1,748 research outputs found
On equilibrium shapes of charged flat drops
Equilibrium shapes of two-dimensional charged, perfectly conducting liquid
drops are governed by a geometric variational problem that involves a perimeter
term modeling line tension and a capacitary term modeling Coulombic repulsion.
Here we give a complete explicit solution to this variational problem. Namely,
we show that at fixed total charge a ball of a particular radius is the unique
global minimizer among all sufficiently regular sets in the plane. For sets
whose area is also fixed, we show that balls are the only minimizers if the
charge is less than or equal to a critical charge, while for larger charge
minimizers do not exist. Analogous results hold for drops whose potential,
rather than charge, is fixed
Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web
We study the topology of the Megaparsec Cosmic Web in terms of the
scale-dependent Betti numbers, which formalize the topological information
content of the cosmic mass distribution. While the Betti numbers do not fully
quantify topology, they extend the information beyond conventional cosmological
studies of topology in terms of genus and Euler characteristic. The richer
information content of Betti numbers goes along the availability of fast
algorithms to compute them.
For continuous density fields, we determine the scale-dependence of Betti
numbers by invoking the cosmologically familiar filtration of sublevel or
superlevel sets defined by density thresholds. For the discrete galaxy
distribution, however, the analysis is based on the alpha shapes of the
particles. These simplicial complexes constitute an ordered sequence of nested
subsets of the Delaunay tessellation, a filtration defined by the scale
parameter, . As they are homotopy equivalent to the sublevel sets of
the distance field, they are an excellent tool for assessing the topological
structure of a discrete point distribution. In order to develop an intuitive
understanding for the behavior of Betti numbers as a function of , and
their relation to the morphological patterns in the Cosmic Web, we first study
them within the context of simple heuristic Voronoi clustering models.
Subsequently, we address the topology of structures emerging in the standard
LCDM scenario and in cosmological scenarios with alternative dark energy
content. The evolution and scale-dependence of the Betti numbers is shown to
reflect the hierarchical evolution of the Cosmic Web and yields a promising
measure of cosmological parameters. We also discuss the expected Betti numbers
as a function of the density threshold for superlevel sets of a Gaussian random
field.Comment: 42 pages, 14 figure
Microscopic Study of Slablike and Rodlike Nuclei: Quantum Molecular Dynamics Approach
Structure of cold dense matter at subnuclear densities is investigated by
quantum molecular dynamics (QMD) simulations. We succeeded in showing that the
phases with slab-like and rod-like nuclei etc. can be formed dynamically from
hot uniform nuclear matter without any assumptions on nuclear shape. We also
observe intermediate phases, which has complicated nuclear shapes. Geometrical
structures of matter are analyzed with Minkowski functionals, and it is found
out that intermediate phases can be characterized as ones with negative Euler
characteristic. Our result suggests the existence of these kinds of phases in
addition to the simple ``pasta'' phases in neutron star crusts.Comment: 6 pages, 4 figures, RevTex4; to be published in Phys. Rev. C Rapid
Communication (accepted version
Mixed-integer convex representability
Motivated by recent advances in solution methods for mixed-integer convex
optimization (MICP), we study the fundamental and open question of which sets
can be represented exactly as feasible regions of MICP problems. We establish
several results in this direction, including the first complete
characterization for the mixed-binary case and a simple necessary condition for
the general case. We use the latter to derive the first non-representability
results for various non-convex sets such as the set of rank-1 matrices and the
set of prime numbers. Finally, in correspondence with the seminal work on
mixed-integer linear representability by Jeroslow and Lowe, we study the
representability question under rationality assumptions. Under these
rationality assumptions, we establish that representable sets obey strong
regularity properties such as periodicity, and we provide a complete
characterization of representable subsets of the natural numbers and of
representable compact sets. Interestingly, in the case of subsets of natural
numbers, our results provide a clear separation between the mathematical
modeling power of mixed-integer linear and mixed-integer convex optimization.
In the case of compact sets, our results imply that using unbounded integer
variables is necessary only for modeling unbounded sets
A unified flow approach to smooth, even -Minkowski problems
We study long-time existence and asymptotic behaviour for a class of
anisotropic, expanding curvature flows. For this we adapt new curvature
estimates, which were developed by Guan, Ren and Wang to treat some stationary
prescribed curvature problems. As an application we give a unified flow
approach to the existence of smooth, even -Minkowski problems in
for Comment: 21 pages. Comments are welcom
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