4,641 research outputs found
A Survey of the Differential Geometry of Discrete Curves
Discretization of curves is an ancient topic. Even discretization of curves
with an eye toward differential geometry is over a century old. However there
is no general theory or methodology in the literature, despite the ubiquitous
use of discrete curves in mathematics and science. There are conflicting
definitions of even basic concepts such as discrete curvature {\kappa},
discrete torsion {\tau}, or discrete Frenet frame.Comment: 19 pages, 16 figure
Mixing in turbulent jets: scalar measures and isosurface geometry
Experiments have been conducted to investigate mixing and the geometry of scalar isosurfaces in turbulent jets. Specifically, we have obtained high-resolution, high-signal-to-noise-ratio images of the jet-fluid concentration in the far field of round, liquid-phase, turbulent jets, in the Reynolds number range 4.5 × 10^3 ≤ Re ≤ 18 × 10^3, using laser-induced-fluorescence imaging techniques. Analysis of these data indicates that this Reynolds-number range spans a mixing transition in the far field of turbulent jets. This is manifested in the probability-density function of the scalar field, as well as in measures of the scalar isosurfaces. Classical as well as fractal measures of these isosurfaces have been computed, from small to large spatial scales, and are found to be functions of both scalar threshold and Reynolds number. The coverage of level sets of jet-fluid concentration in the two-dimensional images is found to possess a scale-dependent-fractal dimension that increases continuously with increasing scale, from near unity, at the smallest scales, to 2, at the largest scales. The geometry of the scalar isosurfaces is, therefore, more complex than power-law fractal, exhibiting an increasing complexity with increasing scale. This behaviour necessitates a scale-dependent generalization of power-law-fractal geometry. A connection between scale-dependent-fractal geometry and the distribution of scales is established and used to compute the distribution of spatial scales in the flow
Work functions, ionization potentials, and in-between: Scaling relations based on the image charge model
We revisit a model in which the ionization energy of a metal particle is
associated with the work done by the image charge force in moving the electron
from infinity to a small cut-off distance just outside the surface. We show
that this model can be compactly, and productively, employed to study the size
dependence of electron removal energies over the range encompassing bulk
surfaces, finite clusters, and individual atoms. It accounts in a
straightforward manner for the empirically known correlation between the atomic
ionization potential (IP) and the metal work function (WF), IP/WF2. We
formulate simple expressions for the model parameters, requiring only a single
property (the atomic polarizability or the nearest neighbor distance) as input.
Without any additional adjustable parameters, the model yields both the IP and
the WF within 10% for all metallic elements, as well as matches the size
evolution of the ionization potentials of finite metal clusters for a large
fraction of the experimental data. The parametrization takes advantage of a
remarkably constant numerical correlation between the nearest-neighbor distance
in a crystal, the cube root of the atomic polarizability, and the image force
cutoff length. The paper also includes an analytical derivation of the relation
of the outer radius of a cluster of close-packed spheres to its geometric
structure.Comment: Original submission: 8 pages with 7 figures incorporated in the text.
Revised submission (added one more paragraph about alloy work functions): 18
double spaced pages + 8 separate figures. Accepted for publication in PR
Depletion interactions of non-spherical colloidal particles in polymer solutions
We consider anisotropic colloidal particles immersed in a solution of long,
flexible, and nonadsorbing polymers. For the dumbbell shapes of recently
synthesized particles consisting of two intersecting spheres and for
lens-shaped particles with spherical surfaces we calculate the isotropic and
anisotropic interaction parameters that determine the immersion free energy and
the orientation-dependent depletion interaction between particles that are
induced by the polymers. Exact results are obtained for random-walk like
(ideal) polymer chains
Enhancements to the GW space-time method
We describe the following new features which significantly enhance the power
of the recently developed real-space imaginary-time GW scheme (Rieger et al.,
Comp. Phys. Commun. 117, 211 (1999)) for the calculation of self-energies and
related quantities of solids: (i) to fit the smoothly decaying time/energy
tails of the dynamically screened Coulomb interaction and other quantities to
model functions, treating only the remaining time/energy region close to zero
numerically and performing the Fourier transformation from time to energy and
vice versa by a combination of analytic integration of the tails and
Gauss-Legendre quadrature of the remaining part and (ii) to accelerate the
convergence of the band sum in the calculation of the Green's function by
replacing higher unoccupied eigenstates by free electron states (plane waves).
These improvements make the calculation of larger systems (surfaces, clusters,
defects etc.) accessible.Comment: 10 pages, 6 figure
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