907 research outputs found
A Superlinear Lower Bound on the Number of 5-Holes
Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position.
Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h_5(n) have been of order Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), obtaining the first superlinear lower bound on h_5(n).
The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted
The Minrank of Random Graphs
The minrank of a graph is the minimum rank of a matrix that can be
obtained from the adjacency matrix of by switching some ones to zeros
(i.e., deleting edges) and then setting all diagonal entries to one. This
quantity is closely related to the fundamental information-theoretic problems
of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and
distributed storage, and to Valiant's approach for proving superlinear circuit
lower bounds (Valiant, Boolean Function Complexity '92).
We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs
for all regimes of . In particular, for any constant ,
we show that with high probability,
where is chosen from . This bound gives a near quadratic
improvement over the previous best lower bound of (Haviv and
Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky
and Stav (FOCS '07). Our lower bound matches the well-known upper bound
obtained by the "clique covering" solution, and settles the linear index coding
problem for random graphs.
Finally, our result suggests a new avenue of attack, via derandomization, on
Valiant's approach for proving superlinear lower bounds for logarithmic-depth
semilinear circuits
On automorphism groups of Toeplitz subshifts
In this article we study automorphisms of Toeplitz subshifts. Such groups are
abelian and any finitely generated torsion subgroup is finite and cyclic. When
the complexity is non superlinear, we prove that the automorphism group is,
modulo a finite cyclic group, generated by a unique root of the shift. In the
subquadratic complexity case, we show that the automorphism group modulo the
torsion is generated by the roots of the shift map and that the result of the
non superlinear case is optimal. Namely, for any we construct
examples of minimal Toeplitz subshifts with complexity bounded by whose automorphism groups are not finitely generated. Finally,
we observe the coalescence and the automorphism group give no restriction on
the complexity since we provide a family of coalescent Toeplitz subshifts with
positive entropy such that their automorphism groups are arbitrary finitely
generated infinite abelian groups with cyclic torsion subgroup (eventually
restricted to powers of the shift)
Molecular Star Formation Rate Indicators in Galaxies
We derive a physical model for the observed relations between star formation
rate (SFR) and molecular line (CO and HCN) emission in galaxies, and show how
these observed relations are reflective of the underlying star formation law.
We do this by combining 3D non-LTE radiative transfer calculations with
hydrodynamic simulations of isolated disk galaxies and galaxy mergers. We
demonstrate that the observed SFR-molecular line relations are driven by the
relationship between molecular line emission and gas density, and anchored by
the index of the underlying Schmidt law controlling the SFR in the galaxy.
Lines with low critical densities (e.g. CO J=1-0) are typically thermalized and
trace the gas density faithfully. In these cases, the SFR will be related to
line luminosity with an index similar to the Schmidt law index. Lines with high
critical densities greater than the mean density of most of the emitting clouds
in a galaxy (e.g. CO J=3-2, HCN J=1-0) will have only a small amount of
thermalized gas, and consequently a superlinear relationship between molecular
line luminosity and mean gas density. This results in a SFR-line luminosity
index less than the Schmidt index for high critical density tracers. One
observational consequence of this is a significant redistribution of light from
the small pockets of dense, thermalized gas to diffuse gas along the line of
sight, and prodigious emission from subthermally excited gas. At the highest
star formation rates, the SFR-Lmol slope tends to the Schmidt index, regardless
of the molecular transition. The fundamental relation is the Kennicutt-Schmidt
law, rather than the relation between SFR and molecular line luminosity. We use
these results to make imminently testable predictions for the SFR-molecular
line relations of unobserved transitions.Comment: ApJ Accepted - Results remain same as previous version. Content
clarified with Referee's comment
Stochastic Coagulation and the Timescale for Runaway Growth
We study the stochastic coagulation equation using simplified models and
efficient Monte Carlo simulations. It is known that (i) runaway growth occurs
if the two-body coalescence kernel rises faster than linearly in the mass of
the heavier particle; and (ii) for such kernels, runaway is instantaneous in
the limit that the number of particles tends to infinity at fixed collision
time per particle. Superlinear kernels arise in astrophysical systems where
gravitational focusing is important, such as the coalescence of planetesimals
to form planets or of stars to form supermassive black holes. We find that the
time required for runaway decreases as a power of the logarithm of the the
initial number of particles. Astrophysical implications are briefly discussed.Comment: 16 pages, 4 figures, 1 appendi
Correlated electron-hole plasma in organometal perovskites
Organic-inorganic perovskites are a class of solution-processed semiconductors holding promise for the realization of low-cost efficient solar cells and on-chip lasers. Despite the recent attention they have attracted, fundamental aspects of the photophysics underlying device operation still remain elusive. Here we use photoluminescence and transmission spectroscopy to show that photoexcitations give rise to a conducting plasma of unbound but Coulomb-correlated electron-hole pairs at all excitations of interest for light-energy conversion and stimulated optical amplification. The conductive nature of the photoexcited plasma has crucial consequences for perovskite-based devices: in solar cells, it ensures efficient charge separation and ambipolar transport while, concerning lasing, it provides a low threshold for light amplification and justifies a favourable outlook for the demonstration of an electrically driven laser. We find a significant trap density, whose cross-section for carrier capture is however low, yielding a minor impact on device performance
Exact Distance Oracles for Planar Graphs
We present new and improved data structures that answer exact node-to-node
distance queries in planar graphs. Such data structures are also known as
distance oracles. For any directed planar graph on n nodes with non-negative
lengths we obtain the following:
* Given a desired space allocation , we show how to
construct in time a data structure of size that answers
distance queries in time per query.
As a consequence, we obtain an improvement over the fastest algorithm for
k-many distances in planar graphs whenever .
* We provide a linear-space exact distance oracle for planar graphs with
query time for any constant eps>0. This is the first such data
structure with provable sublinear query time.
* For edge lengths at least one, we provide an exact distance oracle of space
such that for any pair of nodes at distance D the query time is
. Comparable query performance had been observed
experimentally but has never been explained theoretically.
Our data structures are based on the following new tool: given a
non-self-crossing cycle C with nodes, we can preprocess G in
time to produce a data structure of size that can
answer the following queries in time: for a query node u, output
the distance from u to all the nodes of C. This data structure builds on and
extends a related data structure of Klein (SODA'05), which reports distances to
the boundary of a face, rather than a cycle.
The best distance oracles for planar graphs until the current work are due to
Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For
and space , we essentially improve the query
time from to .Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on
Discrete Algorithms, SODA 201
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