The $\varphi\dim$ of Cyclic Nakayama Algebras
K. Igusa and G. Todorov introduced $\varphi$ function which generalizes the notion of projective dimension. We study the behavior of the $\varphi$ functions for cyclic Nakayama algebras of infinite global dimension. We prove that it is always an even number. In particular we show that $\varphi$ dimension is 2 if and only if algebra satisfies certain symmetry conditions. Also we give a sharp upper bound for $\varphi\dim$ in terms of number of monomial relations which describes the algebra.