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Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder O (2) × Γ × ℤ 2 -equivariant degree theory. Here, O (2) is related to the reversal symmetry combined with the autonomous form of the system, Γ reflects extra spacial symmetries of the system, and ℤ 2 is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with Γ = D 6 — the dihedral group of order 12. [ABSTRACT FROM AUTHOR]

We consider a generalized type of unique factorization of the positive integers with restrictions on the exponents and view them as a family of arithmetic convolutions and divisibility relations, similar to the convolutions defined by Narkewicz [On a class of arithmetical convolutions, Colloq. Math.10 (1963) 81–94]. We introduce special types of multiplicativity corresponding to these convolutions, and discuss algebraic properties of the associated arithmetic convolutions and analogs of the Möbius functions. We also prove asymptotics for analogs of the totient function, totient summatory function, and divisor summatory function. [ABSTRACT FROM AUTHOR]