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Pingzhi Yuan

  1. Davenport constant with weights.

    Втр, 09/15/2009 - 11:27 — SomNambul
    Authors: Pingzhi Yuan, Xiangneng Zeng
    Subjects: Number Theory
    Abstract

    For the cyclic group $G=\mathbb{Z}/n\mathbb{Z}$ and any non-empty
    $A\in\mathbb{Z}$. We define the Davenport constant of $G$ with weight $A$,
    denoted by $D_A(n)$, to be the least natural number $k$ such that for any
    sequence $(x_1, ..., x_k)$ with $x_i\in G$, there exists a non-empty
    subsequence $(x_{j_1}, ..., x_{j_l})$ and $a_1, ..., a_l\in A$ such that
    $\sum_{i=1}^l a_ix_{j_i} = 0$.

  2. On the transcendence of some infinite sums.

    Втр, 09/15/2009 - 11:27 — SomNambul
    Authors: Juan Li, Pingzhi Yuan
    Subjects: Number Theory
    Abstract

    In this paper we investigate the infinite convergent sum
    $T=\sum_{n=0}^\infty\frac{P(n)}{Q(n)}$, where

    $P(x)\in\bar{\mathbb{Q}}[x]$, $Q(x)\in\mathbb{Q}[x]$ and $Q(x)$ has only
    simple rational zeros. N. Saradha and R. Tijdeman have obtained sufficient and
    necessary conditions for the transcendence of $T$ if the degree of $Q(x)$ is 3.
    In this paper we give sufficient and necessary conditions for the transcendence
    of $T$ if the degree of $Q(x)$ is 4 and $Q(x)$ is reduced.

  3. Subsequence Sums of Zero-sum free Sequences II.

    Пнд, 09/14/2009 - 09:01 — SomNambul
    Authors: Pingzhi Yuan
    Subjects: Number Theory
    Abstract

    Let $G$ be a finite abelian group, and let $S$ be a sequence over $G$. Let

    $f(S)$ denote the number of elements in $G$ which can be expressed as the sum
    over a nonempty subsequence of $S$. In this paper, we determine all the
    sequences $S$ that contains no zero-sum subsequences and $f(S)\leq 2|S|-1$.

  4. Coefficients of cyclotomic polynomials.

    Втр, 09/08/2009 - 12:16 — SomNambul
    Authors: Pingzhi Yuan
    Subjects: Number Theory
    Abstract

    Let $a(n, k)$ be the $k$-th coefficient of the $n$-th cyclotomic polynomial.
    Recently, Ji, Li and Moree \cite{JLM09} proved that for any integer $m\ge1$,
    $\{a(mn, k)| n, k\in\mathbb{N}\}=\mathbb{Z}$. In this paper, we improve this
    result and prove that for any integers $s>t\ge0$,

    $$\{a(ns+t, k)| n, k\in\mathbb{N}\}=\mathbb{Z}.$$

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