Alternative Error Bounds for Some Numerical Quadrature Rules

Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.

Altered white matter microarchitecture in Parkinson’s disease:a voxel-based meta-analysis of diffusion tensor imaging studies

This study aimed to define the most consistent white matter microarchitecture pattern in Parkinson’s disease(PD)reflected by fractional anisotropy(FA),addressing clinical profiles and methodology-related heterogeneity.Web-based publication databases were searched to conduct a meta-analysis of whole-brain diffusion tensor imaging studies comparing PD with healthy controls(HC)using the anisotropic effect size–signed differential mapping.A total of 808 patients with PD and 760 HC coming from 27 databases were finally included.Subgroup analyses were conducted considering heterogeneity with respect to medication status,disease stage,analysis methods,and the number of diffusion directions in acquisition.Compared with HC,patients with PD had decreased FA in the left middle cerebellar peduncle,corpus callosum(CC),left inferior fronto-occipital fasciculus,and right inferior longitudinal fasciculus.Most of the main results remained unchanged in subgroup metaanalyses of medicated patients,early stage patients,voxel-based analysis,and acquisition with˂30 diffusion directions.The subgroup meta-analysis of medication-free patients showed FA decrease in the right olfactory cortex.The cerebellum and CC,associated with typical motor impairment,showed the most consistent FA decreases in PD.Medication status,analysis approaches,and the number of diffusion directions have an important impact on the findings,needing careful evaluation in future meta-analyses.

Voxel-based meta-analysis of grey matter changes in Alzheimer’s disease

Background:Voxel-based morphometry(VBM)using structural brain MRI has been widely used for the assessment of impairment in Alzheimer’s disease(AD),but previous studies in VBM studies on AD remain inconsistent.Objective:We conducted meta-analyses to integrate the reported studies to determine the consistent grey matter alterations in AD based on VBM method.Methods:The PubMed,ISI Web of Science,EMBASE and Medline database were searched for articles between 1995 and June 2014.Manual searches were also conducted,and authors of studies were contacted for additional data.Coordinates were extracted from clusters with significant grey matter difference between AD patients and healthy controls(HC).Meta-analysis was performed using a new improved voxel-based meta-analytic method,Effect Size Signed Differential Mapping(ES-SDM).Results:Thirty data-sets comprising 960 subjects with AD and 1195 HC met inclusion criteria.Grey matter volume(GMV)reduction at 334 coordinates in AD and no GMV increase were found in the current meta-analysis.Significant reductions in GMV were robustly localized in the limbic regions(left parahippocampl gyrus and left posterior cingulate gyrus).In addition,there were GM decreases in right fusiform gyrus and right superior frontal gyrus.The findings remain largely unchanged in the jackknife sensitivity analyses.Conclusions:Our meta-analysis clearly identified GMV atrophy in AD.These findings confirm that the most prominent and replicable structural abnormalities in AD are in the limbic regions and contributes to the understanding of pathophysiology underlying AD.

Rational maps as Schwarzian primitives

We examine when a meromorphic quadratic differential φ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of φ around each pole c, the most singular term should take the form(1- d2)/(2(z- c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles(i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko(2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by φ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative.Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.