Owing to the improvements of y-ray and X-ray Compton spectrometers during the last few years, measurements of Compton spectra of single crystals in a shorter time and with higher angular resolution have become feasible. Therefore, in addition to the current studies, an increased number of investigations of a greater variety of crystals with large sets of directional data will soon be performed. This work is at first concerned with the accurate and efficient computation of associated Legendre functions of the first kind, Pf, with high / and m and for the whole range of their real argument, which are needed for the description of data with high angular resolution. We further show that electron momentum densities obey Laue-class symmetries in the case of crystals and point-group symmetries with an inversion centre in general, in absence of external magnetic fields. All necessary information about the polyhedral harmonics belonging to the totally symmetric representations of these point groups is given, with particular emphasis on the group O h for which the hexoctahedral harmonics with I ^ 20 are displayed graphically. In addition, the locations of the extrema on the unit sphere are tabulated.

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Spherical Harmonics having Polyhedral Symmetry

Cited by 10 publications

References 0 publications

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry

Three-Dimensional Reciprocal Form Factors and Momentum Densities of Electrons from Compton Experiments I. Symmetry-Adapted Series Expansion of the Electron Momentum Density

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