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Sporulating bacteria need to temporally coordinate DNA replication, chromosome partitioning and sporulation initiation. Recent work has shown that one aspect of this coordination lies with the interdependent subcellular localization of two proteins, Spo0J and Soj, and in the Spo0J-dependent spatial oscillation of Soj. Asymptotics are obtained for the number Buy Tiffany Jewelry Cheap of n × n symmetric non-negative integer matrices subject to the following constraints: (i) each row sum Tiffany Outlet Online Shop is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled graphs. Assume that A is a liner transformation from Cn into Cn. The gap of A is defined as the gap between two subspaces GA and G0, the graph of A and the graph of zero. We prove that the gap of A is θ(A) = ‖A‖/√1 + ‖A‖2. Here ‖A‖ is the uniform norm of A. A binary Byzantine agreement algorithm can be extended to produce a multivalued Byzantine agreement algorithm. The resulting multivalued algorithm is cheaper than previously published algorithms when the cost of transmitting values from the multivalued domain is significant.