Minimal homogeneous Steiner 2-(v,3) trades

A Steiner 2-$(v,3)$ trade is a pair $(T_1,T_2)$ of disjoint partial Steiner triple systems, each on the same set of $v$ points, such that each pair of points occurs in $T_1$ if and only if it occurs in $T_2$. A Steiner 2-$(v,3)$ trade is called d-homogeneous if each point occurs in exactly d blocks of $T_1$ (or $T_2$). In this paper we construct minimal d-homogeneous Steiner 2-$(v,3)$ trades of foundation $v$ and volume $\mathit{dv}/3$ for sufficiently large values of $v$. (Specifically, $v>3(1.75d^2+3)$ if $v$ is divisible by 3 and $v>d(4^{d/3+1}+1)$ otherwise.)     

Properties of Beurling-type submodules via Agler decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2({\mathbb{D}}^2)$. Specifically, we study Beurling-type submodules – namely submodules of the form $\theta H^2({\mathbb{D}}^2)$ for θ inner – using properties of Agler decompositions of θ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2({\mathbb{D}}^2)?\theta H^2({\mathbb{D}}^2)$. Results include characterizations (in terms of θ) of when a commutator $[S_{z_j}^{?},S_{z_j}]$ has rank n and when subspaces associated to Agler decompositions are reducing for $S_{z_1}$ and $S_{z_2}$. We include several open questions.     

Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant

$\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u?\mathrm{?}?(u{\chi }_1(w)\mathrm{?}w)+{\mu }_{1}u(1?u?a_{1}v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v?\mathrm{?}?(v{\chi }_2(w)\mathrm{?}w)+{\mu }_{2}v(1?a_{2}u?v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w?(\alpha u+\beta v)w,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary, where the initial data $(u_0,v_0)\in {(C^0(\stackrel{￣}{\mathrm{\Omega }}))}^2$ and $w_0\in W^{1,\infty }(\mathrm{\Omega })$ are non-negative and the parameters $d_1,d_2,d_3>0$, ${\mu }_1,{\mu }_2>0$, $a_1,a_2>0$ and $\alpha ,\beta >0$. The chemotactic function ${\chi }_i(w)$ ($i=1,2$) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for $i=1,2$,(i) ${\chi }_i(w)={\chi }_{0,i}>0$ and

${6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}<\frac{\pi \sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}?\frac{2\sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}\arctan ?\frac{d_i?d_3}{2}\sqrt{\frac{n+1}{d_i{d}_3}};$

(ii) $0<{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}\le \frac{d_3}{3(n+1){6{\chi }_{i}6}_{L^{\infty }[0,{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}]}}\min ?\{\frac{2d_i}{d_i+d_3},1\}$.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If $a_1,a_2\in (0,1)$ and $u_0\ne 0\ne v_0$, then any global bounded solution exponentially converge to $(\frac{1?a_1}{1?a_1{a}_2},\frac{1?a_2}{1?a_1{a}_2},0)$ as $t\to \infty$;? If $a_1>1>a_2>0$ and $v_0\ne 0$, then any global bounded solution exponentially converge to $(0,1,0)$ as $t\to \infty$;? If $a_1=1>a_2>0$ and $v_0\ne 0$, then any global bounded solution algebraically converge to $(0,1,0)$ as $t\to \infty$.