In this note, we show that the set of all commuting d-tuples of commuting n × n matrices that are contained is an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber’s theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also stu...

While classical ergodic theory deals largely with single ergodic transformations or flows (i.e. with actions of N,Z,R+ or R on measure spaces), many of the lattice models in statistical mechanics (such as dimer models) have multi-dimensional symmetry groups: they carry actions of Z or R with d > 1. However, the transition from Zor R-actions to multi-parameter ergodic theory presents considerabl...

BACKGROUND Few studies have assessed the predictors of changes in commuting. This study investigated the associations between physical environmental characteristics and changes in active commuting. METHODS Adults from the population-based European Prospective Investigation into Cancer (EPIC)-Norfolk cohort self-reported commuting patterns in 2000 and 2007. Active commuters were defined as tho...

INTRODUCTION Active commuting is associated with a reduced risk of several chronic diseases, but few studies have considered institutional factors that influence it. We examined the association between cultural and physical workplace supports for active commuting and employee active commuting behavior. METHODS Data were collected from employees (N = 375) in Manhattan, Kansas, via an online su...

BACKGROUND The current study examined the associations between active commuting to school, cognitive performance, and academic achievement in Dutch adolescents. In addition, it was explored whether these associations were moderated by sex and mediated by depressive symptoms. METHODS Students in grades 7 and 9 (N = 270; mean age 13.4 years; 53% boys) were included. Active commuting to school w...

‎in a recent paper c‎. ‎miguel proved that the diameter of the commuting graph of the matrix ring $mathrm{m}_n(mathbb{r})$ is equal to $4$ if either $n=3$ or $ngeq5$‎. ‎but the case $n=4$ remained open‎, ‎since the diameter could be $4$ or $5$‎. ‎in this work we close the problem showing that also in this case the diameter is $4$.

Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring...

BACKGROUND Walking or bicycling to school (ie, active commuting) has shown promise for improving physical activity and preventing obesity in youth. Our objectives were to examine, among US youth, whether active commuting was inversely associated with adiposity and positively associated with moderate-to-vigorous physical activity (MVPA). We also examined whether MVPA mediated the relationships b...

BACKGROUND The recent decline in children's active commuting (walking or biking) to school has become an important public health issue. Recent programs have promoted the positive effects of active commuting on physical activity (PA) and overweight. However, the evidence supporting such interventions among schoolchildren has not been previously evaluated. METHODS This article presents the resu...