0 0.1 0.2 -6 -5 -4 -3 -2 -1 0 s 0 0.1 0.2 0 0.1 0.2 s$_{2}$ * s$_{im}$ * s * E (a) (b) (c) FIG. 9: (a) s ∗ im − 2 . 0 obtained from Eq. (16) with s ∗ ex obtained from ref. $^{30}$, (b) s ∗ 2 and (c) from the Einstein Frequency. See the text for more details. In all cases, the entropy is plotted against density for fixed temperatures T ∗ = 0.20, 0.30, 0.40 and 0.50 from bottom to top. free volume effects is better captured by s ∗ $_{r}$. As a summary of the insights obtained from INM analysis and pair correlation entropy into the nested cascade of anomalies picture of water-like liquids, we use the extrema in F$_{im}$ ( ρ ) and s ∗ $_{2}$( ρ ) curves to define additional anomalous regimes within which ( ∂F$_{im}$/∂ρ )$_{T}$ > 0 and ( ∂s ∗ $_{2}$/∂ρ $^{∗}$)$_{T}$ > 0 respectively, as shown in Fig.10. The low density boundaries of anoma- lous regime in t , F$_{im}$ and s ∗ 2 almost coincide, reflecting the onset of the steep repulsive wall. The high-density boundary of the anomalous regime of F$_{im}$ occurs at very high densities. In contrast, the high density limit of the anomalous regime in s ∗ 2 is very close to that defined by t , and also by s ∗ im though the latter is not shown in the Figure. This paper explores the connection between entropy, diffusivity and the potential en- ergy landscape of a core-softened fluid with water-like anomalies using molecular dynamics simulations and instantaneous normal mode analysis. VI. CONCLUSIONS 16