We consider harmonic Bergman functions, i.e., functions which are harmonic and $p$-th integrable. In the present paper, we shall show that when $10p0\infty$, every harmonic Bergman function on a smooth domain is represented as a series using the harmonic Bergman kernel. This representation is called an atomic decomposition.

For $n$-bridge decompositions of links in $S^3$, we propose a practical method to ensure that the Hempel distance is at least two.

Let $\tilde{S}$ be an analytically finite Riemann surface of type $(p,n)$ with $3p+n>3$. Let $x\in \tilde{S}$ and $S=\tilde{S}\backslash \{x\}$. Let $\mbox{Mod}_S^x$ denote the $x$-pointed mapping class group of $S$ and $\mbox{Mod}_{\tilde{S}}$ the mapping class group of $\tilde{S}$. Then the natural projection $J:T(S)\rightarrow T(\tilde{S})$ between Teichmüller spaces induces a group epimorphism $I:\mbox{Mod}_S^x\rightarrow \mbox{Mod}_{\tilde{S}}$. It is well known that for a given Teichmüller disk $\tilde{\Delta}$ in $T(\tilde{S})$, there is a family $\mathscr{F}(\tilde{\Delta})$ of Teichmüller disks $\Delta(z)$ in $T(S)$ parametrized by a hyperbolic plane. If $\tilde{\Delta}$ is invariant under a hyperbolic mapping class $\tilde{\theta}$, then all known hyperbolic mapping classes $\theta\in \mbox{Mod}_S^x$ for which $I(\theta)=\tilde{\theta}$ stem from the construction of $\mathscr{F}(\tilde{\Delta})$. We show that if $\tilde{\theta}$ is represented by a product of Dehn twists along two filling simple closed geodesics, then there exist infinitely many hyperbolic mapping classes $\gamma\in \mbox{Mod}_S^x$ with $I(\gamma)=\tilde{\theta}$ so that their invariant Teichmüller disks are not members of $\mathscr{F}(\tilde{\Delta})$. The result contrasts with the original pattern established by I. Kra.

Under certain conditions we show that such systems are weakly permanent or permanent and for two solutions $u$ and $v$ of such systems, the difference $u-v$ tends to zero at the infinity. Our results give generalizations of previous ones.

We deal with the problem of classifying a new observation vector into one of two known multivariate normal populations. Linear discriminant analysis (LDA) is now widely available. However, for high-dimensional data classification problem, due to the small number of samples and the large number of variables, classical LDA has poor performance corresponding to the singularity and instability of the sample covariance matrix. Recently, Xu et al. suggested modified linear discriminant analysis (MLDA). This method is based on the shrink type estimator of the covariance matrix derived by Ledoit and Wolf. This estimator was proposed under the asymptotic framework ${\rm A}_0:n=O(p)$ and $p=O(n)$ when $p\to\infty$. In this paper, we propose a shrink type estimator under more flexible high-dimensional framework. Using this estimator, we define the new MLDA. Through the numerical simulation, the expected correct classification rate of our MLDA is larger than the ones of other discrimination methods when $p>n$. In addition, we consider the limiting value of the expected probability of misclassification (EPMC) under some assumptions.

By means of the transformations of Sears and Watson for the terminating balanced $_4\phi_3$-series, we investigate the two terminating $q$-Kampé de Fériet series $\Phi^{1:3;\lam}_{1:2;\mu}$ and $\Phi^{2:2;\lam}_{2:1;\mu}$. Several reduction and summation formulae are established. They extend the corresponding known results about the double $q$-Clausen series.

As a mathematical model of $n$-membered ringed hydrocarbon molecules, we consider closed chains in $\R^{3}$. Assume that the bond angle $\theta$ satisfies $\frac{n-4}{n-2}\pi0\theta0\frac{n-2}{n}\pi$ when $n=5,6,7$, and that $\frac{5}{7}\pi \leq \theta0 \frac{3}{4}\pi$ when $n=8$. Then the configuration space $C_{n}$ of the model is homeomorphic to $(n-4)$-dimensional sphere $S^{n-4}$. By this result, it is possible for approximating larger macrocyclic molecules by smaller ones to be more widely applied.

We consider nonnegative solutions of the initial-boundary value problems in cone domains for the reaction-diffusion systems with inhomogeneous terms dependent on space-time coordinates. In this paper we show the condition for the existence of global solutions. Our conditions for the global existence are optimal in view of our nonexistence results in 2009.