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Let$C$_{$k$}$(p)$denote the group of the$k$-th powers (mod$p$)$p$a prime with ($k, p$−1)>1. A new elementary result for the least$k$-th power non-residue is given and the result is applied to finding a new elementary bound for the maximum number of consecutuve integers in any coset of$C$_{$k$}$(p)$.

The moment-entropy inequality shows that a contin- uous random variable with given second moment and maximal Shannon entropy must be Gaussian. Stam’s inequality shows that a continuous random variable with given Fisher information and minimal Shannon entropy must also be Gaussian. The Crame ́r- Rao inequality is a direct consequence of these two inequalities. In this paper the inequalities above are extended to Renyi entropy, p-th moment, and generalized Fisher information. Gen- eralized Gaussian random densities are introduced and shown to be the extremal densities for the new inequalities. An extension of the Crame ́r–Rao inequality is derived as a consequence of these moment and Fisher information inequalities.

The free-living SAR11 clade is a globally abundant group of oceanic Alphaproteobacteria, with small genome sizes and rich genomic A+T content. However, the taxonomy of SAR11 has become controversial recently. Some researchers argue that the position of SAR11 is a sister group to Rickettsiales. Other researchers advocate that SAR11 is located within free-living lineages of Alphaproteobacteria. Here, we use the natural vector representation method to identify the evolutionary origin of the SAR11 clade. This alignment-free method does not depend on any model assumptions. With this approach, the correspondence between proteome sequences and their natural vectors is one-to-one. After fixing a set of proteins, each bacterium is represented by a set of vectors. The Hausdorff distance is then used to compute the dissimilarity distance between two bacteria. The phylogenetic tree can be reconstructed based on these distances. Using our method, we systematically analyze four data sets of alphaproteobacterial proteomes in order to reconstruct the phylogeny of Alphaproteobacteria. From this we can see that the phylogenetic position of the SAR11 group is within a group of other free-living lineages of Alphaproteobacteria.