A special basis for the image of an alternating map and its applications to the size of the Schur multiplier of pp-groups

We consider two vector spaces $U, \ V$ over a field $F$ and an alternating bilinear map $A:U\times U\to V$ such that the \im{A} spans $V$. We construct a special basis for $V$ consisting of elements of \im{A}, and then study several properties of this basis. As an application, we obtain bounds for the size of the Schur multiplier of $p$-groups of nilpotency class $c$, which improves existing bounds

Bazzoni-Glaz Conjecture

In their paper, Bazzoni and Glaz conjecture that the weak global dimension of a Gaussian ring is $0,1$ or $\infty$. In this paper, we prove their conjecture.Comment: arXiv admin note: substantial text overlap with arXiv:1107.044

Low levels of urinary epidermal growth factor predict chronic kidney disease progression in children

Urinary epidermal growth factor (uEGF) has recently been identified as a promising biomarker of chronic kidney disease (CKD) progression in adults with glomerular disease. Low levels of uEGF predict CKD progression and appear to reflect the extent of tubulointerstitial damage. We investigated the relevance of uEGF in pediatric CKD. We performed a post hoc analysis of the Cardiovascular Comorbidity in Children with CKD (4C) study, which prospectively follows children aged 6-17 years with baseline estimated glomerular filtration rate (eGFR) of 10-60 ml/min/1.73 m(2). uEGF levels were measured in archived urine collected within 6 months of enrollment. Congenital abnormalities of the kidney and urinary tract were the most common cause of CKD, with glomerular diseases accounting for <10% of cases. Median eGFR at baseline was 28 ml/min/1.73 m(2), and 288 of 623 participants (46.3%) reached the composite endpoint of CKD progression (50% eGFR loss, eGFR < 10 ml/min/1.73 m(2), or initiation of renal replacement therapy). In a Cox proportional hazards model, higher uEGF/Cr was associated with a decreased risk of CKD progression (HR 0.76; 95% CI 0.69-0.84) independent of age, sex, baseline eGFR, primary kidney disease, proteinuria, and systolic blood pressure. The addition of uEGF/Cr to a model containing these variables resulted in a significant improvement in C-statistics, indicating better prediction of the 1-, 2- and 3-year risk of CKD progression. External validation in a prospective cohort of 222 children with CKD demonstrated comparable results. Thus, uEGF may be a useful biomarker to predict CKD progression in children with CKD