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Content available remote The minimum exponent of the primitive digraphs on the given number of arcs
EN
Primitive digraphs on n vertices, k arcs and girth s are considered. By a(n, k, s) we mean the minimum exponent taken over all such digraphs. We estimate the number a(n, k, s) using the Frobenius number for special values of k and s.
EN
The concept of operator stability on finite-dimensional vector spaces V was generalized in the past into several directions. In particular, operator-semistable and self-decomposable laws and self-similar processes were investigated and the underlying vector space V may be replaced by a simply connected nilpotent Lie group G. This motivates investigations of certain linear subgroups of GL (V) and Aut (G), respectively, the decomposability group of a full probability μ and its compact normal subgroup, the invariance group. Using some basic properties of algebraic groups, the structure of normalizers and centralizers of compact matrix groups is analyzed and applied to the above-mentioned set-up, proving the existence and describing the shape of exponents and of commuting exponents of (operator-) semistable laws. Further applications are mentioned, in particular for operator self-decomposable laws and self-similar processes.
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