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 The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$.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

Composition-Diamond Lemma for Non-associative Algebras over a Commutative Algebra

We establish the Composition-Diamond lemma for non-associative algebras over a free commutative algebra. As an application, we prove that every countably generated non-associative algebra over an arbitrary commutative algebra $K$ can be embedded into a two-generated non-associative algebra over $K$.Comment: 10 page

AdS Black Hole with Phantom Scalar Field

In this paper, we present an AdS black hole solution with Ricci flat horizon in Einstein-phantom scalar theory. The phantom scalar fields just depend on the transverse coordinates $x$ and $y$, and which are parameterized by the parameter $\alpha$. We study the thermodynamics of the AdS phantom black hole. Although its horizon is a Ricci flat Euclidean space, we find that the thermodynamical properties of the black hole solution are qualitatively same as those of AdS Schwarzschild black hole. Namely there exists a minimal temperature, the large black hole is thermodynamically stable , while the smaller one is unstable, so there is a so-called Hawking-Page phase transition between the large black hole and the thermal gas solution in the AdS spacetime in Poincare coordinates. We also calculate the entanglement entropy for a strip geometry dual to the AdS phantom black holes and find that the behavior of the entanglement entropy is qualitatively the same as that of the black hole thermodynamical entropy.Comment: 6 pages, 8 figure

The Effects of Outdoor Air Pollutants on the Costs of Stroke Hospitalizations in China

Stroke, the most frequent cause of severe disability and the second cause of death among adults in the world, brings tremendous mental and economic burden to patients and their families. Emerging evidence indicates that the air pollution mixture contributes to strokes. Knowing the relationship between the air pollution and the hospital costs of stroke can help us predict the costs due to air pollution, provide grounds for the allocation of medical insurance funds, and provide better working arrangements for CDC. However, few studies have examined this connection. We used time series analysis with a generalized additive model to estimate the associations between ambient air pollutions and hospital costs between the period of 2015–2017. We were surprised to find that although same-day air pollutions were positively associated with stroke mortality hospital costs were found to have a negatively association. Suggestive evidence of an association between fine particles and the costs of stroke were found: more serious air pollution increases the risk of stroke, but has a dampening effect on hospital costs. This study is the first step in optimizing medical resources, which is essential for policy making, service planning, and cost-effectiveness analysis of new therapeutic strategies