3,995 research outputs found
On Packing Densities of Set Partitions
We study packing densities for set partitions, which is a generalization of
packing words. We use results from the literature about packing densities for
permutations and words to provide packing densities for set partitions. These
results give us most of the packing densities for partitions of the set
. In the final section we determine the packing density of the set
partition .Comment: 12 pages, to appear in the Permutation Patterns edition of the
Australasian Journal of Combinatoric
Phase field approach to optimal packing problems and related Cheeger clusters
In a fixed domain of we study the asymptotic behaviour of optimal
clusters associated to -Cheeger constants and natural energies like the
sum or maximum: we prove that, as the parameter converges to the
"critical" value , optimal Cheeger clusters
converge to solutions of different packing problems for balls, depending on the
energy under consideration. As well, we propose an efficient phase field
approach based on a multiphase Gamma convergence result of Modica-Mortola type,
in order to compute -Cheeger constants, optimal clusters and, as a
consequence of the asymptotic result, optimal packings. Numerical experiments
are carried over in two and three space dimensions
Thermo-statistical description of gas mixtures from space partitions
The new mathematical framework based on the free energy of pure classical
fluids presented in [R. D. Rohrmann, Physica A 347, 221 (2005)] is extended to
multi-component systems to determine thermodynamic and structural properties of
chemically complex fluids. Presently, the theory focuses on -dimensional
mixtures in the low-density limit (packing factor ). The formalism
combines the free-energy minimization technique with space partitions that
assign an available volume to each particle. is related to the
closeness of the nearest neighbor and provides an useful tool to evaluate the
perturbations experimented by particles in a fluid. The theory shows a close
relationship between statistical geometry and statistical mechanics. New,
unconventional thermodynamic variables and mathematical identities are derived
as a result of the space division. Thermodynamic potentials ,
conjugate variable of the populations of particles class with the
nearest neighbors of class are defined and their relationships with the
usual chemical potentials are established. Systems of hard spheres are
treated as illustrative examples and their thermodynamics functions are derived
analytically. The low-density expressions obtained agree nicely with those of
scaled-particle theory and Percus-Yevick approximation. Several pair
distribution functions are introduced and evaluated. Analytical expressions are
also presented for hard spheres with attractive forces due to K\^ac-tails and
square-well potentials. Finally, we derive general chemical equilibrium
conditions.Comment: 14 pages, 8 figures. Accepted for publication in Physical Review
Sliced rotated sphere packing designs
Space-filling designs are popular choices for computer experiments. A sliced
design is a design that can be partitioned into several subdesigns. We propose
a new type of sliced space-filling design called sliced rotated sphere packing
designs. Their full designs and subdesigns are rotated sphere packing designs.
They are constructed by rescaling, rotating, translating and extracting the
points from a sliced lattice. We provide two fast algorithms to generate such
designs. Furthermore, we propose a strategy to use sliced rotated sphere
packing designs adaptively. Under this strategy, initial runs are uniformly
distributed in the design space, follow-up runs are added by incorporating
information gained from initial runs, and the combined design is space-filling
for any local region. Examples are given to illustrate its potential
application
A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results
We consider the multiphase shape optimization problem
where
is a given constant and is a bounded open set
with Lipschitz boundary. We give some new results concerning the qualitative
properties of the optimal sets and the regularity of the corresponding
eigenfunctions. We also provide numerical results for the optimal partitions
Extremal densities and measures on groups and -spaces and their combinatorial applications
This text contains lecture notes of the course taught to Ph.D. students of
Jagiellonian University in Krakow on 25-28 November, 2013.Comment: 18 page
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