Introduction to the Theory of Fourier's Series and Integrals.

Carslaw, H. S.

doi:10.2307/2972810

Cited by 86 publications

(67 citation statements)

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“…Equations [27] and [28] indicate that the discontinuities in the force density at the two interfaces of the film are equal. This conclusion is valid despite the fact that the [24] adjoining phases have not been assumed to be identical.…”

Section: Pjo(z) = Pjo(z) + Pjw~o(z)mentioning

confidence: 99%

“…Hence, the equality of Eqs. [27] and [28] suggests that these localized [25] differences in the force density can be regarded as a characteristic of the entire film system. This difference is referred to here as the disjoining pressure (II(h)) of a plane parallel infinite film.…”

Section: Pjo(z) = Pjo(z) + Pjw~o(z)mentioning

confidence: 99%

“…then the boundedness of A is guaranteed from the '*tx test" for convergence (27 Spatial coordinate (see Fig. 3 …”

mentioning

confidence: 99%

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Stability of symmetric and unsymmetric thin liquid films to short and long wavelength perturbations

Maldarelli

Jain

Ivanov

et al. 1980

Journal of Colloid and Interface Science

144

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The stability of thin (< 100 nm) symmetrical and unsymmetrical membranes, assimilated with viscous liquids, to short and long wavelength perturbations is investigated. The asymmetry is due to the two different viscous phases surrounding the membrane and to the different interfacial tensions on the two faces of the membrane. The cell membrane is a case for which the present treatment is of significance. The dynamics of the membrane to perturbations is described by the Navier-Stokes equation modified with a body force which accounts for the fact that the range of the interaction forces is larger than the thickness of the film. The body force is computed assuming pair-wise additivity and accounting for the deformation of the interface produced by the perturbation. General dispersion equations are derived, and these equations describe the squeezing and stretching modes of perturbations. The growth coefficient is expressed as a function of the wavelength for various ratios of the viscosities of the two surrounding phases and various values of the two interfacial tensions. In the limiting cases of interfacial tension ratio equal to unity and wavelength large compared to the thickness of the film results of the previous investigators are obtained. For the symmetrical case expressions are derived for the critical and dominant wavelength of the squeezing and stretching modes. An application of the results to a cell membrane shows that the growth of the instability is dominated in this case by the stretching mode since the time scale of growth of the perturbations is four orders of magnitude less than that in the squeezing mode. For unsymmetrical systems the effect of differences between the interfacial tensions on the two faces on the ratio of the amplitudes of perturbations on the two faces is investigated. The results show in what manner differences in interfacial tension convey amplified or damped messages across the membrane.

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Section: Pjo(z) = Pjo(z) + Pjw~o(z)mentioning

confidence: 99%

Section: Pjo(z) = Pjo(z) + Pjw~o(z)mentioning

confidence: 99%

See 1 more Smart Citation

Stability of symmetric and unsymmetric thin liquid films to short and long wavelength perturbations

Maldarelli

Jain

Ivanov

et al. 1980

Journal of Colloid and Interface Science

144

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“…However, there exists the Gibbs Phenomenon (hearafter GP), which refers to the peculiar manner in which the Fourier representation behaves at a jump discontinuity (Gibbs 1898;Carslaw 1921). The Gibbs phenomenon disappear only in the continuum limit (i.e.…”

Section: Spherical Harmonic Transformation and Discontinuitiesmentioning

confidence: 99%

“…The Gibbs phenomenon disappear only in the continuum limit (i.e. infinite pixel resolution and summation up to an infinitely high multipole) (Gibbs 1898;Carslaw 1921;Allen & Mills 2004;Bankman 2000;Proakis & Manolakis 2006;Glassner 1995). Jump discontinuities due to foreground masking lead to the Gibbs phenomenon, making Eq.…”

Section: Spherical Harmonic Transformation and Discontinuitiesmentioning

confidence: 99%

How to make a clean separation between CMB E and B modes with proper foreground masking

Kim¹

2011

A&A

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We investigate the E/B decomposition of CMB polarization on a masked sky. In real space, operators of E and B mode decomposition involve only differentials of CMB polarization. We may, therefore in principle, perform a clean E/B decomposition from incomplete sky data. Since it is impractical to apply second derivatives to observation data, we usually rely on spherical harmonic transformation and inverse transformation, instead of using real-space operators. In spherical harmonic representation, jump discontinuities in a cut sky produces Gibbs phenomenon, unless a spherical harmonic expansion is made up to an infinitely high multipole. By smoothing a foreground mask, we may suppress the Gibbs phenomenon effectively in a similar manner to apodization of a foreground mask discussed in other works. However, we incur foreground contamination by smoothing a foreground mask, because zero-value pixels in the original mask may be rendered non-zero by the smoothing process. In this work, we investigate an optimal foreground mask, which ensures proper foreground masking and suppresses Gibbs phenomenon. We apply our method to a simulated map of the pixel resolution comparable to the Planck satellite. The simulation shows that the leakage power is lower than unlensed CMB B mode power spectrum of tensor-to-scalar ratio r ∼ 1 × 10 −7 . We compare the result with that of the original mask. We find that the leakage power is reduced by a factor of 10 6 ∼ 10 9 at the cost of a sky fraction 0.07, and that that the enhancement is highest at lowest multipoles. We confirm that all the zero-value pixels in the original mask remain zero in our mask. The application of this method to the Planck data will improve the detectability of primordial tensor perturbation.

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Non-standard Neutral Kaon Dynamics from Infinite Statistics

Benatti

Floreanini

1999

Annals of Physics

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The neutral kaon system can be effectively described by non-unitary, dissipative, completely positive dynamics that extend the usual treatment. In the framework of open quantum systems, we show how the origin of these non-standard time evolutions can be traced to the interaction of the kaon system with a large environment. We find that a heat bath of quanta obeying infnite statistics could constitute a realistic example of such an environment. 1999Academic Press

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Introduction to the Theory of Fourier's Series and Integrals.

Cited by 86 publications

References 0 publications

Stability of symmetric and unsymmetric thin liquid films to short and long wavelength perturbations

Stability of symmetric and unsymmetric thin liquid films to short and long wavelength perturbations

How to make a clean separation between CMB E and B modes with proper foreground masking

Non-standard Neutral Kaon Dynamics from Infinite Statistics

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