9,130 research outputs found
Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2
We describe a new approach to triple linking invariants and integrals, aiming
for a simpler, wider and more natural applicability to the search for higher
order helicities of fluid flows and magnetic fields. To each three-component
link in Euclidean 3-space, we associate a geometrically natural generalized
Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking
numbers and Milnor triple linking number that classify the link up to link
homotopy correspond to the Pontryagin invariants that classify its generalized
Gauss map up to homotopy. This can be viewed as a natural extension of the
familiar fact that the linking number of a two-component link in 3-space is the
degree of its associated Gauss map from the 2-torus to the 2-sphere. When the
pairwise linking numbers are all zero, we give an integral formula for the
triple linking number analogous to the Gauss integral for the pairwise linking
numbers, but patterned after J.H.C. Whitehead's integral formula for the Hopf
invariant. The integrand in this formula is geometrically natural in the sense
that it is invariant under orientation-preserving rigid motions of 3-space,
while the integral itself can be viewed as the helicity of a related vector
field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we
did this for three-component links in the 3-sphere. Komendarczyk has applied
this approach in special cases to derive a higher order helicity for magnetic
fields whose ordinary helicity is zero, and to obtain from this nonzero lower
bounds for the field energy.Comment: 22 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1101.337
A new method to compute quasi-local spin and other invariants on marginally trapped surfaces
We accurately compute the scalar 2-curvature, the Weyl scalars, associated
quasi-local spin, mass and higher multipole moments on marginally trapped
surfaces in numerical 3+1 simulations. To determine the quasi-local quantities
we introduce a new method which requires a set of invariant surface integrals,
allowing for surface grids of a few hundred points only. The new technique
circumvents solving the Killing equation and is also an alternative to
approximate Killing vector fields. We apply the method to a perturbed
non-axisymmetric black hole ringing down to Kerr and compare the quasi-local
spin with other methods that use Killing vector fields, coordinate vector
fields, quasinormal ringing and properties of the Kerr metric on the surface.
Interesting is the agreement with the spin of approximate Killing vector fields
during the phase of perturbed axisymmetry. Additionally, we introduce a new
coordinate transformation, adapting spherical coordinates to any two points on
the sphere like the two minima of the scalar 2-curvature on axisymmetric
trapped surfaces.Comment: 22 pages, 5 figure
A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces
In this paper, we study triply periodic surfaces with minimal surface area
under a constraint in the volume fraction of the regions (phases) that the
surface separates. Using a variational level set method formulation, we present
a theoretical characterization of and a numerical algorithm for computing these
surfaces. We use our theoretical and computational formulation to study the
optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume
fractions of the two phases are equal and explore the properties of optimal
structures when the volume fractions of the two phases not equal. Due to the
computational cost of the fully, three-dimensional shape optimization problem,
we implement our numerical simulations using a parallel level set method
software package.Comment: 28 pages, 16 figures, 3 table
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