arXiv subjects

Chuang Xu

Publications and source records attributed to Chuang Xu.

40 records · Page 3Linked to original sources

arXiv2018

Best finite approximations of Benford's Law

For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and Kolmogorov), given any number of atoms, and allowing for additional constraints regarding weights or positions of atoms. As an application, best (constrained or unconstrained) approximations are identified for Benford's Law (logarithmic distribution of significands) and other familiar distributions. The results complement and extend known facts in the literature; they also provide new rigorous benchmarks against which to evaluate empirical observations regarding Benford's Law.

math.PR↗

arXiv2017

Circularly invariant uniformizable probability measures for linear transformations

In this paper, we prove a threshold result on the existence of a circularly invariant uniformizable probability measure (CIUPM) for linear transformations with non-zero slope on the line. We show that there is a threshold constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as $c.$ Moreover, the CIUPM is unique up to translation if the diameter of the support equals $c.$

math.DS↗

arXiv2016

Global Threshold Dynamics of a Stochastic Differential Equation SIS Model

In this paper, we further investigate the global dynamics of a stochastic differential equation SIS (Susceptible-Infected-Susceptible) epidemic model recently proposed in [A. Gray et al., SIAM. J. Appl. Math., 71 (2011), 876-902]. We present a stochastic threshold theorem in term of a \textit{stochastic basic reproduction number} $R_0^S:$ the disease dies out with probability one if $R_0^S<1,$ and the disease is recurrent if $R_0^S\geqslant1.$ We prove the existence and global asymptotic stability of a unique invariant density for the Fokker-Planck equation associated with the SDE SIS model when $R_0^S>1.$ In term of the profile of the invariant density, we define a \textit{persistence basic reproduction number} $R_0^P$ and give a persistence threshold theorem: the disease dies out with large probability if $R_0^P\leqslant1,$ while persists with large probability if $R_0^P>1.$ Comparing the \textit{stochastic disease prevalence} with the \textit{deterministic disease prevalence}, we discover that the stochastic prevalence is bigger than the deterministic prevalence if the deterministic basic reproduction number $R_0^D>2.$ This shows that noise may increase severity of disease. Finally, we study the asymptotic dynamics of the stochastic SIS model as the noise vanishes and establish a sharp connection with the threshold dynamics of the deterministic SIS model in term of a \textit{Limit Stochastic Threshold Theorem}.

math.DS↗

arXiv2016

Strong Allee Effect in A Stochastic Logistic Model with Mate Limitation and Stochastic Immigration

We propose a stochastic logistic model with mate limitation and stochastic immigration. Incorporating stochastic immigration into a continuous time Markov chain model, we derive and analyze the associated master equation. By a standard result, there exists a unique globally stable positive stationary distribution. We show that such stationary distribution admits a bimodal profile which implies that a strong Allee effect exists in the stochastic model. Such strong Allee effect disappears and threshold phenomenon emerges as the total population size goes to infinity. Stochasticity vanishes and the model becomes deterministic as the total population size goes to infinity. This implies that there is only one possible fate (either to die out or survive) for a species constrained to a specific community and whether population eventually goes extinct or persists does not depend on initial population density but on a critical inherent constant determined by birth, death and mate limitation. Such a conclusion interprets differently from the classical ordinary differential equation model and thus a paradox on strong Allee effect occurs. Such paradox illustrates the diffusion theory's dilemma.

math.DS↗