The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<infty)$, then the number of reducible $M$-ideals does not exceed of $frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The...

The genus Mycobacterium contains 188 species including several major human pathogens as well as numerous other environmental species. We report here comprehensive phylogenomics and comparative genomic analyses on 150 genomes of Mycobacterium species to understand their interrelationships. Phylogenetic trees were constructed for the 150 species based on 1941 core proteins for the genus Mycobacte...