Quantization of energy and writhe in self-repelling knots

Probably the most natural energy functional to be considered for knotted strings is that given by electrostatic repulsion. In the absence of counter-charges, a charged, knotted string evolving along the energy gradient of electrostatic repulsion would progressively tighten its knotted domain into a point on a perfectly circular string. However, in the presence of charge screening self-repelling knotted strings can be stabilized. It is known that energy functionals in which repulsive forces between repelling charges grow inversely proportionally to the third or higher power of their relative distance stabilize self-repelling knots. Especially interesting is the case of the third power since the repulsive energy becomes scale invariant and does not change upon Mobius transformations (reflections in spheres) of knotted trajectories. We observe here that knots minimizing their repulsive Mobius energy show quantization of the energy and writhe (measure of chirality) within several tested families of knots.

Do orthopaedic shoes improve local dynamic stability of gait? An observational study in patients with chronic foot and ankle injuries.

BACKGROUND: Complex foot and ankle fractures, such as calcaneum fractures or Lisfranc dislocations, are often associated with a poor outcome, especially in terms of gait capacity. Indeed, degenerative changes often lead to chronic pain and chronic functional limitations. Prescription footwear represents an important therapeutic tool during the rehabilitation process. Local Dynamic Stability (LDS) is the ability of locomotor system to maintain continuous walking by accommodating small perturbations that occur naturally during walking. Because it reflects the degree of control over the gait, LDS has been advocated as a relevant indicator for evaluating different conditions and pathologies. The aim of this study was to analyze changes in LDS induced by orthopaedic shoes in patients with persistent foot and ankle injuries. We hypothesised that footwear adaptation might help patients to improve gait control, which could lead to higher LDS: METHODS: Twenty-five middle-aged inpatients (5 females, 20 males) participated in the study. They were treated for chronic post-traumatic disabilities following ankle and/or foot fractures in a Swiss rehabilitation clinic. During their stay, included inpatients received orthopaedic shoes with custom-made orthoses (insoles). They performed two 30s walking trials with standard shoes and two 30s trials with orthopaedic shoes. A triaxial motion sensor recorded 3D accelerations at the lower back level. LDS was assessed by computing divergence exponents in the acceleration signals (maximal Lyapunov exponents). Pain was evaluated with Visual Analogue Scale (VAS). LDS and pain differences between the trials with standard shoes and the trials with orthopaedic shoes were assessed. RESULTS: Orthopaedic shoes significantly improved LDS in the three axes (medio-lateral: 10% relative change, paired t-test p < 0.001; vertical: 9%, p = 0.03; antero-posterior: 7%, p = 0.04). A significant decrease in pain level (VAS score -29%) was observed. CONCLUSIONS: Footwear adaptation led to pain relief and to improved foot & ankle proprioception. It is likely that that enhancement allows patients to better control foot placement. As a result, higher dynamic stability has been observed. LDS seems therefore a valuable index that could be used in early evaluation of footwear outcome in clinical settings.

Aggregation invariance in general clustering approaches

General clustering deals with weighted objects and fuzzy memberships. We investigate the group- or object-aggregation-invariance properties possessed by the relevant functionals (effective number of groups or objects, centroids, dispersion, mutual object-group information, etc.). The classical squared Euclidean case can be generalized to non-Euclidean distances, as well as to non-linear transformations of the memberships, yielding the c-means clustering algorithm as well as two presumably new procedures, the convex and pairwise convex clustering. Cluster stability and aggregation-invariance of the optimal memberships associated to the various clustering schemes are examined as well.

A random linear functional approach to efficiency bounds

This paper suggests a method for obtaining efficiency bounds in models containing either only infinite-dimensional parameters or both finite- and infinite-dimensional parameters (semiparametric models). The method is based on a theory of random linear functionals applied to the gradient of the log-likelihood functional and is illustrated by computing the lower bound for Cox's regression model