This paper will review the consequences of having limit cycles in the renormalization group. Characteristics such as discrete symmetry invariance and log periodic behavior of observables are expected in systems that exhibit limit cycles in their renormalization group. Some realizations of limit cycles will be explained. Specifically limit cycles in RG flows will be used to explain the Efimov ef...

In this paper, we consider for any free presentation G = F/R of a group G the coinvariance H0(G,R ⊗n ab ) of the n-th tensor power of the relation module Rab and show that the homology group H2n(G,Z) may be identified with the limit of the groups H0(G,R ⊗n ab ), where the limit is taken over the category of these presentations of G. We also consider the free Lie ring generated by the relation m...

The nonperturbative functional renormalization group equation depends on the choice of a regulator function, whose main properties are ``coarse-graining scale'' $k$ and an overall dimensionless amplitude $a$. In this paper we shall discuss limit $a\ensuremath{\rightarrow}0$ with fixed. This is closely related to pseudoregulator that reproduces beta functions $\overline{\mathrm{MS}}$ scheme stud...