On the trace formula for Hecke operators on congruence subgroups, II

In a previous paper, we obtained a general trace formula for double coset operators acting on modular forms for congruence subgroups, expressed as a sum over conjugacy classes. Here we specialize it to the congruence subgroups $\Gamma_0(N)$ and $\Gamma_1(N)$, obtaining explicit formulas in terms of class numbers for the trace of a composition of Hecke and Atkin-Lehner operators. The formulas are among the simplest in the literature, and hold without any restriction on the index of the operators. We give two applications of the trace formula for $\Gamma_1(N)$: we determine explicit trace forms for $\Gamma_0(4)$ with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level $N$ tends to infinity

The Balanced Scorecard: translating strategy into results

In the paper it is shown that the Balanced Business Scorecard (BSC) allows a balance between financial and non financial objectives and performs a quantification of the dimensions of strategy in four areas - financial, customers, processes and potential. Moreover, the BSC provides feedback on internal processes and external outcomes in order to improve further the performance and results in terms of strategy. When the Balanced Scorecard is implemented effectively, can transform strategic planning from an academic exercise in the central nervous of the entire company.strategy, financial objectives, non financial objectives, Balanced Business Scorecard, performance measurement