Contents 1. Introduction 1 2. Preliminaries 2 3. Coarse disjoint unions 3 4. Special ultrametric spaces 5 5. Universal spaces 6 6. Ultrametric groups as universal spaces 7 References 9Abstract. Dranishnikov and Zarichnyi constructed a universal space in the coarse category of spaces of bounded geometry of asymptotic dimension 0. In this paper we construct universal spaces in the coarse category of separable (respectively, proper) metric spaces of asymptotic dimension 0. Our methods provide an alternative proof of Dranishnikov-Zarichnyi result.

In this paper, we define the multi-Stirling numbers of the first kind by means of the multiple logarithm and as a generalization of the Stirling numbers of the first kind. Then we introduce two additional special numbers by using the multiple logarithm, namely the modified multi-Bernoulli numbers as a generalization of the higher-order Bernoulli numbers and the multi-Lah numbers of type 2 as a generalization of the Lah numbers. For both of these special numbers, we derive explicit expressions for them which involve the multi-Stirling numbers of the first kind.

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