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Yufei Zhao

  1. Subsets Characterized by the Number of Missing Sums and Differences.

    Пт, 11/13/2009 - 16:01 — SomNambul
    Authors: Yufei Zhao
    Subjects: Number Theory
    Abstract

    A more sums than differences (MSTD) set is a finite subset S of the integers
    such |S+S|>|S-S|. We show that the probability that a uniform random subset of
    {0, 1, ..., n} is an MSTD set approaches some limit \rho, and \rho > 4 x
    10^{-4}. This improves the previous result of Martin and O'Bryant that there is
    a lower limit of at least 2 x 10^{-7}. Monte Carlo experiments suggest that
    \rho \approx 4.5 x 10^{-4}. We have a deterministic algorithm that can compute
    \rho up to arbitrary precision.

  2. Counting MSTD Sets in Finite Abelian Groups.

    Пт, 11/13/2009 - 16:01 — SomNambul
    Authors: Yufei Zhao
    Subjects: Combinatorics
    Abstract

    In an abelian group G, a more sums than differences (MSTD) set is a subset A
    of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets
    in finite abelian groups, extending previous results of Nathanson. The proof
    contains an application of a recently resolved conjecture of Alon and Kahn on
    the number of independent sets in a regular graph.

  3. Constructing Numerical Semigroups of a Given Genus.

    Втр, 10/13/2009 - 18:46 — SomNambul
    Authors: Yufei Zhao
    Subjects: Combinatorics
    Abstract

    Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros
    conjectured that n_g possesses certain Fibonacci-like properties. Almost all
    previous attempts at proving this conjecture were based on analyzing the
    semigroup tree. We offer a new, simpler approach to counting numerical
    semigroups of a given genus. Our method gives direct constructions of families
    of numerical semigroups, without referring to the generators or the semigroup
    tree. In particular, we give an improved asymptotic lower bound for n_g.

  4. Constructing Numerical Semigroups of a Given Genus.

    Втр, 10/13/2009 - 18:46 — SomNambul
    Authors: Yufei Zhao
    Subjects: Combinatorics
    Abstract

    Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros
    conjectured that n_g possesses certain Fibonacci-like properties. Almost all
    previous attempts at proving this conjecture were based on analyzing the
    semigroup tree. We offer a new, simpler approach to counting numerical
    semigroups of a given genus. Our method gives direct constructions of families
    of numerical semigroups, without referring to the generators or the semigroup
    tree. In particular, we give an improved asymptotic lower bound for n_g.

  5. Constructing MSTD Sets Using Bidirectional Ballot Sequences.

    Втр, 09/01/2009 - 12:47 — SomNambul
    Authors: Yufei Zhao
    Subjects: Combinatorics
    Abstract

    A more sums than differences (MSTD) set is a finite subset S of the integers
    such that |S+S| > |S-S|. We construct a new dense family of MSTD subsets of {0,
    1, 2, ..., n-1}. Our construction gives Theta(2^n/n) MSTD sets, improving the
    previous best construction with Omega(2^n/n^4) MSTD sets by Miller, Orosz, and
    Scheinerman.

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