Ramanujan grammar and Cayley trees. (Grammaire de Ramanujan et arbres de Cayley.)
Let be the Ramsey number of the two graphs and . For let be the double star given by , and . We determine
The affine group of a local field acts on the tree (the Bruhat-Tits building of ) with a fixed point in the space of ends . More generally, we define the affine group of any homogeneous tree as the group of all automorphisms of with a common fixed point in , and establish main asymptotic properties of random products in : (1) law of large numbers and central limit theorem; (2) convergence to and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary...
For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H. Bart, A. P. M. Wagelmans (2000). The proof involves elements from integer programming and employs Farkas' lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred...