Dynamical behavior of solutions of a reaction–diffusion model in river network

In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant $\beta$. We show the existence of two critical values $c_0$ and 2 with $0<c_0<2$, and prove that when $-c_0\le \beta <2$, the population density in every branch of the river goes to 1 as time goes to infinity; when $-2<\beta <-c_0$, then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when $\left|\beta \right|\ge 2$, the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., $\left|\beta \right|<2$), the species will survive in the long run.     

Projective well orders and coanalytic witnesses

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}={\mathrm{?}}_1<2^{{\mathrm{?}}_0}={\mathrm{?}}_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a ${\mathrm{\Pi }}_1^1$ witness and there is a ${\mathrm{\Delta }}_3^1$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a ${\mathrm{\Delta }}_3^1$ well-order of the reals is consistent with $\mathfrak{c}={\mathrm{?}}_2$ and each of the following: $\mathfrak{a}=\mathfrak{u}<\mathfrak{i}$, $\mathfrak{a}=\mathfrak{i}<\mathfrak{u}$, $\mathfrak{a}<\mathfrak{u}=\mathfrak{i}$, where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.     

An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings ${\mathbb{F}}_2+u{\mathbb{F}}_2$ and ${\mathbb{F}}_4+u{\mathbb{F}}_4$; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with $\gamma =5$ in $W_{68,2}$, which is the first instance of such a $\gamma$ value in the literature.     

Global stability of an SAIRS epidemic model with vaccinations,transient immunity and treatment

We consider an SAIRS epidemic model with vaccinations and treatment, where asymptomatic and symptomatic infectious individuals are considered in the transmission of the disease. We found the basic reproduction number, ${ℛ}_0$ and using ${ℛ}_0$, we conducted global stability analysis. We proved when ${ℛ}_0<1$, the disease-free equilibrium is globally stable. If ${ℛ}_0>1$, the disease-free equilibrium in unstable and a unique endemic equilibrium exists. We explored the global stability of the endemic equilibrium and noticed it is globally stable under certain conditions. Moreover, we then considered a special case of the SAIRS model, the SAIR model. We proved the disease-free equilibrium is globally stability when ${ℛ}_0<1$ and the endemic equilibrium is globally stable when ${ℛ}_0>1$. Next, we numerically simulated our analytical results and plotted these for various cases. Finally, we performed sensitivity analysis to tell us how each parameter in the system affects disease transmission.     

Vertex-disjoint cycles in local tournaments

Let $r$ be a positive integer. The Bermond–Thomassen conjecture states that, a digraph of minimum out-degree at least $2r-1$ contains $r$ vertex-disjoint directed cycles. A digraph $D$ is called a local tournament if for every vertex $x$ of $D$, both the out-neighbours and the in-neighbours of $x$ induce tournaments. Note that tournaments form the subclass of local tournaments. In this paper, we verify that the Bermond–Thomassen conjecture holds for local tournaments. In particular, we prove that every local tournament $D$ with ${\delta }^{+}\left(D\right)\ge 2r-1$ contains $r$ disjoint cycles $C_1,C_2,\dots ,C_r$, satisfying that either $C_i$ has the length at most 4 or is a shortest cycle of the original digraph of $D-C_1-\dots -C_{i-1}$ for $1\le i\le r$.