20,361 research outputs found
Asymmetric binary covering codes
An asymmetric binary covering code of length n and radius R is a subset C of
the n-cube Q_n such that every vector x in Q_n can be obtained from some vector
c in C by changing at most R 1's of c to 0's, where R is as small as possible.
K^+(n,R) is defined as the smallest size of such a code. We show K^+(n,R) is of
order 2^n/n^R for constant R, using an asymmetric sphere-covering bound and
probabilistic methods. We show K^+(n,n-R')=R'+1 for constant coradius R' iff
n>=R'(R'+1)/2. These two results are extended to near-constant R and R',
respectively. Various bounds on K^+ are given in terms of the total number of
0's or 1's in a minimal code. The dimension of a minimal asymmetric linear
binary code ([n,R]^+ code) is determined to be min(0,n-R). We conclude by
discussing open problems and techniques to compute explicit values for K^+,
giving a table of best known bounds.Comment: 16 page
Implicit Decomposition for Write-Efficient Connectivity Algorithms
The future of main memory appears to lie in the direction of new technologies
that provide strong capacity-to-performance ratios, but have write operations
that are much more expensive than reads in terms of latency, bandwidth, and
energy. Motivated by this trend, we propose sequential and parallel algorithms
to solve graph connectivity problems using significantly fewer writes than
conventional algorithms. Our primary algorithmic tool is the construction of an
-sized "implicit decomposition" of a bounded-degree graph on
nodes, which combined with read-only access to enables fast answers to
connectivity and biconnectivity queries on . The construction breaks the
linear-write "barrier", resulting in costs that are asymptotically lower than
conventional algorithms while adding only a modest cost to querying time. For
general non-sparse graphs on edges, we also provide the first writes
and operations parallel algorithms for connectivity and biconnectivity.
These algorithms provide insight into how applications can efficiently process
computations on large graphs in systems with read-write asymmetry
Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities
We examine in detail the relative equilibria in the four-vortex problem where
two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and
\Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is
that for m > 0, the convex configurations all contain a line of symmetry,
forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for
all m but the isosceles trapezoid case exists only when m is positive. In fact,
there exist asymmetric convex configurations when m < 0. In contrast to the
Newtonian four-body problem with two equal pairs of masses, where the symmetry
of all convex central configurations is unproven, the equations in the vortex
case are easier to handle, allowing for a complete classification of all
solutions. Precise counts on the number and type of solutions (equivalence
classes) for different values of m, as well as a description of some of the
bifurcations that occur, are provided. Our techniques involve a combination of
analysis and modern and computational algebraic geometry
High-Resolution Measurements of the Dark Matter Halo of NGC 2976: Evidence for a Shallow Density Profile
We have obtained two-dimensional velocity fields of the dwarf spiral galaxy
NGC 2976 in Halpha and CO. The high spatial (~75 pc) and spectral (13 km/s and
2 km/s, respectively) resolution of these observations, along with our
multicolor optical and near-infrared imaging, allow us to measure the shape of
the density profile of the dark matter halo with good precision. We find that
the total (baryonic plus dark matter) mass distribution of NGC 2976 follows a
rho_tot ~ r^(-0.27 +/- 0.09) power law out to a radius of 1.8 kpc, assuming
that the observed radial motions provide no support. The density profile
attributed to the dark halo is even shallower, consistent with a nearly
constant density of dark matter over the entire observed region. A maximal disk
fit yields an upper limit to the K-band stellar mass-to-light ratio (M*/L_K) of
0.09^{+0.15}_{-0.08} M_sun/L_sun,K (including systematic uncertainties), with
the caveat that for M*/L_K > 0.19 M_sun/L_sun,K the dark matter density
increases with radius, which is unphysical. Assuming 0.10 M_sun/L_sun,K <
M*/L_K < 0.19 M_sun/L_sun,K, the dark matter density profile lies between
rho_dm ~ r^-0.17 and rho_dm ~ r^-0.01. Therefore, independent of any
assumptions about the stellar disk or the functional form of the density
profile, NGC 2976 does not contain a cuspy dark matter halo. We also
investigate some of the systematic effects that can hamper rotation curve
studies, and show that 1) longslit rotation curves are far more vulnerable to
systematic errors than two-dimensional velocity fields, 2) NGC 2976 contains
large radial motions at small radii, and 3) the Halpha and CO velocity fields
of NGC 2976 agree within their uncertainties. [slightly abridged]Comment: 30 pages, 4 tables, 13 figures (7 in color; Figures 1 and 3 are
low-resolution to save space). Accepted for publication in ApJ. Version with
full-resolution figures available at
http://astro.berkeley.edu/~bolatto/ngc2976rotation.ps (46 MB
Codes Correcting Two Deletions
In this work, we investigate the problem of constructing codes capable of
correcting two deletions. In particular, we construct a code that requires
redundancy approximately 8 log n + O(log log n) bits of redundancy, where n is
the length of the code. To the best of the author's knowledge, this represents
the best known construction in that it requires the lowest number of redundant
bits for a code correcting two deletions
Optimal design of nanoplasmonic materials using genetic algorithms as a multi-parameter optimization tool
An optimal control approach based on multiple parameter genetic algorithms is
applied to the design of plasmonic nanoconstructs with pre-determined optical
properties and functionalities. We first develop nanoscale metallic lenses that
focus an incident plane wave onto a pre-specified, spatially confined spot. Our
results illustrate the role of symmetry breaking and unravel the principles
that favor dimeric constructs for optimal light localization. Next we design a
periodic array of silver particles to modify the polarization of an incident,
linearly-polarized plane wave in a desired fashion while localizing the light
in space. The results provide insight into the structural features that
determine the birefringence properties of metal nanoparticles and their arrays.
Of the variety of potential applications that may be envisioned, we note the
design of nanoscale light sources with controllable coherence and polarization
properties that could serve for coherent control of molecular or electronic
dynamics in the nanoscale.Comment: 13 pages, 6 figures. submitted to J. Chem. Phy
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