6,555 research outputs found
Relaxed Disk Packing
Motivated by biological questions, we study configurations of equal-sized
disks in the Euclidean plane that neither pack nor cover. Measuring the quality
by the probability that a random point lies in exactly one disk, we show that
the regular hexagonal grid gives the maximum among lattice configurations.Comment: 8 pages => 5 pages of main text plus 3 pages in appendix. Submitted
to CCCG 201
Relaxed disk packing
Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations
Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy
The freezing mechanism suggested for a fluid composed of hard disks [Huerta
et al., Phys. Rev. E, 2006, 74, 061106] is used here to probe the
fluid-to-solid transition in a hard-dumbbell fluid composed of overlapping hard
disks with a variable length between disk centers. Analyzing the trends in the
shape of second maximum of the radial distribution function of the planar
hard-dumbbell fluid it has been found that the type of transition could be
sensitive to the length of hard-dumbbell molecules. From the Monte
Carlo simulations data we show that if a hard-dumbbell length does not exceed
15% of the disk diameter, the fluid-to-solid transition scenario follows the
case of a hard-disk fluid, i.e., the isotropic hard-dumbbell fluid experiences
freezing. However, for a hard-dumbbell length larger than 15% of disk diameter,
there is evidence that fluid-to-solid transition may change to continuous
transition, i.e., such an isotropic hard-dumbbell fluid will avoid freezing.Comment: 9 pages, 7 figure
Relaxed spanners for directed disk graphs
Let be a finite metric space, where is a set of points
and is a distance function defined for these points. Assume that
has a constant doubling dimension and assume that each point
has a disk of radius around it. The disk graph that corresponds
to and is a \emph{directed} graph , whose vertices are
the points of and whose edge set includes a directed edge from to
if . In \cite{PeRo08} we presented an algorithm for
constructing a (1+\eps)-spanner of size O(n/\eps^d \log M), where is
the maximal radius . The current paper presents two results. The first
shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for
metrics of constant doubling dimension it is not possible to guarantee a
spanner whose size is independent of . The second result shows that by
slightly relaxing the requirements and allowing a small perturbation of the
radius assignment, considerably better spanners can be constructed. In
particular, we show that if it is allowed to use edges of the disk graph
I(V,E,r_{1+\eps}), where r_{1+\eps}(p) = (1+\eps)\cdot r(p) for every , then it is possible to get a (1+\eps)-spanner of size O(n/\eps^d) for
. Our algorithm is simple and can be implemented efficiently
Towards frustration of freezing transition in a binary hard-disk mixture
The freezing mechanism, recently suggested for a monodisperse hard-disk fluid
[Huerta et al., Phys. Rev. E, 2006, 74, 061106] is extended here to an
equimolar binary hard-disk mixtures. We are showing that for diameter ratios,
smaller than 1.15 the global orientational order parameter of the binary
mixture behaves like in the case of a monodisperse fluid. Namely, by increasing
the disk number density there is a tendency to form a crystalline-like phase.
However, for diameter ratios larger than 1.15 the binary mixtures behave like a
disordered fluid. We use some of the structural and thermodynamic properties to
compare and discuss the behavior as a function of diameter ratio and packing
fraction.Comment: 9 pages, 4 figure
Binary mixture of hard disks as a model glass former: Caging and uncaging
I have proposed a measure for the cage effect in glass forming systems. A
binary mixture of hard disks is numerically studied as a model glass former. A
network is constructed on the basis of the colliding pairs of disks. A rigidity
matrix is formed from the isostatic (rigid) sub--network, corresponding to a
cage. The determinant of the matrix changes its sign when an uncaging event
occurs. Time evolution of the number of the uncaging events is determined
numerically. I have found that there is a gap in the uncaging timescales
between the cages involving different numbers of disks. Caging of one disk by
two neighboring disks sustains for a longer time as compared with other cages
involving more than one disk. This gap causes two--step relaxation of this
system
Phase Changes in an Inelastic Hard Disk System with a Heat Bath under Weak Gravity for Granular Fluidization
We performed numerical simulations on a two-dimensional inelastic hard disk
system under gravity with a heat bath to study the dynamics of granular
fluidization. Upon increasing the temperature of the heat bath, we found that
three phases, namely, the condensed phase, locally fluidized phase, and
granular turbulent phase, can be distinguished using the maximum packing
fraction and the excitation ratio, or the ratio of the kinetic energy to the
potential energy.It is shown that the system behavior in each phase is very
different from that of an ordinary vibrating bed.Comment: 4 pages, including 5 figure
Electrical networks and Stephenson's conjecture
In this paper, we consider a planar annulus, i.e., a bounded, two-connected,
Jordan domain, endowed with a sequence of triangulations exhausting it. We then
construct a corresponding sequence of maps which converge uniformly on compact
subsets of the domain, to a conformal homeomorphism onto the interior of a
Euclidean annulus bounded by two concentric circles. As an application, we will
affirm a conjecture raised by Ken Stephenson in the 90's which predicts that
the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome
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