Asteroseismology of DAV white dwarf stars and G29-38

Asteroseismology is a powerful tool to detect the inner structure of stars. It is also widely used to study white dwarfs. In this paper, we discuss the asteroseismology work of DAV stars. The detailed period to period fitting method is fully discussed, including the reliability to detect the inner structure of DAV stars. If we assume that all observed modes of some DAV star are the $l$ = 1 ones, the errors of model fitting will be always great. If we assume that the observed modes are composed of $l$ = 1 and 2 modes, the errors of model fitting will be small. However, there will be modes identified as $l$ = 2 without quintuplets observed. G29-38 has been observed spectroscopically and photometrically for many years. Thompson et al. (2008) made $l$ identifications for the star through limb darkening effect. With eleven known $l$ modes, we also do the asteroseismology work for G29-38, which reduces the blind $l$ fittings and is a fair choice. Unfortunately, our two best-fitting models are not in line with the previous atmospheric results. Based on factors of only a few modes observed, stability and identification of eigenmodes, identification of spherical degrees, construction of physical and realistic models and so on, detecting the inner structure of DAV stars by asteroseismology needs further development.Comment: 7pages, 1figur

Security flaw of counterfactual quantum cryptography in practical setting

Recently, counterfactual quantum cryptography proposed by T. G. Noh [Phys. Rev. Lett. 103, 230501 (2009)] becomes an interesting direction in quantum cryptography, and has been realized by some researchers (such as Y. Liu et al's [Phys. Rev. Lett. 109, 030501 (2012)]). However, we find out that it is insecure in practical high lossy channel setting. We analyze the secret key rates in lossy channel under a polarization-splitting-measurement attack. Analysis indicates that the protocol is insecure when the loss rate of the one-way channel exceeds $50$.Comment: In this version, we changed some descriptio

Asteroseismology of DAV star EC14012-1446, mode identification and model fittings

EC14012-1446 was observed by Handler et al. in April 2004, June 2004, May 2005, and April 2007, and by Provencal et al. in 2008. We review the observations together and obtain 34 independent frequencies. According to the frequency splitting and the asymptotic period spacing law, we identify 6 $l$=1 modes, 4 $l$=2 modes, 5 $l$=3 modes, 10 $l$=1 or 2 modes. Grids of white dwarf models are generated by WDEC with H, He, C, O diffusion in a four-parameter space. The core compositions are directly from white dwarf models generated by MESA. The best-fitting model has $M_{*}$=0.710\,$M_{\odot}$, $T_{\rm eff}$=12200\,K, log($M_{\rm He}/M_{*}$)=-2.5, log($M_{\rm H}/M_{*}$)=-7.0, log\,$g$=8.261, and $\phi$=3.185\,s. There are 4, 2, and 1 modes identified as trapped in H envelope for observed $l$=1, 2, and 3 modes, respectively. Trapped modes jump the queue of uniform period spacing.Comment: 9 pages, 6 figures, 7 tables, accepted by MNRAS 10 July 201

Rotational splitting and asteroseismic modelling of the delta Scuti star EE Camelopardalis

According to the rotational splitting law of g modes, the frequency spectra of EE Cam can be disentangled only with oscillation modes of $\ell$ = 0, 1, and 2. Fifteen sets of rotational splits are found, containing five sets of $\ell=1$ multiplets and ten sets of $\ell=2$ multiplets. The rotational period of EE Cam is deduced to be $P_{\rm rot}$ = $1.84_{-0.05}^{+0.07}$ days. When we do model fittings, we use two nonradial oscillation modes ($f_{11}$ and $f_{32}$) and the fundamental radial mode $f_{1}$. The fitting results show that $\chi^{2}$ of the best-fitting model is much smaller than those of other theoretical models. The physical parameters of the best-fitting model are $M$ = 2.04 $M_{\odot}$, $Z=0.028$, $T_{\rm eff}$ = 6433 K, $\log L/L_{\odot}$ = 1.416, $R$ = 4.12 $R_{\odot}$, $\log g$ = 3.518, and $\chi^{2}$ = 0.00035. Furthermore, we find $f_{11}$ and $f_{32}$ are mixed mode, which mainly characterize the features of the helium core. The fundamental radial mode $f_{1}$ mainly restrict the features of the stellar envelope. Finally, the acoustic radius $\tau_{0}$ and the period separation $\Pi_{0}$ are determined to be 5.80 hr and 463.7 s respectively, and the size of the helium core of EE Cam is estimated to be $M_{\rm He}$ = 0.181 $M_{\odot}$ and $R_{\rm He}$ = 0.0796 $R_{\odot}$.Comment: 9 pages, 5 figures, 4 tables. accepted for publication in The Astrophysical Journa

A direct blowing-up and rescaling argument on the fractional Laplacian equation

In this paper, we develop a direct {\em blowing-up and rescaling} argument for a nonlinear equation involving the fractional Laplacian operator. Instead of using the conventional extension method introduced by Caffarelli and Silvestre, we work directly on the nonlocal operator. Using the integral defining the nonlocal elliptic operator, by an elementary approach, we carry on a {\em blowing-up and rescaling} argument directly on nonlocal equations and thus obtain a priori estimates on the positive solutions for a semi-linear equation involving the fractional Laplacian. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators

A direct method of moving planes for the fractional Laplacian

In this paper, we develop a direct method of moving planes for the fractional Laplacian. Instead of conventional extension method introduced by Caffarelli and Silvestre, we work directly on the non-local operator. Using the integral defining the fractional Laplacian, by an elementary approach, we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space, such as strong maximum principles for anti-symmetric functions, narrow region principles, and decay at infinity. Then, using a simple example, a semi-linear equation involving the fractional Laplacian, we illustrate how this new method of moving planes can be employed to obtain symmetry and non-existence of positive solutions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities.Comment: 36 page

A General Construction of Binary Sequences with Optimal Autocorrelation

A general construction of binary sequences with low autocorrelation are considered in the paper. Based on recent progresses about this topic and this construction, several classes of binary sequences with optimal autocorrelation and other low autocorrelation are presented

Direct Method of Moving Spheres on Fractional Order Equations

In this paper, we introduce a direct method of moving spheres for the nonlocal fractional Laplacian $(-\triangle)^{\alpha/2}$ for $0<\alpha<2$, in which a key ingredient is the narrow region maximum principle. As immediate applications, we classify the non-negative solutions for a semilinear equation involving the fractional Laplacian in $\mathbb{R}^n$; we prove a non-existence result for prescribing $Q_{\alpha}$ curvature equation on $\mathbb{S}^n$; then by combining the direct method of moving planes and moving spheres, we establish a Liouville type theorem on the half Euclidean space. We expect to see more applications of this method to many other equations involving non-local operators

On Menon-Sury's identity with several Dirichlet characters

The Menon-Sury's identity is as follows: \begin{equation*} \sum_{\substack{1 \leq a, b_1, b_2, \ldots, b_r \leq n\\\mathrm{gcd}(a,n)=1}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)=\varphi(n) \sigma_r(n), \end{equation*} where $\varphi$ is Euler's totient function and $\sigma_r(n)=\sum_{d\mid n}{d^r}$. Recently, Li, Hu and Kim \cite{L-K} extended the above identity to a multi-variable case with a Dirichlet character, that is, they proved \begin{equation*} \sum_{\substack{a\in\Bbb Z_n^\ast \\ b_1, \ldots, b_r\in\Bbb Z_n}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)\chi(a)=\varphi(n)\sigma_r{\left(\frac{n}{d}\right)}, \end{equation*} where $\chi$ is a Dirichlet character modulo $n$ and $d$ is the conductor of $\chi$. In this paper, we explicitly compute the sum \begin{equation*}\sum_{\substack{a_1, \ldots, a_s\in\Bbb Z_n^\ast \\ b_1, ..., b_r\in\Bbb Z_n}}\gcd(a_1-1, \ldots, a_s-1,b_1, \ldots, b_r, n)\chi_{1}(a_1) \cdots \chi_{s}(a_s).\end{equation*} where $\chi_{i} (1\leq i\leq s)$ are Dirichlet characters mod $n$ with conductor $d_i$. A special but common case of our main result reads like this : \begin{equation*}\sum_{\substack{a_1, \ldots, a_s\in\Bbb Z_n^\ast \\ b_1, ..., b_r\in\Bbb Z_n}}\gcd(a_1-1, \ldots, a_s-1,b_1, \ldots, b_r, n)\chi_{1}(a_1) \cdots \chi_{s}(a_s)=\varphi(n)\sigma_{s+r-1}\left(\frac{n}{d}\right)\end{equation*} if $d$ and $n$ have exactly the same prime factors, where $d={\rm lcm}(d_1,\ldots,d_s)$ is the least common multiple of $d_1,\ldots,d_s$. Our result generalizes the above Menon-Sury's identity and Li-Hu-Kim's identity.Comment: 12 page

Some Liouville theorems for the fractional Laplacian

In this paper, we prove the following result. Let $\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$\left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\ \displaystyle\underset{|x| \to \infty}{\underline{\lim}} \frac{u(x)}{|x|^{\gamma}} \geq 0 , \end{array} \right.$$ for some $0 \leq \gamma \leq 1$ and $\gamma < \alpha$. Then $u$ must be constant throughout $\mathbb{R}^n$. This is a Liouville Theorem for $\alpha$-harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using potential theory. The other is by Fourier analysis as a corollary of the fact that the only $\alpha$-harmonic functions are affine.Comment: 19 page