1,720,981 research outputs found
Recommended from our members
Analysis of the internal electric fields of pristine ice crystals and aggregate snowflakes, and their effect on scattering
Full text linkThe discrete dipole approximation is used to explore the internal electric fields of plane-wave-illuminated ice particles, and analyse their differential scattering cross sections. The results are displayed for monocrystals and aggregates of size parameters x=2 and x=10. We show that the field is relatively uniform for x=2, but for monocrystals of x=10 there is a complex internal structure. For a hexagonal plate, this structure is a combination of two components: a "distorted" plane wave, with wavefronts aligned perpendicular to the incident wave close to the centre of the plate, and curved forward near the particle boundary; and a standing wave, internally reflected around the perimeter. The former is due to the transverse component of the field i.e., the component perpendicular to the incident wave, and the latter is due to the component parallel to the incident direction. Focussing of the field towards the forward side of the particle is observed. As the particle complexity is increased due to aggregation, the field becomes smoother and less focussing is seen. For complex aggregates, the individual monomers act independently of one another, suggesting simplified methods of calculating scattering from such particles. The influence of the internal fields on far-field scattering is explored. It is demonstrated that scattering in the forward and backward directions is dominated by the transverse component. The parallel component contributes to sidescattering, with its influence on total scattering decreasing with particle complexity. We propose that this is due to the inability of complex particles to maintain a standing wave, diminishing much of the sidescattering observed for monocrystals. Comparisons of the far-field scattering properties of complex aggregates using the discrete dipole and Rayleigh-Gans approximations are also presented for x=2 and x=10, along with results obtained using a soft sphere approximation
Numerical Quadrature for Singular Integrals on Fractals
Full text linkWe present and analyse numerical quadrature rules for evaluating regular and
singular integrals on self-similar fractal sets. The integration domain
is assumed to be the compact attractor of an iterated function
system of contracting similarities satisfying the open set condition.
Integration is with respect to any ``invariant'' (also known as ``balanced'' or
``self-similar'') measure supported on , including in particular the
Hausdorff measure restricted to , where is the
Hausdorff dimension of . Both single and double integrals are
considered. Our focus is on composite quadrature rules in which integrals over
are decomposed into sums of integrals over suitable partitions of
into self-similar subsets. For certain singular integrands of
logarithmic or algebraic type we show how in the context of such a partitioning
the invariance property of the measure can be exploited to express the singular
integral exactly in terms of regular integrals. For the evaluation of these
regular integrals we adopt a composite barycentre rule, which for sufficiently
regular integrands exhibits second-order convergence with respect to the
maximum diameter of the subsets. As an application we show how this approach,
combined with a singularity-subtraction technique, can be used to accurately
evaluate the singular double integrals that arise in Hausdorff-measure Galerkin
boundary element methods for acoustic wave scattering by fractal screens
SPURIOUS QUASI-RESONANCES IN BOUNDARY INTEGRAL EQUATIONS FOR THE HELMHOLTZ TRANSMISSION PROBLEM
Full text linkWe consider the Helmholtz transmission problem with piecewise-constant material coefficients and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur and propose modified boundary integral equations that are not affected by the
Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions
Full text linkWe consider the Helmholtz transmission problem with one penetrable
star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of
the wavenumbers, we prove bounds on the solution in terms of the data, with
these bounds explicit in all parameters. In particular, the (weighted)
norm of the solution is bounded by the norm of the source term,
independently of the wavenumber. These bounds then imply the existence of a
resonance-free strip beneath the real axis. The main novelty is that the only
comparable results currently in the literature are for smooth, convex obstacles
with strictly positive curvature, while here we assume only Lipschitz
regularity and star-shapedness with respect to a point. Furthermore, our bounds
are obtained using identities first introduced by Morawetz (essentially
integration by parts), whereas the existing bounds use the much-more
sophisticated technology of microlocal analysis and propagation of
singularities. We also recap existing results that show that if the assumption
on the wavenumbers is lifted, then no bound with polynomial dependence on the
wavenumber is possible.Comment: 26 pages, 2 figure
Density results for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets
No full textWe investigate two density questions for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set Ω ⊂ R^n, D(Ω) is dense in {u ∈ H^s(Rn) : supp u ⊂ Ω} whenever ∂Ω has zero Lebesgue measure and Ω is “thick” (in the sense of Triebel); and (ii) for a d-set Γ ⊂ R^n (0 < d < n), {u ∈ H^s1(R^n) : supp u ⊂ Γ} is dense in {u ∈ H^s2(R^n) : supp u ⊂ Γ} whenever −(n−d)/2−m−1 < s2 ≤ s1 < −(n−d)/2−m for some m ∈ N_0. For (ii), we provide concrete examples, for any m ∈ N_0, where density fails when s1 and s2
are on opposite sides of −(n−d)/2−m. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether {u ∈ H^s(R^n) : supp u ⊂ Γ} = {0} for a given closed set Γ ⊂ Rn and s ∈ R. They also both arise naturally in the study of boundary integral equation formulations of acoustic wave scattering by fractal screens
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Recommended from our members
Corrigendum to "interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples" [Mathematika 61 (2015), 414-443]
Full text linkSince we published the paper [4] in 2015, the quantitative results we derive therein, and the summary we provide of results in the literature on interpolation spaces, have been of use in our own work (for example, [5]) and elsewhere (for example, [13]). But the paper as published is marred by inaccuracies which we correct in this note, including the inaccuracy flagged in [13, p. 1768]. We use throughout the notations of [4]. As in [4], we intend primarily that vector space, Banach space, and Hilbert space should be read as their complex versions. But, except where we deal with complex interpolation, the results below apply equally in the real case, with minor changes to the statements and proofs
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
- …
