Hamiltonian submanifolds of regular polytopes

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\leq 9$ are now decided.Comment: 26 pages, 4 figure

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result

Mental contrasting as a behaviour change technique: a systematic review protocol paper of effects, mediators and moderators on health

Background Mental contrasting is a self-regulation strategy that is required for strong goal commitment. In mental contrasting, individuals firstly imagine a desired future or health goal that contrasted with the reality proceeding the goal state, which after reflection is viewed as an obstacle (Oettingen et al. J Pers Soc Psychol 80:736–753, 2001). Mentally contrasting a positive future with reality enables individuals to translate positive attitudes and high efficacy into strong goal commitment. Methods A systematic review of the literature is proposed to explore the efficacy of mental contrasting as a behaviour change technique (Michie et al., Ann Behav Med 46: 81-95, 2013) for health. The review also aims to identify the effects of mental contrasting on health-related behaviour, as well as identifying mediator and moderator variables. Discussion This will be the first systematic review of mental contrasting as a health behaviour change technique. With sufficient studies, a meta-analysis will be conducted with sensitivity and sub group analyses. If meta-analysis is not appropriate, a narrative synthesis of the reviewed studies will be conducted. Systematic review registration Review protocol registered on PROSPERO reference CRD42016034202.N/

Permutonestohedra

There are several real spherical models associated with a root arrangement, depending on the choice of a building set. The connected components of these models are manifolds with corners which can be glued together to obtain the corresponding real De Concini–Procesi models. In this paper, starting from any root system with finite Coxeter group W and any W -invariant building set, we describe an explicit realization of the real spherical model as a union of polytopes (nestohedra) that lie inside the chambers of the arrangement. The main point of this realization is that the convex hull of these nestohedra is a larger polytope, a permutonestohedron, equipped with an action of W or also, depending on the building set, of Aut ( ). The permutonestohedra are natural generalizations of Kapranov’s permutoassociahedra

Flanker performance in female college students with ADHD: a diffusion model analysis

Attention-deficit hyperactivity disorder (ADHD) is characterized by poor adaptation to environmental demands, which leads to various everyday life problems. The present study had four aims: (1) to compare performance in a flanker task in female college students with and without ADHD (N = 39) in a classical analyses of reaction time and error rate and studying the underlying processes using a diffusion model, (2) to compare the amount of focused attention, (3) to explore the adaptation of focused attention, and (4) to relate adaptation to psychological functioning. The study followed a 2-between (group: ADHD vs. control) × 2-within (flanker conflict: incongruent vs. congruent) × 2-within (conflict frequency: 20 vs. 80 %) design. Compared to a control group, the ADHD group displayed prolonged response times accompanied by fewer errors in a flanker task. Results from the diffusion model analyses revealed that the members of the ADHD group showed deficits in non-decisional processes (i.e., higher non-decision time) and leaned more toward accuracy than participants without ADHD (i.e., setting higher boundaries). The ADHD group showed a more focused attention and less adaptation to the task conditions which is related to psychological functioning. Deficient non-decisional processes and poor adaptation are in line with theories of ADHD and presumably typical for the ADHD population, although this has not been shown using a diffusion model. However, we assume that the cautious strategy of trading speed of for accuracy is specific to the subgroup of female college students with ADHD and might be interpreted as a compensation mechanism