arXiv subjects

Rong-Hua Wang

Publications and source records attributed to Rong-Hua Wang.

21 records · Page 2Linked to original sources

arXiv2014

An Algorithm for Deciding the Summability of Bivariate Rational Functions

Let $Δ_x f(x,y)=f(x+1,y)-f(x,y)$ and $Δ_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$ such that $f(x,y)=Δ_x g(x,y) + Δ_y h(x,y)$. Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method in the sense of algorithms, we need to solve two problems. The first is to determine the shift equivalence of two bivariate polynomials. We solve this problem by presenting an algorithm for computing the dispersion sets of any two bivariate polynomials. The second is to solve a univariate difference equation in an algebraically closed field. By considering the irreducible factorization of the denominator of $f(x,y)$ in a general field, we present a new criterion which requires only finding a rational solution of a bivariate difference equation. This goal can be achieved by deriving a universal denominator of the rational solutions and a degree bound on the numerator. Combining these two algorithms, we can decide the summability of a bivariate rational function.

cs.SC↗

arXiv2014

Ramanujan-type Congruences for Overpartitions Modulo 5

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\overline{p}(4n+3)\equiv 0 \pmod{8}$ for $n\geq 0$. They also conjectured that $\overline{p}(40n+35)\equiv 0 \pmod{40}$ for $n\geq 0$. Chen and Xia proved this conjecture by using the $(p,k)$-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that $\overline{p}(5n)\equiv (-1)^{n}\overline{p}(4\cdot 5n) \pmod{5}$ for $n \geq 0$ and $\overline{p}(n)\equiv (-1)^{n}\overline{p}(4n)\pmod{8}$ for $n \geq 0$ by using the relation of the generating function of $\overline{p}(5n)$ modulo $5$ found by Treneer and the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg. As a consequence, we deduce that $\overline{p}(4^k(40n+35))\equiv 0 \pmod{40}$ for $n,k\geq 0$. Furthermore, applying the Hecke operator on $ϕ(q)^3$ and the fact that $ϕ(q)^3$ is a Hecke eigenform, we obtain an infinite family of congrences $\overline{p}(4^k \cdot5\ell^2n)\equiv 0 \pmod{5}$, where $k\ge 0$ and $\ell$ is a prime such that $\ell\equiv3 \pmod{5}$ and $\left(\frac{-n}{\ell}\right)=-1$. Moreover, we show that $\overline{p}(5^{2}n)\equiv \overline{p}(5^{4}n) \pmod{5}$ for $n \ge 0$. So we are led to the congruences $\overline{p}\big(4^k5^{2i+3}(5n\pm1)\big)\equiv 0 \pmod{5}$ for $n, k, i\ge 0$. In this way, we obtain various Ramanujan-type congruences for $\overline{p}(n)$ modulo $5$ such as $\overline{p}(45(3n+1))\equiv 0 \pmod{5}$ and $\overline{p}(125(5n\pm 1))\equiv 0 \pmod{5}$ for $n\geq 0$.

math.CO↗

arXiv2008

Persistence, extinction and spatio-temporal synchronization of SIRS cellular automata models

Spatially explicit models have been widely used in today's mathematical ecology and epidemiology to study persistence and extinction of populations as well as their spatial patterns. Here we extend the earlier work--static dispersal between neighbouring individuals to mobility of individuals as well as multi-patches environment. As is commonly found, the basic reproductive ratio is maximized for the evolutionary stable strategy (ESS) on diseases' persistence in mean-field theory. This has important implications, as it implies that for a wide range of parameters that infection rate will tend maximum. This is opposite with present results obtained in spatial explicit models that infection rate is limited by upper bound. We observe the emergence of trade-offs of extinction and persistence on the parameters of the infection period and infection rate and show the extinction time having a linear relationship with respect to system size. We further find that the higher mobility can pronouncedly promote the persistence of spread of epidemics, i.e., the phase transition occurs from extinction domain to persistence domain, and the spirals' wavelength increases as the mobility increasing and ultimately, it will saturate at a certain value. Furthermore, for multi-patches case, we find that the lower coupling strength leads to anti-phase oscillation of infected fraction, while higher coupling strength corresponds to in-phase oscillation.

q-bio.PE↗