Abstract. An exponent matrix is an n×n matrix A = (a ij ) over N 0 satisfying (1) a ii = 0 for all i = 1, . . . , n and (2) a ij + a jk ≥ a ik for all pairwise distinct i, j, k ∈ {1, . . . , n}. In the present paper we study the set E n of all non-negative n × n exponent matrices as an algebra with the operations ⊕ of component-wise maximum and ⊙ of component-wise addition. We provide a basis of the algebra (E n , ⊕, ⊙, 0) and give a row and a column decompositions of a matrix A ∈ E n with respect to this basis. This structure result determines all n × n tiled orders over a fixed discrete valuation ring. We also study automorphisms of E n with respect to each of the operations ⊕ and ⊙ and prove that Aut(E n , ⊙) = Aut(E n , ⊕) = Aut(E n , ⊙, ⊕, 0) ≃ S n × C 2 , n > 2.
We consider in this article the properties of topological conjugacy of the tent-like maps [Formula: see text], which are piecewise linear and whose graph consists of straight line segments, extending from [Formula: see text] to [Formula: see text] to [Formula: see text], where [Formula: see text] is a parameter. For any point [Formula: see text], we reduce the calculation of the derivative of the conjugacy [Formula: see text] of functions [Formula: see text] and [Formula: see text] to the limit of a recurrently defined sequence, which is defined by [Formula: see text]. In the case [Formula: see text], this result is reduced to a study of some properties of the binary expansion of [Formula: see text].
Abstract. We introduce a notion of Gorenstein quiver associated with a Gorenstein matrix. We study properties of such quivers. In particular, we show that any such quiver is strongly connected and simply laced. We use Perron-Frobenius theory of non-negative matrices for characterization of isomorphic Gorenstein quivers.
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