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Chunlin Wang

Publications and source records attributed to Chunlin Wang.

26 records · Page 2Linked to original sources
arXiv2014

The elementary symmetric functions of a reciprocal polynomial sequence

Erdös and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $n\geq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1), 1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $m\geq2$ being an integer and $n=1$.

math.NT↗
arXiv2013

Uniform lower bound for the least common multiple of a polynomial sequence

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $s\ge 2$ being an integer and $n=1$, where $\lceil n/2\rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.

math.NT↗
arXiv2011

On the integrality of the elementary symmetric functions of $1, 1/3, ..., 1/(2n-1)$

Erdos and Niven proved that for any positive integers $m$ and $d$, there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/m,1/(m+d), ..., 1/(m+nd)$ are integers. Recently, Chen and Tang proved that if $n\ge 4$, then none of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we show that if $n\ge 2$, then none of the elementary symmetric functions of $1, 1/3, ..., 1/(2n-1)$ is an integer.

math.NT↗
arXiv2008

On the Asymptotic Normality of the Conditional Maximum Likelihood Estimators for the Truncated Regression Model and the Tobit Model

In this paper, we study the asymptotic normality of the conditional maximum likelihood (ML) estimators for the truncated regression model and the Tobit model. We show that under the general setting assumed in his book, the conjectures made by Hayashi (2000) \footnote{see page 516, and page 520 of Hayashi (2000).} about the asymptotic normality of the conditional ML estimators for both models are true, namely, a sufficient condition is the nonsingularity of $\mathbf{x_tx'_t}$.

math.ST↗
arXiv2008

Viral population estimation using pyrosequencing

The diversity of virus populations within single infected hosts presents a major difficulty for the natural immune response as well as for vaccine design and antiviral drug therapy. Recently developed pyrophosphate based sequencing technologies (pyrosequencing) can be used for quantifying this diversity by ultra-deep sequencing of virus samples. We present computational methods for the analysis of such sequence data and apply these techniques to pyrosequencing data obtained from HIV populations within patients harboring drug resistant virus strains. Our main result is the estimation of the population structure of the sample from the pyrosequencing reads. This inference is based on a statistical approach to error correction, followed by a combinatorial algorithm for constructing a minimal set of haplotypes that explain the data. Using this set of explaining haplotypes, we apply a statistical model to infer the frequencies of the haplotypes in the population via an EM algorithm. We demonstrate that pyrosequencing reads allow for effective population reconstruction by extensive simulations and by comparison to 165 sequences obtained directly from clonal sequencing of four independent, diverse HIV populations. Thus, pyrosequencing can be used for cost-effective estimation of the structure of virus populations, promising new insights into viral evolutionary dynamics and disease control strategies.

q-bio.PE↗
arXiv2006

On the Estimates of the Density of Feynman-Kac Semigroups of $α$-Stable-like Processes

Suppose that $α\in (0,2)$ and that $X$ is an $α$-stable-like process on $\R^d$. Let $F$ be a function on $\R^d$ belonging to the class $\bf{J_{d,α}}$ (see Introduction) and $A_{t}^{F}$ be $\sum_{s \le t}F(X_{s-},X_{s}), t> 0$, a discontinuous additive functional of $X$. With neither $F$ nor $X$ being symmetric, under certain conditions, we show that the Feynman-Kac semigroup $\{S_{t}^{F}:t \ge 0\}$ defined by $$ S_{t}^{F}f(x)=\mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density $q$ and that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that $$C_{1}e^{-C_{2}t}t^{-\frac{d}α}(1 \wedge \frac{t^{\frac{1}α}}{|x-y|})^{d+α} \leq q(t,x,y) \leq C_{3}e^{C_{4}t}t^{-\frac{d}α}(1 \wedge \frac{t^{\frac{1}α}}{|x-y|})^{d+α}$$ for all $(t,x,y)\in (0,\infty) \times \R^d \times \R^d$.

math.PR↗
arXiv2006

Estimation of Parameters of Stable Distributions

In this paper, we propose a method based on GMM (the generalized method of moments) to estimate the parameters of stable distributions with $0<α<2$. We don't assume symmetry for stable distributions.

math.ST↗