20,361 research outputs found

    Asymmetric binary covering codes

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    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

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    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 o(n)o(n)-sized "implicit decomposition" of a bounded-degree graph GG on nn nodes, which combined with read-only access to GG enables fast answers to connectivity and biconnectivity queries on GG. 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 mm edges, we also provide the first o(m)o(m) writes and O(m)O(m) 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

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    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

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    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

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    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

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    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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