1,146,017 research outputs found
Measured descent: A new embedding method for finite metrics
We devise a new embedding technique, which we call measured descent, based on
decomposing a metric space locally, at varying speeds, according to the density
of some probability measure. This provides a refined and unified framework for
the two primary methods of constructing Frechet embeddings for finite metrics,
due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space
(X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where
alpha_X is a geometric estimate on the decomposability of X. As an immediate
corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding,
where \lambda_X is the doubling constant of X. Since \lambda_X\le n, this
result recovers Bourgain's theorem, but when the metric X is, in a sense,
``low-dimensional,'' improved bounds are achieved.
Our embeddings are volume-respecting for subsets of arbitrary size. One
consequence is the existence of (k, O(log n)) volume-respecting embeddings for
all 1 \leq k \leq n, which is the best possible, and answers positively a
question posed by U. Feige. Our techniques are also used to answer positively a
question of Y. Rabinovich, showing that any weighted n-point planar graph
embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the
dimension is optimal, and improves upon the previously known bound of O((log
n)^2).Comment: 17 pages. No figures. Appeared in FOCS '04. To appeaer in Geometric &
Functional Analysis. This version fixes a subtle error in Section 2.
The Evolution of the Galaxy Stellar Mass Function at z= 4-8: A Steepening Low-mass-end Slope with Increasing Redshift
We present galaxy stellar mass functions (GSMFs) at 4-8 from a
rest-frame ultraviolet (UV) selected sample of 4500 galaxies, found via
photometric redshifts over an area of 280 arcmin in the CANDELS/GOODS
fields and the Hubble Ultra Deep Field. The deepest Spitzer/IRAC data
yet-to-date and the relatively large volume allow us to place a better
constraint at both the low- and high-mass ends of the GSMFs compared to
previous space-based studies from pre-CANDELS observations. Supplemented by a
stacking analysis, we find a linear correlation between the rest-frame UV
absolute magnitude at 1500 \AA\ () and logarithmic stellar mass
() that holds for galaxies with . We
use simulations to validate our method of measuring the slope of the - relation, finding that the bias is minimized with a hybrid
technique combining photometry of individual bright galaxies with stacked
photometry for faint galaxies. The resultant measured slopes do not
significantly evolve over 4-8, while the normalization of the trend
exhibits a weak evolution toward lower masses at higher redshift. We combine
the - distribution with observed rest-frame UV luminosity
functions at each redshift to derive the GSMFs, finding that the low-mass-end
slope becomes steeper with increasing redshift from
at to at
. The inferred stellar mass density, when integrated over
-, increases by a factor of
between and and is in good agreement with the time integral of the
cosmic star formation rate density.Comment: 27 pages, 17 figures, ApJ, in pres
Balls-in-boxes condensation on networks
We discuss two different regimes of condensate formation in zero-range
processes on networks: on a q-regular network, where the condensate is formed
as a result of a spontaneous symmetry breaking, and on an irregular network,
where the symmetry of the partition function is explicitly broken. In the
latter case we consider a minimal irregularity of the q-regular network
introduced by a single Q-node with degree Q>q. The statics and dynamics of the
condensation depends on the parameter log(Q/q), which controls the exponential
fall-off of the distribution of particles on regular nodes and the typical time
scale for melting of the condensate on the Q-node which increases exponentially
with the system size . This behavior is different than that on a q-regular
network where log(Q/q)=0 and where the condensation results from the
spontaneous symmetry breaking of the partition function, which is invariant
under a permutation of particle occupation numbers on the q-nodes of the
network. In this case the typical time scale for condensate melting is known to
increase typically as a power of the system size.Comment: 7 pages, 3 figures, submitted to the "Chaos" focus issue on
"Optimization in Networks" (scheduled to appear as Volume 17, No. 2, 2007
Graph diameter in long-range percolation
We study the asymptotic growth of the diameter of a graph obtained by adding
sparse "long" edges to a square box in . We focus on the cases when an
edge between and is added with probability decaying with the Euclidean
distance as when . For we show
that the graph diameter for the graph reduced to a box of side scales like
where . In particular, the
diameter grows about as fast as the typical graph distance between two vertices
at distance . We also show that a ball of radius in the intrinsic metric
on the (infinite) graph will roughly coincide with a ball of radius
in the Euclidean metric.Comment: 17 pages, extends the results of arXiv:math.PR/0304418 to graph
diameter, substantially revised and corrected, added a result on volume
growth asymptoti
Far-UV FUSE spectroscopy of the OVI resonance doublet in Sand2 (WO)
We present Far-Ultraviolet Spectroscopic Explorer (FUSE) spectroscopy of Sand
2, a LMC WO-type Wolf-Rayet star, revealing the OVI resonance P Cygni doublet
at 1032-38A. These data are combined with HST/FOS ultraviolet and Mt Stromlo
2.3m optical spectroscopy, and analysed using a spherical, non-LTE,
line-blanketed code. Our study reveals exceptional stellar parameters:
T*=150,000K, v_inf=4100 km/s, log (L/Lo)=5.3, and Mdot=10^-5 Mo/yr if we adopt
a volume filling factor of 10%. Elemental abundances of C/He=0.7+-0.2 and
O/He=0.15(-0.05+0.10) by number qualitatively support previous recombination
line studies. We confirm that Sand 2 is more chemically enriched in carbon than
LMC WC stars, and is expected to undergo a supernova explosion within the next
50,000 yr.Comment: 17 pages, 4 figures, AASTeX preprint format. This paper will appear
in a special issue of ApJ Letters devoted to the first scientific results
from the FUSE missio
Approaching the isoperimetric problem in via the hyperbolic log-convex density conjecture
We prove that geodesic balls centered at some base point are isoperimetric in
the real hyperbolic space endowed with a smooth, radial,
strictly log-convex density on the volume and perimeter. This is an analogue of
the result by G. R. Chambers for log-convex densities on . As an
application we prove that in any rank one symmetric space of non-compact type,
geodesic balls are isoperimetric in a class of sets enjoying a suitable notion
of radial symmetry.Comment: 17 pages, 5 figures. Added references. Generalized Definition 1.2 to
the octonionic case, and simplified the argument in Section
Linear-Time Algorithms for Computing Maximum-Density Sequence Segments with Bioinformatics Applications
We study an abstract optimization problem arising from biomolecular sequence
analysis. For a sequence A of pairs (a_i,w_i) for i = 1,..,n and w_i>0, a
segment A(i,j) is a consecutive subsequence of A starting with index i and
ending with index j. The width of A(i,j) is w(i,j) = sum_{i <= k <= j} w_k, and
the density is (sum_{i<= k <= j} a_k)/ w(i,j). The maximum-density segment
problem takes A and two values L and U as input and asks for a segment of A
with the largest possible density among those of width at least L and at most
U. When U is unbounded, we provide a relatively simple, O(n)-time algorithm,
improving upon the O(n \log L)-time algorithm by Lin, Jiang and Chao. When both
L and U are specified, there are no previous nontrivial results. We solve the
problem in O(n) time if w_i=1 for all i, and more generally in
O(n+n\log(U-L+1)) time when w_i>=1 for all i.Comment: 23 pages, 13 figures. A significant portion of these results appeared
under the title, "Fast Algorithms for Finding Maximum-Density Segments of a
Sequence with Applications to Bioinformatics," in Proceedings of the Second
Workshop on Algorithms in Bioinformatics (WABI), volume 2452 of Lecture Notes
in Computer Science (Springer-Verlag, Berlin), R. Guigo and D. Gusfield
editors, 2002, pp. 157--17
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