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.     

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.     

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space ${\mathsf{h}}_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of ${\mathsf{h}}_p$ for all $0<p<1$, and provide a constructive proof of the atomic decomposition for $p=1$ which resolves a main problem on the subject left open for the last twelve years. We also study ${(p,\infty )}_c$-atoms, and show that every ${(p,2)}_c$-atom can be decomposed into a sum of ${(p,\infty )}_c$-atoms; consequently, for every $0<p\le 1$, the ${(p,q)}_c$-atoms lead to the same atomic space for all $2\le q\le \infty$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space ${\mathsf{h}}_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.     

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.