On higher order boundary derivatives of an analytic self-map of the unit disk

Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and sufficient conditions are given for the existence of a Schur-class function with the prescribed nontangential boundary expansion $f\left(z\right)=s_0+s_1(z?t_0)+?+s_{{}_N}{(z?t_0)}^N+o\left({|z?t_0|}^N\right)$ at a given point $t_0$ on the unit circle.     

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.     

Normal family of meromorphic functions sharing holomorphic functions and the converse of the bloch principle

In 1996, C. C. Yang and P. C. Hu [8] showed that: Let f be a transcendental meromorphic function on the complex plane, and a = 0 be a complex number; then assume that n ≥ 2, n_1, ···, n_k are nonnegative integers such that n_1+ ··· + n_k ≥1; thus f~n(f′)~(n_1)···(f~(k))~(n_k)-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥ 2. Namely, we prove that f~n(f′)~(n_1)···(f~(k))~(n_k)-a(z)has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k ≥ 2, and a(z) ≡ 0 is a small function of f and n ≥ 2, n_1, ···, n_k are nonnegative integers satisfying n1+ ··· + n k ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by J. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y.Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloch's principle.     

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0\in F_2\left[x\right]$. If $f_0$ is of degree $n=2^l?m$, where $m$ is odd and $l$ is a nonnegative integer, after an initial finite sequence of polynomials $f_0,f_1,\dots ,f_s$, with $s\le l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.     

Pointwise asymptotic behavior of modulated periodic reaction–diffusion waves

By working with the periodic resolvent kernel and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction–diffusion equations. With our linearized estimates together with a nonlinear iteration scheme developed by Johnson–Zumbrun, we obtain $L^p$-behavior ($p?1$) of a nonlinear solution to a perturbation equation of a reaction–diffusion equation with respect to initial data in $L^1\cap H^2$ recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations $|u_0|?E_0{e}^{?{|x|}^2/M}$, ${|u_0|}_{H^2}?E_0$ and $|u_0|?E_0{(1+|x|)}^{?r}$, $r>2$, ${|u_0|}_{H^2}?E_0$ respectively, $E_0>0$ sufficiently small and $M>1$ sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.