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The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict ∞-categories. This result is central to the homotopy theory of strict ∞-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely ∞-categorical methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the most popular book of recreational mathematics published in the second half of the 18th century: The Nouvelles Récréations by Guyot. We indicate the motivations of the author, a simple postman, and the conditions which led him to write this book. We describe the spirit of the book and the public at which it aims. The success of the Nouvelles Récréations illustrates the rise of a science in polite society whose main goal is to amaze and amuse. Then, we examine the place of mathematics in this project and analyze the repertoire of problems and tricks. We focus on problems of combinatorics proposed by Guyot, like anagrams and card shuffles, which inspired some real mathematical work on the part of Monge and Gergonne. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we first characterize the subsets of the Bruhat–Tits tree of , K a complete valued field, that are the sets of fixed points of a subgroup G of . When G is irreducible, the are the “strips” in the tree. We then evaluate the form of the strip , in particular in terms of algebraic invariants of the group G. We give two applications to representation theory, one about a new generalization of Ribet's lemma on extensions, the other about the trace-convergence of a sequence of representations. [Copyright &y& Elsevier]