Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles

The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the "Simplicial Steinitz problem". It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of "short sets"

Polytopality of simple games

The Bier sphere $Bier(\mathcal{G}) = Bier(K) = K\ast_\Delta K^\circ$ and the canonical fan $Fan(\Gamma) = Fan(K)$ are combinatorial/geometric companions of a simple game $\mathcal{G} = (P,\Gamma)$ (equivalently the associated simplicial complex $K$), where $P$ is the set of players, $\Gamma\subseteq 2^P$ is the set of wining coalitions, and $K = 2^P\setminus \Gamma$ is the simplicial complex of losing coalitions. We characterize roughly weighted majority games as the games $\Gamma$ such that $Bier(\mathcal{G})$ (respectively $Fan(\Gamma)$) is canonically polytopal (canonically pseudo-polytopal) and show, by an experimental/theoretical argument, that all simple games with at most five players are polytopal

Validation of the pendulum test in the assessment of muscle tone in persons with cerebral palsy

Introduction: Cerebral palsy (CP) is often accompanied by motor limitations, abnormal movements and spasticity. Precise and reproducible assessment of spasticity is essential for the selection and follow up of the therapeutic protocol. In clinics the spasticity is most often characterized with the estimate of resistance to a manual flexion and extension of a particular joint and use of a modified Ashworth scale (MAS). The MAS grading depends on the subjective assessment by the examiner. Aim: The possible alternative is to use the pendulum test (PT) determined parameters for the quantitative assessment of the spasticity. Material and Methods: We used a new instrument from the company 3F - Fit Fabricando Faber comprising inertial measurement units at the shank and thigh, and two EMG recording units to record electromyographic signals from the hamstrings and quadriceps muscles. The study included 48 subjects diagnosed with cerebral palsy. Results: The analysis of results showed that the new measure based on the PT recordings is highly corelated with the muscle tone in CP patients, and automatically distinguishes the type of spasticity (flexion or extension), relaxation rate and spasticity strength. Conclusion: The new measure showed sensitivity of the PT and indicates great potential of using the PT in the protocols for better control of spasticity in CP patients.The conference took place in the Anatomy Institute of Padova Universit

Extending the straight leg raise test for improved clinical evaluation of sciatica : reliability of hip internal rotation or ankle dorsiflexion

Background The straight leg raise (SLR) is the most commonly applied physical tests on patients with sciatica, but the sensitivity and specificity ratings for disc hernia and neural compression leave areas for improvement. Hip internal rotation tensions the lumbosacral nerve roots and ankle dorsiflexion tensions the sciatic nerve along its course. We added these movements to the SLR (extended SLR = ESLR) as structural differentiators and tested inter-rater reliability in patients with LBP, with and without sciatica. Methods Forty subjects were recruited to the study by the study controller (SC), 20 in the sciatic group and in the control group. Two independent examiners (E1&E2) performed the ESLR and did not communicate to the subjects other than needed to determine the outcome of the ESLR. First, SLR was performed traditionally until first responses were evoked. At this hip flexion angle, a location-specific structural differentiation was performed to confirm whether the emerged responses were of neural origin. Cohen's Kappa score (CK) for interrater reliability was calculated for ESLR result in detection of sciatic patients. Also, the examiners' ESLR results were compared to the traditional SLR results. Results The interrater agreement between Examiner 1 and Examiner 2 for the ESLR was 0.85 (p <0.001, 95%CI: 0.71-0.99) translating to almost perfect agreement as measured by Cohen's Kappa When the ESLR was compared to the traditional SLR, the overall agreement rate was 75% (30/40). Kappa values between the traditional SLR and the E1's or E2's ESLR results were 0.50 (p <0.0001; 95%CI 0.27-0.73) and 0.54 (p <0.0001; 95%CI 0.30-0.77), respectively. Conclusions ESLR with the addition of location-specific structural differentiation is a reliable and repeatable tool in discerning neural symptoms from musculoskeletal in patients with radiating low back pain. We recommend adding these movements to the standard SLR with aim of improving diagnostic ability.Peer reviewe