Randi\'c energy and Randi\'c eigenvalues

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$ otherwise. The normalized signless Laplacian matrix $\mathcal{Q}$ is defined as $\mathcal{Q} =I+\bf{R}$, where $I$ is the identity matrix. The Randi\'c energy is the sum of absolute values of the eigenvalues of $\bf{R}$. In this paper, we find a relation between the normalized signless Laplacian eigenvalues of $G$ and the Randi\'c energy of its subdivided graph $S(G)$. We also give a necessary and sufficient condition for a graph to have exactly $k$ and distinct Randi\'c eigenvalues.Comment: 7 page

On Two Simple and Effective Procedures for High Dimensional Classification of General Populations

In this paper, we generalize two criteria, the determinant-based and trace-based criteria proposed by Saranadasa (1993), to general populations for high dimensional classification. These two criteria compare some distances between a new observation and several different known groups. The determinant-based criterion performs well for correlated variables by integrating the covariance structure and is competitive to many other existing rules. The criterion however requires the measurement dimension be smaller than the sample size. The trace-based criterion in contrast, is an independence rule and effective in the "large dimension-small sample size" scenario. An appealing property of these two criteria is that their implementation is straightforward and there is no need for preliminary variable selection or use of turning parameters. Their asymptotic misclassification probabilities are derived using the theory of large dimensional random matrices. Their competitive performances are illustrated by intensive Monte Carlo experiments and a real data analysis.Comment: 5 figures; 22 pages. To appear in "Statistical Papers

Testing the Sphericity of a covariance matrix when the dimension is much larger than the sample size

This paper focuses on the prominent sphericity test when the dimension $p$ is much lager than sample size $n$. The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when $p/n\rightarrow\infty, n\rightarrow\infty$ is well established in this paper. Meanwhile, John's test has been found to possess the powerful {\it dimension-proof} property, which keeps exactly the same limiting distribution under the null with any $(n,p)$-asymptotic, i.e. $p/n\rightarrow[0,\infty]$, $n\rightarrow\infty$. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison

Quasinonlocal coupling of nonlocal diffusions

We developed a new self-adjoint, consistent, and stable coupling strategy for nonlocal diffusion models, inspired by the quasinonlocal atomistic-to-continuum method for crystalline solids. The proposed coupling model is coercive with respect to the energy norms induced by the nonlocal diffusion kernels as well as the $L^2$ norm, and it satisfies the maximum principle. A finite difference approximation is used to discretize the coupled system, which inherits the property from the continuous formulation. Furthermore, we design a numerical example which shows the discrepancy between the fully nonlocal and fully local diffusions, whereas the result of the coupled diffusion agrees with that of the fully nonlocal diffusion.Comment: 28 pages, 3 figures, ams.or