Large blocking sets in PG(2,q2)

Minimal blocking sets in $\mathrm{PG}(2,q^2)$ have size at most $q^3+1$. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most $q^3+1-(p-3)/2$, if $q=p$, $p\ge 67$, or $q=p^h$, $p>7$, $h>1$. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets).     

Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $?h^2\mathrm{\Delta }+V(|x|)?E$ in dimension $n\ge 2$, where $h,E>0$, and $V:[0,\infty )\to \mathbb{R}$ is $L^{\infty }$ and compactly supported. The weighted resolvent norm grows no faster than $\exp ?(C{h}^{?1})$, while an exterior weighted norm grows $～h^{?1}$. We introduce a new method based on the Mellin transform to handle the two-dimensional case.     

Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production

This paper deals with the chemotaxis-growth system: $u_t=\mathrm{\Delta }u?\mathrm{?}?(u\mathrm{?}v)+\mu u(1?u)$, $v_t=\mathrm{\Delta }v?v+w$, $\tau w_t+\delta w=u$ in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^3$ with zero-flux boundary conditions, where μ, δ, and τ are given positive parameters. It is shown that the solution $(u,v,w)$ exponentially stabilizes to the constant stationary solution $(1,\frac{1}{\delta },\frac{1}{\delta })$ in the norm of $L^{\infty }(\mathrm{\Omega })$ as $t\to \infty$ provided that $\mu >0$ and any given nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$ fulfills $u_0?0$, which extends the condition $\mu >\frac{1}{8{\delta }^2}$ in [8].     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.     

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.