4,606 research outputs found

    Simultaneous Representation of Proper and Unit Interval Graphs

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    In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary

    Statistics of self-avoiding walks on randomly diluted lattice

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    A comprehensive numerical study of self-avoiding walks (SAW's) on randomly diluted lattices in two and three dimensions is carried out. The critical exponents ν\nu and χ\chi are calculated for various different occupation probabilities, disorder configuration ensembles, and walk weighting schemes. These results are analyzed and compared with those previously available. Various subtleties in the calculation and definition of these exponents are discussed. Precise numerical values are given for these exponents in most cases, and many new properties are recognized for them.Comment: 34 pages (+ 12 figures), REVTEX 3.

    The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary

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    We consider the dimer-monomer problem for the rectangular lattice. By mapping the problem into one of close-packed dimers on an extended lattice, we rederive the Tzeng-Wu solution for a single monomer on the boundary by evaluating a Pfaffian. We also clarify the mathematical content of the Tzeng-Wu solution by identifying it as the product of the nonzero eigenvalues of the Kasteleyn matrix.Comment: 4 Pages to appear in the Physical Review E (2006

    Pseudorandom Number Generators and the Square Site Percolation Threshold

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    A select collection of pseudorandom number generators is applied to a Monte Carlo study of the two dimensional square site percolation model. A generator suitable for high precision calculations is identified from an application specific test of randomness. After extended computation and analysis, an ostensibly reliable value of pc = 0.59274598(4) is obtained for the percolation threshold.Comment: 11 pages, 6 figure

    Is the Melting Pot Still Hot? Explaining the Resurgence of Immigrant Segregation

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    This paper uses decennial Census data to examine trends in immigrant segregation in the United States between 1910 and 2000. Immigrant segregation declined in the first half of the century, but has been rising steadily over the past three decades. Analysis of restricted access 1990 Census microdata suggests that this rise would be even more striking if the native-born children of immigrants could be consistently excluded from the analysis. We analyze panel and cross-sectional variation in immigrant segregation, as well as housing price patterns across metropolitan areas, to test four hypotheses of immigrant segregation. Immigration itself has surged in recent decades, but the tendency for newly arrived immigrants to be younger and of lower socioeconomic status explains very little of the recent rise in immigrant segregation. We also find little evidence of increased nativism in the housing market. Evidence instead points to changes in urban form, manifested in particular as native-driven suburbanization and the decline of public transit as a transportation mode, as a central explanation for the new immigrant segregation.

    PAC Learning, VC Dimension, and the Arithmetic Hierarchy

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    We compute that the index set of PAC-learnable concept classes is mm-complete Σ30\Sigma^0_3 within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable Π10\Pi^0_1 classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension
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