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 $j=2\text{,}\dots \text{,}n-1$.     

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $PSL(2,\mathbb{Z})?\mathbb{H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma ?\mathbb{H}$ and odd and grotesque continued fractions, where $\Gamma ?{\mathbb{Z}}_{3}1{\mathbb{Z}}_3$ is the index two subgroup of $PSL(2,\mathbb{Z})$ generated by the order three elements $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill ?1\hfill & \hfill 1\hfill \end{array}\right)$, and having an ideal quadrilateral as fundamental domain.A similar connection between geodesics on $\Theta ?\mathbb{H}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $PSL(2,\mathbb{Z})$ generated by $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$.     

Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

One dimensional Dirac operators $L_{bc}\left(v\right)y=i\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?1\hfill \end{array}\right)\frac{dy}{dx}+v\left(x\right)y\text{,}y=\left(\begin{array}{c}\hfill y_1\hfill \\ \hfill y_2\hfill \end{array}\right)\text{,}x\in [0,\pi ]\text{,}$ considered with $L^2$-potentials $v\left(x\right)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill P\left(x\right)\hfill \\ \hfill Q\left(x\right)\hfill & \hfill 0\hfill \end{array}\right)$ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc$, the spectrum of the free operator $L_{bc}\left(0\right)$ is simple while the spectrum of $L_{bc}\left(v\right)$ is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval $[0,\pi ]$. Analogous results are obtained for regular but not strictly regular $bc$.     

Characterizations of perturbations of spectra of 2×2 upper triangular operator matrices

When $A\in B\left(H\right)$ and $B\in B\left(K\right)$ are given, we denote by $M_C$ the operator acting on the infinite dimensional separable Hilbert space $H\oplus K$ of the form $M_C=\left(\begin{array}{cc}\hfill A\hfill & \hfill C\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right)$. In this paper, we first give some necessary and sufficient conditions for $M_C$ to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some $C\in B(K,H)$, which extend the corresponding results in Cao et al. (2006) [4], Cao and Meng (2005) [5], Hwang and Lee (2001) [12] and Li and Du (2006) [15]. Then we present some counter-examples.     

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.     

Isolation number versus Boolean rank

Let $\mathbb{B}$ be the binary Boolean algebra. The Boolean rank, or factorization rank, of a matrix A in $M_{m\text{,}n}(\mathbb{B})$ is the smallest k such that A can be factored as an $m\times k$ times a $k\times n$ matrix. The isolation number of a matrix, A, is the largest number of entries equal to 1 in the matrix such that no two ones are in the same row, no two ones are in the same column, and no two ones are in a submatrix of A of the form $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$. It is known that the isolation number of A is always at most the Boolean rank. This paper investigates for each k, if the isolation number of A is k what are some of the possible values of the Boolean rank of A.     

Competitive exclusion for a two-species chemotaxis system with two chemicals

In this paper we consider the following competitive two-species chemotaxis system with two chemicals in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$ with $n\ge 1$, where ${\chi }_i\ge 0$, $a_i\ge 0$ and ${\mu }_i>0$ $(i=1,2)$. For the case $a_1>1>a_2\ge 0$, it will be proved that if ${\chi }_1{\chi }_2<{\mu }_1{\mu }_2$, ${\chi }_1\le a_1{\mu }_1$ and ${\chi }_2<{\mu }_2$, then the initial–boundary value problem with homogeneous Neumann boundary condition admits a unique global bounded solution and $(u,v,w,z)\to (0,1,1,0)$ uniformly on $\bar{\Omega }$ as $t\to \infty$.