7,992 research outputs found
Constructing a polynomial whose nodal set is the three-twist knot
We describe a procedure that creates an explicit complex-valued polynomial
function of three-dimensional space, whose nodal lines are the three-twist knot
. The construction generalizes a similar approach for lemniscate knots: a
braid representation is engineered from finite Fourier series and then
considered as the nodal set of a certain complex polynomial which depends on an
additional parameter. For sufficiently small values of this parameter, the
nodal lines form the three-twist knot. Further mathematical properties of this
map are explored, including the relationship of the phase critical points with
the Morse-Novikov number, which is nonzero as this knot is not fibred. We also
find analogous functions for other knots with six crossings. The particular
function we find, and the general procedure, should be useful for designing
knotted fields of particular knot types in various physical systems.Comment: 19 pages, 6 figure
Limits to superweak amplification of beam shifts
The magnitudes of beam shifts (Goos-H\"anchen and Imbert-Fedorov, spatial and
angular) are greatly enhanced when a reflected light beam is postselected by an
analyzer, by analogy with superweak measurements in quantum theory.
Particularly strong enhancements can be expected close to angles at which no
light is transmitted for a fixed initial and final polarizations. We derive a
formula for the angular and spatial shifts at such angles (which includes the
Brewster angle), and we show that their maximum size is limited by higher-order
terms from the reflection coefficients occurring in the Artmann shift formula.Comment: 3 pages, 2 figures, Optics Letters styl
Topological aberration of optical vortex beams and singularimetry of dielectric interfaces
The splitting of a high-order optical vortex into a constellation of unit
vortices, upon total reflection, is described and analyzed. The vortex
constellation generalizes, in a local sense, the familiar longitudinal
Goos-H\"anchen and transverse Imbert-Federov shifts of the centroid of a
reflected optical beam. The centroid shift is related to the centre of the
constellation, whose geometry otherwise depends on higher-order terms in an
expansion of the reflection matrix. We present an approximation of the field
around the constellation of increasing order as an Appell sequence of complex
polynomials whose roots are the vortices, and explain the results by an analogy
with the theory of optical aberration.Comment: 5 pages, 3 figures, REVTeX 4.
Propagation-invariant beams with quantum pendulum spectra: from Bessel beams to Gaussian beam-beams
We describe a new class of propagation-invariant light beams with Fourier
transform given by an eigenfunction of the quantum mechanical pendulum. These
beams, whose spectra (restricted to a circle) are doubly-periodic Mathieu
functions in azimuth, depend on a field strength parameter. When the parameter
is zero, pendulum beams are Bessel beams, and as the parameter approaches
infinity, they resemble transversely propagating one-dimensional Gaussian
wavepackets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an
operator which interpolates between the squared angular momentum operator and
the linear momentum operator. The analysis reveals connections with Mathieu
beams, and insight into the paraxial approximation.Comment: 4 pages, 3 figures, Optics Letters styl
Analysis and design of planar and non-planar wings for induced drag minimization
Improvements in the aerodynamic efficiency of commercial transport aircraft will reduce fuel usage with subsequent reduced cost, both monetary and environmental. To this end, the current research is aimed at reducing the overall drag of these aircraft with specific emphasis on reducing the drag generated by the lifting surfaces. The ultimate goal of this program is to create a wing design methodology which optimizes the geometry of the wing for lowest total drag within the constraints of a particular design specification. The components of drag which must be considered include profile drag, and wave drag. Profile drag is dependent upon, among other things, the airfoil section and the total wetted area. Induced drag, which is manifested as energy left in the wake by the trailing vortex system is mostly a function of wing span, but also depends on other geometric wing parameters. Wave drag of the wing, important in the transonic flight regime, is largely affected by the airfoil section, wing sweep, and so forth. The optimization problem is that of assessing the various parameters which contribute to the different components of wing drag, and determining the wing geometry which generates the best overall performance for a given aircraft mission. The primary thrust of the research effort to date was in the study of induced drag. Results from the study are presented
Assessing Information Bias and Food Safety
Imperfect information can lead to market failure and be an external factor impacting managers of agribusiness firms. A matrix method approach to content analysis was conducted by independent judges based upon established typologies. Food safety articles from consumer publications were examined, and information received by consumers was found to be biased.food safety, information bias, consumers, media, Food Consumption/Nutrition/Food Safety, Marketing, Q10, Q13, Q16,
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