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

How Can Geography and Mobile Phones Contribute to Psychotherapy?

Interdisciplinary relationships between Geography and Psychotherapy are an opportunity for innovation. Indeed, scientific works found on bibliographic databases and concerning this theme are scarce. Geographical sub-fields, such as the Geography of Emotions or Psychoanalytical Geography have started to emerge, theorizing about and interpreting feelings, emotions, moods, sufferings, of the chronically ill or diversified social groups and sites. But a less theoretical and more practical approach, in the sense of proposing, predicting and intervening, is lacking; as well as research into the possibilities offered by communication technologies and mobile phones. In the present work, we present the results of a review of the most relevant scientific works published internationally; we reflect on the contributions of Geography and mobile phones to psychosocial therapies and define the orientation and questions that should be posed in future research, from the point of view of geography and regarding psychotherapy. We conclude that the production of georeferenced data via mobile phones concerning the daily lives of people opens great possibilities for cognitive behavioural therapy and mental health. They allow for the development of personalized mood maps that locate the places where a person experiences greater or lesser stress on a daily basis; they allow for a cartography of emotions, a cognitive cartography of the places we access physically or through the Internet, of our feelings and psychosocial experiences. They open the door to the possibility of offering personalized psychotherapy treatments focusing on the ecological-environmental analysis of the places frequented by the person on a daily basis

Examining Time to Rearrest by Drug Treatment Experience of Drug Court Eligible Offenders

This study explores the relationship between drug treatment experience and time until rearrest among a sample of drug court eligible offenders. The subjects for this work were randomized into an evaluation of the Baltimore City Drug Treatment Court. Of the 235 subjects who participated in this evaluation, 128 received some form of drug treatment during the three year follow-up period. Treated subject's time until rearrest was compared to the 107 subjects who did not receive drug treatment. Treatment experience was measured two different ways: first modality received and number of treatment episodes experienced. Kaplan-Meier survival estimates showed that the treated groups had longer survival times than the non-treated group. Cox regression analyses were then conducted to determine what explained this finding. Results showed that days of treatment and days of supervision were all significant predictors of time until rearrest, while first treatment modality and number of treatment experiences did not predict this outcome. Implications of these findings for policy makers and researchers are discussed

Bifurcations in Discretized Reaction-Diffusion Equations

This pennutation is defined by the braid of the equilibria in the space of (x, u, ux) and determines the attractor up to connection equivalence

Orbit Equivalence of Global Attractors for S1-Equivariant Parabolic Equations

nuloWe consider the global attractor Af for the semiflow generated by a scalar semilinear parabolic equation of the form ut = uxx + f(u, ux), defined on the circle, x 2 S1. Using a characterization of the period maps for planar Hamiltonian systems of the form u00 + g(u) = 0 we discuss questions related to the topological equivalence between global attractors