scholarly journals Topology of the Nodal Set of Random Equivariant Spherical Harmonics on 𝕊3

2020 ◽  
Author(s):  
Junehyuk Jung ◽  
Steve Zelditch
Keyword(s):  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Nodal Set

Abstract We show that real and imaginary parts of equivariant spherical harmonics on ${{\mathbb{S}}}^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left (\frac{N^2 - m^2}{2} + N\right ) $. Hence, if $\frac{m}{N}= c $ for fixed $0 < c < 1$, then the genus has order $N^3$.

scholarly journals On some expressions for the radial and axial components of the magnetic force in the interior of solenoids of circular cross-section

1899 ◽  
Vol 64 (402-411) ◽  
pp. 192-202 ◽  
Keyword(s):  
Cross Section ◽  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Magnetic Force ◽  
Circular Cross Section ◽  
Total Force ◽  
Harmonic Method ◽  
Zonal Harmonics ◽  
Derivatives Of ◽  
Spherical Harmonic Method

In the present paper, certain expressions are arrived at, in terms of zonal spherical harmonics and their first derivatives, by which the values of the two components of the magnetic force may be calculated for any point in the interior of a coil, and hence the total force may be found both in magnitude and direction. The resulting series suffer from the well-known defect in the spherical harmonic method, in that they are not very rapidly converging for points near the boundary of the space for which they apply. A table of the values of the first derivatives of the first seven zonal harmonics is added.

Algorithm 1018: FaVeST—Fast Vector Spherical Harmonic Transforms

2021 ◽  
Vol 47 (4) ◽  
pp. 1-24
Author(s):  
Quoc T. Le Gia ◽  
Ming Li ◽  
Yu Guang Wang
Keyword(s):  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Fourier Coefficients ◽  
Computational Cost ◽  
Tangent Vector ◽  
Tangent Vector Field ◽  
Vector Spherical Harmonics ◽  
Numerical Examples ◽  
Vector Spherical Harmonic ◽  
Tangent Fields

Vector spherical harmonics on the unit sphere of ℝ 3 have broad applications in geophysics, quantum mechanics, and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this article, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to N log √ N for N number of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡ M for M evaluation points, has cost proportional to M log √ M . Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.

scholarly journals Efficient Compression of Far Field Matrices in Multipole Algorithms based on Spherical Harmonics and Radiating Modes

2012 ◽  
Vol 1 (2) ◽  
pp. 5
Author(s):  
A. Schroeder ◽  
H.-D. Bruens ◽  
C. Schuster
Keyword(s):  
Spherical Harmonics ◽  
Electromagnetic Scattering ◽  
Spherical Harmonic ◽  
Fast Multipole Method ◽  
Near Field ◽  
Surface Current ◽  
Far Field ◽  
Radiation Patterns ◽  
Electromagnetic Problems ◽  
Harmonic Representation

This paper proposes a compression of far field matrices in the fast multipole method and its multilevel extension for electromagnetic problems. The compression is based on a spherical harmonic representation of radiation patterns in conjunction with a radiating mode expression of the surface current. The method is applied to study near field effects and the far field of an antenna placed on a ship surface. Furthermore, the electromagnetic scattering of an electrically large plate is investigated. It is demonstrated, that the proposed technique leads to a significant memory saving, making multipole algorithms even more efficient without compromising the accuracy.

Analysis of radiative scattering for multiple sphere configurations

1991 ◽  
Vol 433 (1889) ◽  
pp. 599-614 ◽  
Keyword(s):  
Spherical Harmonics ◽  
Scattered Field ◽  
Spherical Harmonic ◽  
Recurrence Relations ◽  
Solution Technique ◽  
Addition Theorems ◽  
Vector Spherical Harmonics ◽  
Vector Spherical Harmonic ◽  
Arbitrary Configuration ◽  
Radiative Scattering

An analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented. The analysis builds upon the previously developed superposition solution, in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble. The addition theorems for vector spherical harmonics, which transform harmonics from one coordinate system into another, are rederived, and simple recurrence relations for the addition coefficients are developed. The relations allow for a very efficient implementation of the ‘order of scattering’ solution technique for determining the scattered field coefficients for each sphere.

scholarly journals Spherical harmonic analysis of cortical complexity in autism and dyslexia

2012 ◽  
Vol 3 (1) ◽  
Author(s):  
Emily Williams ◽  
Ayman El-Baz ◽  
Matthew Nitzken ◽  
Andrew Switala ◽  
Manuel Casanova
Keyword(s):  
Harmonic Analysis ◽  
Spherical Harmonics ◽  
Neuronal Migration ◽  
Spherical Harmonic ◽  
Spherical Harmonic Analysis ◽  
Extreme Phenotype ◽  
The Brain ◽  
Established Technique

AbstractAlterations in gyral form and complexity have been consistently noted in both autism and dyslexia. In this present study, we apply spherical harmonics, an established technique which we have exapted to estimate surface complexity of the brain, in order to identify abnormalities in gyrification between autistics, dyslexics, and controls. On the order of absolute surface complexity, autism exhibits the most extreme phenotype, controls occupy the intermediate ranges, and dyslexics exhibit lesser surface complexity. Here, we synthesize our findings which demarcate these three groups and review how factors controlling neocortical proliferation and neuronal migration may lead to these distinctive phenotypes.

scholarly journals An Innovative Slepian Approach to Invert GRACE KBRR for Localized Hydrological Information at the Sub-Basin Scale

2021 ◽  
Vol 13 (9) ◽  
pp. 1824
Author(s):  
Guillaume Ramillien ◽  
Lucía Seoane ◽  
José Darrozes
Keyword(s):  
River Basin ◽  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Water Mass ◽  
Function Analysis ◽  
Computational Time ◽  
Drainage Basins ◽  
Mass Fluxes ◽  
Basin Scale ◽  
Slepian Functions

GRACE spherical harmonics are well-adapted for representation of hydrological signals in river drainage basins of large size such as the Amazon or Mississippi basins. However, when one needs to study smaller drainage basins, one comes up against the low spatial resolution of the solutions in spherical harmonics. To overcome this limitation, we propose a new approach based on Slepian functions which can reduce the energy loss by integrating information in the spatial, spectral and time domains. Another advantage of these regionally-defined functions is the reduction of the problem dimensions compared to the spherical harmonic parameters. This also induces a drastic reduction of the computational time. These Slepian functions are used to invert the GRACE satellite data to restore the water mass fluxes of different hydro-climatologic environments in Africa. We apply them to two African drainage basins chosen for their size of medium scale and their geometric specificities: the Congo river basin with a quasi-isotropic shape and the Nile river basin with an anisotropic and more complex shape. Time series of Slepian coefficients have been estimated from real along-track GRACE geopotential differences for about ten years, and these coefficients are in agreement with both the spherical harmonic solutions provided by the official centers CSR, GFZ, JPL and the GLDAS model used for validation. The Slepian function analysis highlights the water mass variations at sub-basin scales in both basins.

Spherical Harmonics: A New Method for Estimating Maximum Reach Distance

1979 ◽  
Vol 23 (1) ◽  
pp. 196-200 ◽  
Author(s):  
L.E. Boydstun ◽  
T.J. Armstrong ◽  
F.L. Bookstein ◽  
D. Kessel
Keyword(s):  
Regression Analysis ◽  
Spherical Harmonics ◽  
Standard Error ◽  
Spherical Harmonic ◽  
Potential Model ◽  
Three Dimensional ◽  
New Method ◽  
Model Parameters ◽  
Data Smoothing ◽  
Residual Plots

Traditional methodology for estimating maximum reach distances has relied primarily on regression analysis or visual data smoothing. This paper evaluates alternative maximum reach estimation techniques including splines, periodic splines, and spherical harmonics. The results indicate that spherical harmonics are most suited for reach estimation due to the spherical nature of the data. A spherical harmonic maximum reach model is developed based on R2 and standard error values and residual plots. Three dimensional computer plots of the residuals provide a clear picture of potential model biases. Model parameters are obtained for 5th, 50th, and 95th percentiles for individuals and populations in the constrained-back seating posture.

scholarly journals Linearizations of the Spherical Harmonic Discrete Ordinate Method (SHDOM)

Atmosphere ◽  
2019 ◽  
Vol 10 (6) ◽  
pp. 292 ◽  
Author(s):  
Adrian Doicu ◽  
Dmitry S. Efremenko
Keyword(s):  
Radiative Transfer ◽  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Adaptive Grid ◽  
Test Problems ◽  
Discrete Ordinate Method ◽  
Discrete Ordinate ◽  
Adjoint Approach ◽  
Transfer Problems

Linearizations of the spherical harmonic discrete ordinate method (SHDOM) by means of a forward and a forward-adjoint approach are presented. Essentially, SHDOM is specialized for derivative calculations and radiative transfer problems involving the delta-M approximation, the TMS correction, and the adaptive grid splitting, while practical formulas for computing the derivatives in the spherical harmonics space are derived. The accuracies and efficiencies of the proposed methods are analyzed for several test problems.

scholarly journals ON GENERALIZED MONOPOLE SPHERICAL HARMONICS AND THE WAVE EQUATION OF A CHARGED MASSIVE KERR BLACK HOLE

2010 ◽  
Vol 25 (38) ◽  
pp. 3191-3200
Author(s):  
SHABNAM BEHESHTI ◽  
FLOYD L. WILLIAMS
Keyword(s):  
Black Hole ◽  
Wave Equation ◽  
Scalar Field ◽  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Kerr Black Hole ◽  
Complex Scalar ◽  
Linearly Independent ◽  
Complex Scalar Field

We find linearly independent solutions of the Goncharov–Firsova equation in the case of a massive complex scalar field on a Kerr black hole. The solutions generalize, in some sense, the classical monopole spherical harmonic solutions previously studied in the massless cases.

Sign in / Sign up

Export Citation Format

Share Document