Green’s function and image system for the Laplace operator in the prolate spheroidal geometry

Xue, Changfeng; Deng, Shuiquan

doi:10.1063/1.4974156

Cited by 12 publications

(18 citation statements)

References 16 publications

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“…By reasoning as we did in the case of Dirichlet-Green's function [11], the vanishing of the = 0 term in (s) implies that the total strength (or in electrostatics, the total surface "charge") on k is zero, and the vanishing of the = 1 terms in (s) implies that the distribution on k is symmetric with its centroid at the origin. Such an image system is graphically illustrated in Figure 2.…”

Section: Constant Nonzero Boundary Condition the Interiormentioning

confidence: 90%

“…There are all types of images, from isolated point images to continuous distributions of images on lines, curves, and surfaces to combinations of these images. While image theories for DirichletGreen's functions have been studied quite extensively in the literature [3][4][5][6][7][8][9][10][11], much less work has been done with image theories for Neumann-Green's functions.…”

Section: Introductionmentioning

confidence: 99%

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Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

Xue

Edmiston

Deng

2018

Advances in Mathematical Physics

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Interior and exterior Neumann functions for the Laplace operator are derived in terms of prolate spheroidal harmonics with the homogeneous, constant, and nonconstant inhomogeneous boundary conditions. For the interior Neumann functions, an image system is developed to consist of a point image, a line image extending from the point image to infinity along the radial coordinate curve, and a symmetric surface image on the confocal prolate spheroid that passes through the point image. On the other hand, for the exterior Neumann functions, an image system is developed to consist of a point image, a focal line image of uniform density, another line image extending from the point image to the focal line along the radial coordinate curve, and also a symmetric surface image on the confocal prolate spheroid that passes through the point image.

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Section: Constant Nonzero Boundary Condition the Interiormentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

Xue

Edmiston

Deng

2018

Advances in Mathematical Physics

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“…For all points on the surface of our spheroidal cell ξ stays constant and we denote it by ξ s = a / f . The coordinates of the release point in the vicinity of the cell pole, are given by: ( ξ b = max( b / f , 1), η b = min( b / f , 1), 0). The solution for the steady state cytoplasmic concentration is given as a sum of the inhomogeneous solution and homogeneous solutions of equation (3), with the coefficients chosen to satisfy the boundary condition [34], where, P n is the Legendre polynomial of order n , and Q n is the Legendre function of second kind of order n . The concentration profile along the polar axis of the cell is given by the integral, which we integrate numerically to obtain the results shown in Figures 4C and D.…”

Section: Cylindrical Cellmentioning

confidence: 99%

“…The solution for the steady state cytoplasmic concentration is given as a sum of the inhomogeneous solution and homogeneous solutions of equation (3), with the coefficients chosen to satisfy the boundary condition [34], where, P n is the Legendre polynomial of order n , and Q n is the Legendre function of second kind of order n .…”

Section: Cylindrical Cellmentioning

confidence: 99%

How to Assemble a Scale-Invariant Gradient

Datta

Ghosh

Kondev

2021

Preprint

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Intracellular protein gradients serve a variety of functions, such as the establishment of cell polarity and to provide positional information for gene expression in developing embryos. Given that cell size in a population can vary considerably, for the protein gradients to work properly they often have to be scaled to the size of the cell. Here we examine a model of protein gradient formation within a cell that relies on cytoplasmic diffusion and cortical transport of proteins toward a cell pole. We show that the shape of the protein gradient is determined solely by the cell geometry. Furthermore, we show that the length scale over which the protein concentration in the gradient varies is determined by the linear dimensions of the cell, independent of the diffusion constant or the transport speed. This gradient provides scale-invariant positional information within a cell, which can be used for assembly of intracellular structures whose size is scaled to the linear dimensions of the cell, as was recently reported for the cytokinetic ring and actin cables in budding yeast cells.

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“…For the non-rotational ellipsoid, the first proposed image solution consisted of a point charge plus a surface charge on an interior ellipsoid [44]. This image formulation was then presented for the particular case of the prolate spheroid in [45]. But the spheroidal surface charge image cloaks the true form of the image lying inside; the potential is actually finite at all but one point of the surface image, so it is theoretically possible to analytically continue the potential within this boundary.…”

Section: Conducting Spheroidmentioning

confidence: 99%

Electrostatic images and T-matrices for simple geometries

Majic¹

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<p>This thesis is concerned with electrostatic boundary problems and how their solutions behave depending on the chosen basis of harmonic functions and the location of the fundamental singularities of the potential. The first part deals with the method of images for simple geometries where the exact nature of the image/fundamental singularity is unknown; essentially a study of analytic continuation for Laplace's equation in 3 dimensions. For the sphere, spheroid and cylinder, new deductions are made on the location of the images of point charges and their linear or surface charge densities, by using different harmonic series solutions that reveal the image. The second part looks for analytic expressions for the T-matrix for electromagnetic scattering of simple objects in the low frequency limit. In this formalism the incident and scattered fields are expanded on an orthogonal basis such as spherical harmonics, and the T-matrix is the transformation between the coefficients of these series, providing the general solution of any electromagnetic scattering problem by a given particle at a given wavelength. For the spheroid, bispherical system and torus, the natural basis of harmonic functions for the geometry of the scatterer are used to determine T-matrix expressed in that basis, which is then transformed onto a basis of canonical spherical harmonics via the linear relationships between different bases of harmonic functions.</p>

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Green’s function and image system for the Laplace operator in the prolate spheroidal geometry

Cited by 12 publications

References 16 publications

Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

How to Assemble a Scale-Invariant Gradient

Electrostatic images and T-matrices for simple geometries

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