Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number field generated by coefficients of f in Qbar. Let A^d be the dimension-d affine space over Qbar, identified with the space of coefficients of degree-d monic polynomials. Let NP(f mod P) denote the p-adic Newton polygon of L(f mod P;T). Let HP(A^d) denote the p-adic Hodge polygon of A^d. We prove that there is a Zariski dense open subset U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_{p-->oo} NP(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.

Hui June Zhu

The structural and electronic properties of fluorine (F)-doped BN nanotubes (BNNTs) are studied using density functional methods. Our results indicate that F atoms prefer to substitute N atoms, resulting in substantial changes of BN layers. However, F substitutional doping results in no shallow impurity states. The adsorption of F atoms on B sites is more stable than that on N sites. BNNTs with adsorbed F atoms are p-type semiconductors, suggesting the electronic conduction in F-doped multiwalled BNNTs with large conductivity observed experimentally might be of p-type due to the adsorbed F atoms, but not n-type as supposed before.

H. J. Xiang Jinlong Yang J. G. Hou Qingshi Zhu

Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphsim $f$. We define the u-pressure $P^u(f, \varphi)$ of $f$ at a continuous function $\varphi$ via the dynamics of $f$ on local unstable leaves. A variational principle for unstable pressure $P^u(f, \varphi)$, which states that $P^u(f, \varphi)$ is the supremum of the sum of the unstable entropy and the integral of $\varphi$ taken over all invariant measures, is obtained. U-equilibrium states at which the supremum in the variational principle attains and their relation to Gibbs u-states are studied. Differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Fr\'{e}chet differentiability and their relations to u-equilibrium states, are also considered.

Huyi Hu Weisheng Wu Yujun Zhu

This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\Delta(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable.

Ringi Kim Seog-Jin Kim Xuding Zhu

We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.

Jürgen Herzog Guangjun Zhu

The generalized Tur\'{a}n number $ex(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $F$ be a linear forest consisting of $k$ paths of orders $\ell_1,\ell_2,...,\ell_k$. In this paper, by characterizing the structure of the $F$-free graph with large minimum degree, we determine the value of $ex(n,K_s,F)$ for $n=\Omega\left(|F|^s\right)$ and $k\geq 2$ except some $\ell_i=3$, and the corresponding extremal graphs. The special case when $s=2$ of our result improves some results of Bushaw and Kettle (2011) and Lidick\'{y} et al. (2013) on the classical Tur\'{a}n number for linear forests.

Xiutao Zhu Yaojun Chen

We study the $L^p$-mean distortion functionals, \[{\cal E}_p[f] = \int_\mathbb Y K^p_f(z) \; dz, \] for Sobolev homeomorphisms $f: \overline{\mathbb Y}\xrightarrow{\rm onto} \overline{\mathbb X}$ where $\mathbb X$ and $\mathbb Y$ are bounded simply connected Lipschitz domains, and $f$ coincides with a given boundary map $f_0 \colon \partial \mathbb Y \to \partial \mathbb X$. Here, $K_f(z)$ denotes the pointwise distortion function of $f$. It is conjectured that for every $1 < p < \infty$, the functional $\mathcal{E}_p$ admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique.

Yizhe Zhu

Let $f$ and $g$ be functions, not identically zero, in the Fock space $F^2$ of $C_n$. We show that the product $T_fT_{\bar g}$ of Toeplitz operators on $F^2$ is bounded if and only if $f(z)=e^{q(z)}$ and $g(z)=ce^{-q(z)}$, where $c$ is a nonzero constant and $q$ is a linear polynomial.

Hon Rae Cho Jong-Do Park Kehe Zhu

We compute effective bounds for $\alpha(n)$, the Fourier coefficients of Ramunujan's mock theta function $f(q)$ utilizing a finite algebraic formula due to Bruinier and Schwagenscheidt. We then use these bounds to prove two conjectures of Hou and Jagadeesan on the convexity and maximal multiplicative properties of the even and odd partition rank counting functions.

Kevin Gomez Eric Zhu

In this paper we describe the irreducible decomposition of the facet ideal $\F(\Delta_{m,n})$ of the chessboard complex $\Delta_{m,n}$ with $n\geq m$. We also provide some lower bounds for depth and regularity of the facet ideal $\F(\Delta_{m,n})$. When $m\leq 3$, we prove that these lower bounds can be obtained.

Chengyao Jiang Yakun Zhao Hong Wang Guangjun Zhu