kkth power residue chains of global fields

In 1974, Vegh proved that if $k$ is a prime and $m$ a positive integer, there is an $m$ term permutation chain of $k$th power residue for infinitely many primes [E.Vegh, $k$th power residue chains, J.Number Theory, 9(1977), 179-181]. In fact, his proof showed that $1,2,2^2,...,2^{m-1}$ is an $m$ term permutation chain of $k$th power residue for infinitely many primes. In this paper, we prove that for any "possible" $m$ term sequence $r_1,r_2,...,r_m$, there are infinitely many primes $p$ making it an $m$ term permutation chain of $k$th power residue modulo $p$, where $k$ is an arbitrary positive integer [See Theorem 1.2]. From our result, we see that Vegh's theorem holds for any positive integer $k$, not only for prime numbers. In fact, we prove our result in more generality where the integer ring $\Z$ is replaced by any $S$-integer ring of global fields (i.e. algebraic number fields or algebraic function fields over finite fields).Comment: 4 page

A systematic literature review of cloud computing in eHealth

Cloud computing in eHealth is an emerging area for only few years. There needs to identify the state of the art and pinpoint challenges and possible directions for researchers and applications developers. Based on this need, we have conducted a systematic review of cloud computing in eHealth. We searched ACM Digital Library, IEEE Xplore, Inspec, ISI Web of Science and Springer as well as relevant open-access journals for relevant articles. A total of 237 studies were first searched, of which 44 papers met the Include Criteria. The studies identified three types of studied areas about cloud computing in eHealth, namely (1) cloud-based eHealth framework design (n=13); (2) applications of cloud computing (n=17); and (3) security or privacy control mechanisms of healthcare data in the cloud (n=14). Most of the studies in the review were about designs and concept-proof. Only very few studies have evaluated their research in the real world, which may indicate that the application of cloud computing in eHealth is still very immature. However, our presented review could pinpoint that a hybrid cloud platform with mixed access control and security protection mechanisms will be a main research area for developing citizen centred home-based healthcare applications

The genus fields of Artin-Schreier extensions

Let $q$ be a power of a prime number $p$. Let $k=\mathbb{F}_{q}(t)$ be the rational function field with constant field $\mathbb{F}_{q}$. Let $K=k(\alpha)$ be an Artin-Schreier extension of $k$. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of $K$ . Using these results we study the $p$-part of the ideal class group of the integral closure of $\mathbb{F}_{q}[t]$ in $K$. And we also give an analogy of R$\acute{e}$dei-Reichardt's formulae for $K$.Comment: 9 pages, Corrected typo

Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model

Let \mathbf{Y}=\mathbf{X}\bolds{\Theta}\mathbf{Z}'+\bolds{\mathcal {E}} be the growth curve model with \bolds{\mathcal{E}} distributed with mean $\mathbf{0}$ and covariance \mathbf{I}_n\otimes\bolds{\Sigma}, where \bolds{\Theta}, \bolds{\Sigma} are unknown matrices of parameters and $\mathbf{X}$, $\mathbf{Z}$ are known matrices. For the estimable parametric transformation of the form \bolds {\gamma}=\mathbf{C}\bolds{\Theta}\mathbf{D}' with given $\mathbf{C}$ and $\mathbf{D}$, the two-stage generalized least-squares estimator \hat{\bolds \gamma}(\mathbf{Y}) defined in (7) converges in probability to \bolds\gamma as the sample size $n$ tends to infinity and, further, \sqrt{n}[\hat{\bolds{\gamma}}(\mathbf{Y})-\bolds {\gamma}] converges in distribution to the multivariate normal distribution \ma thcal{N}(\mathbf{0},(\mathbf{C}\mathbf{R}^{-1}\mathbf{C}')\otimes(\mat hbf{D}(\mathbf{Z}'\bolds{\Sigma}^{-1}\mathbf{Z})^{-1}\mathbf{D}')) under the condition that $\lim_{n\to\infty}\mathbf{X}'\mathbf{X}/n=\mathbf{R}$ for some positive definite matrix $\mathbf{R}$. Moreover, the unbiased and invariant quadratic estimator \hat{\bolds{\Sigma}}(\mathbf{Y}) defined in (6) is also proved to be consistent with the second-order parameter matrix \bolds{\Sigma}.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ128 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A Bayesian marked spatial point processes model for basketball shot chart

The success rate of a basketball shot may be higher at locations where a player makes more shots. For a marked spatial point process, this means that the mark and the intensity are associated. We propose a Bayesian joint model for the mark and the intensity of marked point processes, where the intensity is incorporated in the mark model as a covariate. Inferences are done with a Markov chain Monte Carlo algorithm. Two Bayesian model comparison criteria, the Deviance Information Criterion and the Logarithm of the Pseudo-Marginal Likelihood, were used to assess the model. The performances of the proposed methods were examined in extensive simulation studies. The proposed methods were applied to the shot charts of four players (Curry, Harden, Durant, and James) in the 2017--2018 regular season of the National Basketball Association to analyze their shot intensity in the field and the field goal percentage in detail. Application to the top 50 most frequent shooters in the season suggests that the field goal percentage and the shot intensity are positively associated for a majority of the players. The fitted parameters were used as inputs in a secondary analysis to cluster the players into different groups