A proof of the Tn conjecture: Centralizers,Jacobians and spectrally arbitrary sign patterns

We prove the so-called $T_n$ conjecture: for every real-monic polynomial $p(x)$ of degree $n?2$ there exists an n by n matrix with sign pattern$T_n=\left[\begin{array}{ccccc}\hfill -\hfill & \hfill +\hfill & \hfill 0\hfill & \hfill ?\hfill & \hfill 0\hfill \\ \hfill -\hfill & \hfill 0\hfill & \hfill ?\hfill & \hfill \hfill & \hfill ?\hfill \\ \hfill 0\hfill & \hfill ?\hfill & \hfill ?\hfill & \hfill ?\hfill & \hfill 0\hfill \\ \hfill ?\hfill & \hfill \hfill & \hfill ?\hfill & \hfill 0\hfill & \hfill +\hfill \\ \hfill 0\hfill & \hfill ?\hfill & \hfill 0\hfill & \hfill -\hfill & \hfill +\hfill \end{array}\right]\text{,}$whose characteristic polynomial is $p(x)$. The proof converts the problem of determining the nonsingularity of a certain Jacobi matrix to the problem of proving the non-existence of a nonzero matrix B that commutes with a nilpotent matrix with sign pattern $T_n$ and has zeros in positions $(1\text{,}1)$, and $(j+1\text{,}j)$ for $j=2\text{,}\dots \text{,}n-1$.     

W01,1 solutions in some borderline cases of Calderon–Zygmund theory

In this paper we study the existence of $W_0^{1,1}(\Omega )$ distributional solutions of Dirichlet problems whose simplest example is$\{\begin{array}{cc}?\mathrm{div}\left({|\mathrm{?}u|}^{p?2}\mathrm{?}u\right)=f(x),\hfill & \text{in}\Omega ;\hfill \\ u=0,\hfill & \text{on}?\Omega .\hfill \end{array}$     

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

The quasilinear chemotaxis–haptotaxis system

$\{\begin{array}{cc}{u}_t=\mathrm{?}?(D(u)\mathrm{?}u)?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)\hfill & \hfill \\ \phantom{u_t=}+\mu u(1?u?w),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}$

is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Here $\chi >0$, $\xi >0$ and $\mu >0$, $D(u)\ge c_D{u}^{m?1}$ for all $u>0$ with some $c_D>0$ and $D(u)>0$ for all $u\ge 0$. It is shown that if the ratio $\frac{\chi }{\mu }$ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.     

Hyperinvariant subspaces for operators having a compact part

It is well known that if $T=A\oplus B$, where A is compact, then T has a nontrivial hyperinvariant subspace. In this paper, we try to solve the hyperinvariant subspace problem for operators which have a compact part. Our main result is that if A is compact, then either $\left(\begin{array}{cc}A\hfill & \hfill ?\hfill \\ 0\hfill & \hfill B\hfill \end{array}\right)$ or $\left(\begin{array}{cc}A\hfill & \hfill 0\hfill \\ ?\hfill & \hfill B\hfill \end{array}\right)$ has a nontrivial hyperinvariant subspace.     

A characterization of well-posedness for abstract Cauchy problems with finite delay

Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay,

$\{\begin{array}{cc}{u}^{\prime }(t)=Au(t)+F{u}_t,\hfill & \hfill t>0;\hfill \\ u(0)=x;\hfill \\ u(t)=?(t),\hfill & \hfill ?r\le t<0\hfill \end{array},$

solely in terms of a strongly continuous one-parameter family ${\{G(t)\}}_{t\ge 0}$ of bounded linear operators that satisfy the functional equation

$G(t+s)x=G(t)G(s)x+\underset{?r}{\overset{0}{\int }}G(t+m)[SG(s+?)x](m)dm$

for all $t,s\ge 0,x\in X$. In case $F\equiv 0$ this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the $C_0$-semigroup generated by A.