26,005 research outputs found
On a problem of Erd\H{o}s and Rothschild on edges in triangles
Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by
H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which
is contained in at least one triangle, must contain an edge that is in at least
H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether
for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all
sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed
C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best
possible in terms of the range for C, as it is known that every N-vertex graph
with more than (N^2)/4 edges contains an edge that is in at least N/6
triangles.Comment: 8 page
High-resolution absorption spectroscopy of the circumgalactic medium of the Milky Way
In this article we discuss the importance of high-resolution absorption
spectroscopy for our understanding of the distribution and physical nature of
the gaseous circumgalactic medium (CGM) that surrounds the Milky Way.
Observational and theoretical studies indicate a high complexity of the gas
kinematics and an extreme multi-phase nature of the CGM in low-redshift
galaxies. High-precision absorption-line measurements of the Milky Way's gas
environment thus are essential to explore fundamental parameters of
circumgalactic gas in the local Universe, such as mass, chemical composition,
and spatial distribution. We shortly review important characteristics of the
Milky Way's CGM and discuss recent results from our multi-wavelength
observations of the Magellanic Stream. Finally, we discuss the potential of
studying the warm-hot phase of the Milky Way's CGM by searching for extremely
weak [FeX] l6374.5 and [FeIVX] l5302.9 absorption in optical QSO spectra.Comment: 7 pages, 4 figures; accepted for publication in Astronomical Notes
(paper version of a talk presented at the 10th Thinkshop, Potsdam, 2013
Numerical Description of Dilute Particle-Laden FLows by a Quadrature-Based Moment Method
The numerical simulation of gas-particle flows is divided into two families of methods. In Euler-Lagrange methods individual particle trajectories are computed, whereas in Euler-Euler methods particles are characterized by statistical descriptors. Lagrangian methods are very precise but their computational cost increases with instationarity and particle volume fraction. In Eulerian methods (also called moment methods) the particle-phase computational cost is comparable to that of the fluid phase but requires strong simplificaions. Existing Eulerian models consider unimodal or close-to-equilibrium particle velocity distributions and then fail when the actual distribution is far from equilibrium. Quadrature-based Eulerian methods introduce a new reconstruction of the velocity distribution, written as a sum of delta functions in phase space constrained to give the right values for selected low-order moments. Two of the quadrature-based Eulerian methods, differing by the reconstruction algorithm, are the focus of this work. Computational results for two academic cases (crossing jets, Taylor-Green flow) are compared to those of a Lagrangian method (considered as the reference solution) and of an existing second-order moment method. With the quadrature-based Eulerian methods, significant qualitative improvement is noticed compared to the second-order moment method in the two test cases
A Survey of O VI, C III, and H I in Highly Ionized High-Velocity Clouds
(ABRIDGED) We present a Far-Ultraviolet Spectroscopic Explorer survey of
highly ionized high-velocity clouds (HVCs) in 66 extragalactic sight lines. We
find a total of 63 high-velocity O VI absorbers, 16 with 21 cm-emitting H I
counterparts and 47 ``highly ionized'' absorbers without 21 cm emission. 11 of
these high-velocity O VI absorbers are positive-velocity wings (broad O VI
features extending asymmetrically to velocities of up to 300 km/s). The highly
ionized HVC population is characterized by =38+/-10 km/s and <log
N_a(O VI)>=13.83+/-0.36. We find that 81% (30/37) of high-velocity O VI
absorbers have clear accompanying C III absorption, and 76% (29/38) have
accompanying H I absorption in the Lyman series. The lower average width of the
high-velocity H I absorbers implies the H I lines arise in a separate, lower
temperature phase than the O VI. We find that the shape of the wing profiles is
well reproduced by a radiatively cooling, vertical outflow. However, the
outflow has to be patchy and out of ionization equilibrium. An alternative
model, consistent with the observations, is one where the highly ionized HVCs
represent the low N(H I) tail of the HVC population, with the O VI formed at
the interfaces around the embedded H I cores. Though we cannot rule out a Local
Group explanation, we favor a Galactic origin. This is based on the recent
evidence that both H I HVCs and the million-degree gas detected in X-ray
absorption are Galactic phenomena. Since the highly ionized HVCs appear to
trace the interface between these two Galactic phases, it follows that highly
ionized HVCs are Galactic themselves. However, the non-detection of
high-velocity O VI in halo star spectra implies that any Galactic high-velocity
O VI exists at z-distances beyond a few kpc.Comment: 36 pages, 14 figures (3 in color), accepted to ApJS. Some figures
downgraded to limit file siz
On two problems in graph Ramsey theory
We study two classical problems in graph Ramsey theory, that of determining
the Ramsey number of bounded-degree graphs and that of estimating the induced
Ramsey number for a graph with a given number of vertices.
The Ramsey number r(H) of a graph H is the least positive integer N such that
every two-coloring of the edges of the complete graph contains a
monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi
and Trotter states that there exists a constant c(\Delta) such that r(H) \leq
c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The
important open question is to determine the constant c(\Delta). The best
results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are
constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta
\log^2 \Delta}. We improve this upper bound, showing that there is a constant c
for which c(\Delta) \leq 2^{c \Delta \log \Delta}.
The induced Ramsey number r_{ind}(H) of a graph H is the least positive
integer N for which there exists a graph G on N vertices such that every
two-coloring of the edges of G contains an induced monochromatic copy of H.
Erd\H{o}s conjectured the existence of a constant c such that, for any graph H
on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this
conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an
earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in
the exponent.Comment: 18 page
Multiphase Plasma in Sub-Damped Lyman Alpha Systems: A Hidden Metal Reservoir
We present a VLT/UVES spectrum of a proximate sub-damped Lyman-alpha
(sub-DLA) system at z=2.65618 toward the quasar Q0331-4505
(z_qso=2.6785+/-0.0030). Absorption lines of O I, Si II, Si III, Si IV, C II, C
III, C IV, Fe II, Al II, and O VI are seen in the sub-DLA, which has a neutral
hydrogen column density log N(H I)=19.82+/-0.05. The absorber is at a velocity
of 1820+/-250 km/s from the quasar; however, its low metallicity
[O/H]=-1.64+/-0.07, lack of partial coverage, lack of temporal variations
between observations taken in 2003 and 2006, and non-detection of N V imply the
absorber is not a genuine intrinsic system. By measuring the O VI column
density and assuming equal metallicities in the neutral and ionized gas, we
determine the column density of hot ionized hydrogen in this sub-DLA, and in
two other sub-DLAs with O VI drawn from the literature. Coupling this with
determinations of the typical amount of warm ionized hydrogen in sub-DLAs, we
confirm that sub-DLAs are a more important metal reservoir than DLAs, in total
comprising at least 6-22% of the metal budget at z~2.5.Comment: 5 pages, 3 color figures, accepted for publication in ApJ
Bounds for graph regularity and removal lemmas
We show, for any positive integer k, that there exists a graph in which any
equitable partition of its vertices into k parts has at least ck^2/\log^* k
pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute
constants. This bound is tight up to the constant c and addresses a question of
Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma.
In order to gain some control over irregular pairs, another regularity lemma,
known as the strong regularity lemma, was developed by Alon, Fischer,
Krivelevich, and Szegedy. For this lemma, we prove a lower bound of
wowzer-type, which is one level higher in the Ackermann hierarchy than the
tower function, on the number of parts in the strong regularity lemma,
essentially matching the upper bound. On the other hand, for the induced graph
removal lemma, the standard application of the strong regularity lemma, we find
a different proof which yields a tower-type bound.
We also discuss bounds on several related regularity lemmas, including the
weak regularity lemma of Frieze and Kannan and the recently established regular
approximation theorem. In particular, we show that a weak partition with
approximation parameter \epsilon may require as many as
2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and
solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page
Structure of Pion Photoproduction Amplitudes
We derive and apply the finite energy sum rules to pion photoproduction. We
evaluate the low energy part of the sum rules using several state-of-the-art
models. We show how the differences in the low energy side of the sum rules
might originate from different quantum number assignments of baryon resonances.
We interpret the observed features in the low energy side of the sum rules with
the expectation from Regge theory. Finally, we present a model, in terms of a
Regge-pole expansion, that matches the sum rules and the high-energy
observables.Comment: 19 pages, 15 figures and 4 table
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