The spans of five star-like simple closed curves

Let X be a continuum, that is a compact, connected, nonempty metric space. The span of X is the least upper bound of the set of real numbers r which satisfy the following conditions: there exists a continuum, C, contained in X × X such that d(x,y) is larger than or equal to r for all (x,y) in C and p1(C) = p2(C), where p1, p2 are the usual projection maps. The following question has been asked. If X and Y are two simple closed curves in the plane and Y is contained in the bounded component of the plane minus X, then is the span of X larger than the span of Y? We define a set of simple closed curves, which we refer to as being five star-like. We answer this question in the affirmative when X is one of these simple closed curves. We calculate the spans of the simple closed curves in this collection and consider the spans of various geometric objects related to these simple closed curves

Validation of a novel Montreal Cognitive Assessment scoring algorithm in non-demented Parkinson's disease patients

Introduction The early diagnosis of mild cognitive impairment (PD-MCI) in Parkinson's disease (PD) is essential as it increases the future risk for PD dementia (PDD). Recently, a novel weighting algorithm for the Montreal Cognitive Assessment (MoCA) subtests has been reported, to best discriminate between those with and without cognitive impairment in PD. The aim of our study was to validate this scoring algorithm in a large sample of non-demented PD patients, hypothesizing that the weighted MoCA would have a higher diagnostic accuracy for PD-MCI than the original MoCA. Methods In 202 non-demented PD patients, we evaluated cognitive status, clinical and demographic data, as well as the MoCA with a weighted and unweighted score. Receiver operating characteristic (ROC) curve analysis was used to evaluate discriminative ability of the MoCA. Group comparisons and ROC analysis were performed for PD-MCI classifications with a cut-off <= 1, 1.5, and 2 standard deviation (SD) below appropriate norms. Results PD-MCI patients scored lower on the weighted than the original MoCA version (p < 0.001) compared to PD patients with normal cognitive function. Areas under the curve only differed significantly for the 2 SD cut-off, leading to an increased sensitivity of the weighted MoCA score (72.9% vs. 70.5%) and specificity compared to the original version (79.0% vs. 65.4%). Conclusions Our results indicate better discriminant power for the weighted MoCA compared to the original for more advanced stages of PD-MCI (2 SD cut-off). Future studies are needed to evaluate the predictive value of the weighted MoCA for PDD