Uniform asymptotic expansions of double integrals

Handelsman, Richard A.

doi:10.1016/0022-247x(69)90060-2

Cited by 263 publications

(432 citation statements)

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“…These procedures have to be implemented carefully because of the Stokes phenomenon [6,8] in the asymptotic expansion of the Hankel functions with complex argument. However, as long as the latter condition is satisfied, these estimates are valid.…”

Section: A Zero Deformation-case Of the Rotationally Symmetric Dielementioning

confidence: 99%

Modes of wave-chaotic dielectric resonators

et al. 2005

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Dielectric optical micro-resonators and micro-lasers represent a realization of a wave-chaotic system, where the lack of symmetry in the resonator shape leads to non-integrable ray dynamics. Modes of such resonators display a rich spatial structure, and cannot be classified through mode indices which would require additional constants of motion in the ray dynamics. Understanding and controlling the emission properties of such resonators requires the investigation of the correspondence between classical phase space structures of the ray motion inside the resonator and resonant solutions of the wave equations. We first discuss the breakdown of the conventional eikonal approximation in the short wavelength limit, and motivate the use of phase-space ray tracing and phase space distributions. Next, we introduce an efficient numerical method to calculate the quasi-bound modes of dielectric resonators, which requires only two diagonalizations per N states, where N is approximately equal to the number of half-wavelengths along the perimeter. The relationship between classical phase space structures and modes is displayed via the Husimi projection technique. Observables related to the emission pattern of the resonator are calculated with high efficiency.

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Section: A Zero Deformation-case Of the Rotationally Symmetric Dielementioning

confidence: 99%

Modes of wave-chaotic dielectric resonators

et al. 2005

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“…Our chief result is that a sufficiently smooth real-valued function on R d that dies off at infinity sufficiently rapidly can be written in the form (2). The class of functions for which the theorem is proved depends on the parity of d, as does the choice of w.…”

Section: The Representation Theoremmentioning

confidence: 98%

Integral combinations of Heavisides

Kainen

Kůrková

Vogt

2010

Mathematische Nachrichten

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Abstract:A sufficiently smooth function of d variables that decays fast enough at infinity can be represented pointwise by an integral combination of Heaviside plane waves (i.e., characteristic functions of closed half-spaces). The weight function in such a representation depends on the derivatives of the represented function. The representation is proved here by elementary techniques with separate arguments for even and odd d, and unifies and extends various results in the literature. An outline of the paper follows. Section 2 reviews neural networks and establishes notation, Section 3 discusses Green's functions and Green's second identity, and Section 4 describes functions of controlled decay and states our main theorem. We consider the 1-dimensional case in Section 5, provide necessary lemmas in Section 6, and prove the main theorem in Section 7. Section 8 has extensions and refinements of our representation and its relation to known results. The paper ends with a brief discussion. Keywords Feedforward neural networksFeedforward neural networks compute functions determined by the type of units and their interconnections. Each computational unit depends on two vector variables (an input and a parameter ), and is given by a function φ : R p × R d → R, where p and d are the dimensions of the parameter and input space respectively and R denotes the set of real numbers.One-hidden-layer networks, with hidden units based on a fixed function φ and a single linear output unit, yield functions f :where n is the number of hidden units, and w i ∈ R are the output weights and a i ∈ R p the input parameters of the i-th unit for i = 1, ..., n.A perceptron is a computational unit based on a function of the form φ ((v, b)

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“…Since we are interested in large vortices, we use the stationary phase approximation [18] on the angular integral. There are two stationary phase points at ϕ = 0, ϕ = π which give an oscillatory contribution.…”

Section: Asymptotic Solutionsmentioning

confidence: 99%