Ordering the braid groups

We give an explicit geometric argument that Artin's braid group $B_n$ is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a braid which is positive (respectively negative) in our order has consistently positive (respectively negative) exponent in the smallest braid generator which occurs. It follows that our ordering is identical to that of Dehornoy, constructed by very different means, and we recover Dehornoy's main theorem that any braid can be put into such a form using either positive or negative exponent in the smallest generator but not both. Our definition of order is strongly connected with Mosher's normal form and this leads to an algorithm to decide whether a given braid is positive, trivial, or negative which is quadratic in the length of the braid word.Comment: 24 pages, 10 figure

Dietary fat and not calcium supplementation or dairy product consumption is associated with changes in anthropometrics during a randomized, placebo-controlled energy-restriction trial

Insufficient calcium intake has been proposed to cause unbalanced energy partitioning leading to obesity. However, weight loss interventions including dietary calcium or dairy product consumption have not reported changes in lipid metabolism measured by the plasma lipidome

Globotrioasylsphingosine levels and optical coherence tomography angiography in fabry disease patients

Background: To date, there are no studies associating the dried blood spot (DBS) levels of globotrioasylsphingosine (lysoGb3) with quantitative optical coherence tomography angiography (OCTA) parameters in Fabry disease (FD) patients. Here, we aimed to investigate the association between OCTA vessel density (VD), vessel length density (VLD) with DBS lysoGb3. Methods: A retrospective, single center analysis of all consecutive FD patients enrolled at the Department of Ophthalmology of the University Hospital of Zurich from December 1st, 2017 to September 9th, 2020. An association between VD and VLD detected by OCTA and lysoGb3 was investigated using a linear mixed model. Results: A total of 57 FD patients (23 male, 34 female; 109 eyes) were included. Forty-one patients suffered from the classic phenotype and 16 from the later-onset phenotype. Lys-oGb3 inversely correlated with VD and VLD in both the superficial (VD: p = 0.034; VLD: p = 0.02) and deep capillary plexus (VD: p = 0.017; VLD: p = 0.018) in the overall FD cohort. Conclusions: Our study shows an association between lysoGb3 and OCTA VD and VLD. This supports the hypothesis that quantitative OCTA parameters might be useful as diagnostic biomarkers for evaluating sys-temic involvement in FD, and possibly other diseases

Abraham Lincoln and His Family

Photo reproduction of a composite portrait of five members of the Lincoln Family gathered around a table.https://scholarsjunction.msstate.edu/fvw-artifacts/4440/thumbnail.jp