Canonical Problems in Scattering and Potential theory Part II

Vinogradov, S. S.; Smith, P.D.; Vinogradova, E. D.

doi:10.1201/9780849387067

Cited by 60 publications

(56 citation statements)

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“…The narrow escape problem in three dimensions has been studied in the first paper of this series [1], where is was converted to a mixed DirichletNeumann boundary value problem for the Poisson equation in the domain. This is a well known problem of classical electrostatics (e.g., the electrified disk problem [2]), elasticity (punch problems), diffusion and conductance theory, hydrodynamics, and acoustics [3]- [7]. It dates back to Helmholtz [8] and Lord Rayleigh [9] and has been extensively studied in the literature for special geometries.…”

Section: Introductionmentioning

confidence: 99%

Narrow Escape, Part II: The Circular Disk

2006

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We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ω a , which is absorbing. As ε = |∂Ω a |/|∂Ω| decreases to zero the mean time to absorption in ∂Ω a , denoted Eτ , becomes infinite. The narrow escape problem is to find an asymptotic expansion of Eτ for ε ≪ 1. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general smooth domain, Eτ = |Ω| Dπ log 1 ε + O(1) , (D is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is E[τ | x(0) = 0] = R 2 D log 1 ε + log 2 + 1 4 + O(ε) . The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because log 1 ε is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into ∂Ω a at the endpoints of ∂Ω a , and find the value of the flux near the center of the window.

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Section: Introductionmentioning

confidence: 99%

Narrow Escape, Part II: The Circular Disk

2006

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“…At microwave frequencies the more popular choice for the design of efficient reflectors is a stepped-index Luneburg Lens (LL) with attached metallic spherical cap [2][3][4][5][6]. In this paper we consider the possibility of replacing the relatively-expensive-to-manufacture LL by the cheaper SL.…”

Section: Introductionmentioning

confidence: 99%

“…The calculation of the radar cross-section (RCS) for SLRs employs previously constructed algorithms which are valid for stepped-index LLRs [5,6]. When a spherical cap is attached to the surface of a SL, no dielectric shells exist outside of the core and we use the limiting case of algorithms developed in [5,6]. The rigorous approach, developed in [5] and [6], tackles the case of the normal incidence.…”

Section: Introductionmentioning

confidence: 99%

“…When a spherical cap is attached to the surface of a SL, no dielectric shells exist outside of the core and we use the limiting case of algorithms developed in [5,6]. The rigorous approach, developed in [5] and [6], tackles the case of the normal incidence. Recently, we proposed a highly efficient numerical method which enables us to treat the discussed problem for any incidence angle [14].…”

Section: Introductionmentioning

confidence: 99%

“…The objective of this paper is to demonstrate the competitiveness of SLRs against the LLRs. The calculation of the radar cross-section (RCS) for SLRs employs previously constructed algorithms which are valid for stepped-index LLRs [5,6]. When a spherical cap is attached to the surface of a SL, no dielectric shells exist outside of the core and we use the limiting case of algorithms developed in [5,6].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Radar Cross-Section Studies of Spherical Lens Reflectors

Vinogradov¹,

Smith²,

Kot³

et al. 2007

PIER

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Abstract-The reflectivity of a Spherical Lens Reflector is investigated. The scattering of an electromagnetic plane wave by a Spherical Lens Reflector is treated as a classical boundary value problem for Maxwell's equations. No restrictions are imposed on the electrical size of reflectors and the angular size of the metallic spherical cap. The competitiveness of the Spherical Lens Reflector against the Luneberg Lens Reflector is demonstrated. It has been found that Spherical Lens Reflectors with relative dielectric constant in the range 3.4 ≤ ε r ≤ 3.7 possess better spectral performance than 3-or 5-layer Luneberg Lens Reflectors in a wide frequency range.

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Method of analytical regularization in computational photonics

Nosich¹

2016

Radio Science

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We discuss the advantages of the conversion of electromagnetic field problems to the Fredholm second‐kind integral equations (analytical regularization) and Fredholm second‐kind infinite‐matrix equations (analytical preconditioning). Special attention is paid to specific features of the characterization of metals and dielectrics in the optical range and their effect on the problem formulation and on the methods applicable to the mentioned conversion.

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Canonical Problems in Scattering and Potential theory Part II

Cited by 60 publications

References 0 publications

Narrow Escape, Part II: The Circular Disk

Narrow Escape, Part II: The Circular Disk

Radar Cross-Section Studies of Spherical Lens Reflectors

Method of analytical regularization in computational photonics

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