We prove the positivity conjecture for all skew-symmetric cluster algebras.

A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of "natural" bases in them. One of the approaches to this problem was developed several years ago by P. Sherman -A. Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of K. Lee -R. Schiffler and D. Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.

Eight bacterial strains identified as P1, P2, Y1, Y2, W1, W2, G, and R were isolated from a longterm laboratory culture of the green alga Chlorella ellipsoidea. Although it is unknown how these bacterial strains have been maintained with the C. ellipsoidea culture, all appeared to promote the growth of C. ellipsoidea. Co-inoculation of each bacterial strain with C. ellipsoidea resulted in 0.5-3 times greater algal growth than that of C. ellipsoidea alone. The most effective bacterium (i.e., strain P1) was selected and further characterized. Biochemical analysis and transmission electron microscopy revealed that strain P1 is closely related to the genus Brevundimonas. Sequence analysis of the 16S rRNA of strain P1 showed 99.9 and 99.4% nucleotide sequence identity to that of B. nasdae and B. vesicularis, respectively. In addition to the growth promotion of C. ellipsoidea by strain P1, the growth of strain P1 was also significantly enhanced by co-culturing with C. ellipsoidea, indicating a symbiotic relationship between the bacterium and alga. Scanning electron microscopy showed the direct adhesion of strain P1 cells to the surface of C. ellipsoidea cells, as well as the development of abundant crinkles on the surface of co-cultured C. ellipsoidea cells.

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