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 The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$.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

Muon g-2 Anomaly confronted with the higgs global data in the Left-Right Twin Higgs Models

We will examine the Left-Right Twin Higgs(LRTH) Models as a solution of muon g-2 anomaly with the background of the Higgs global fit data. In the calculation, the joint constrains from the theory, the precision electroweak data, the 125 GeV Higgs data, the leptonic flavor changing decay \mu \to e\gamma decays, and the constraints m_{\nu_R}>m_T>m_{W_H} are all considered. And with the small mass of the \phi^0, the direct searches from the $h\to \phi^0\phi^0$ channels can impose stringent upper limits on Br(h\to \phi^0\phi^0) and can reduce the allowed region of m_{\phi^0} and f. It is concluded that the muon g-2 anomaly can be explained in the region of 200 GeV \leq M\leq 500 GeV, 700 GeV \leq f\leq 1100 GeV, 13 GeV \leq m_{\phi^0}\leq 55 GeV, 100 GeV \leq m_{\phi^\pm}\leq 900 GeV, and m_{\nu_R}\geq 15 TeV after imposing all the constraints mentioned above.Comment: 18 pages, 4 figures, some typos modifie

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

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