Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$u_t = u_{xx} + f(x,u,u_x)\,,$$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria $v_t=0$ are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors $\mathcal{A}$ as 3-cell templates $\mathcal{C}$. The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $x=0$ and $x=1$, respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors $\mathcal{A}$ with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.Comment: 73+(ii) pages, 40 figures, 14 table; see also parts 1 and 2 under arxiv:1611.02003 and arxiv:1704.0034

To Notify or Not to Notify?:Do Organizations Comply with U.S. Data Breach Notification Laws? An Empirical Study

Data Breach Notification Laws (DBNLs) oblige organizations to notify personal data breaches. In theory, DBNLs mitigate damage after a data breach and incentivize companies to invest in information security. The regulatory enforcement of the DBNL is based on deterrence, because penalties are imposed, varying from $1,000 to$750,000 between states. It is uncertain whether DBNLs are deterrent enough to prevent organizations from concealing data breaches, especially because organizations suffer reputational costs from a notification. This study empirically tests compliance, by relating the adoption and characteristics of different U.S. DBNLs to actual observed data breach notifications based on the privacy breach clearinghouse dataset (2005-2012). After the adoption of the law, a 50% increase of notifications is observed. But, the absolute number of notifications is low, merely 0.05% of the U.S. companies notified. This indicates low compliance, possibly caused by high costs of notifying and low costs of concealing a notification. Unexpectedly, higher sanctions did not have an effect, but limited commensurability of the different sanctioning regimes prohibits a permanent statement. This paper recommends enhancing DBNLs by increasing both the benefits of notifying and deterrence. Benefits are increased by incorporating rewards for good behavior by assisting companies in mitigating damage and continuously reward companies that are compliant by sharing knowledge about threats. Deterrence is increased by higher penalties and more stringent enforcement