The variety C(3, n) of commuting triples of n × n matrices over C is shown to be irreducible for n = 7. It had been proved that C(3, n) is reducible for n ≥ 30, but irreducible for n ≤ 6. Guralnick and Omladič have conjectured that it is reducible for n > 7.

We consider the following problem: What are possible sizes of Jordan blocks for a pair of commuting nilpotent matrices? Or equivalently, for which pairs of nilpotent orbits of matrices (under similarity) there exists a pair of matrices, one from each orbit, that commute. The answer to the question could be considered as a generalization of Gerstenhaber– Hesselink theorem on the partial order of...

BACKGROUND We studied the effect of key development and expansion of an off-road multipurpose trail system in Minneapolis, Minnesota between 2000 and 2007 to understand whether infrastructure investments are associated with increases in commuting by bicycle. METHODS We used repeated measures regression on tract-level (N = 116 tracts) data to examine changes in bicycle commuting between 2000 a...

We study the variant of the k-local hamiltonian problem which is a natural generalization of k-CSPs, in which the hamiltonian terms all commute. More specifically, we consider a hamiltonian H = ∑i Hi over n qubits, where each Hi acts non-trivially on O(log n) qubits and all the terms Hi commute, and show the following 1. We show that a specific case of O(log n) local commuting hamiltonians over...

OBJECTIVE To examine whether a relationship exists between active commuting and physical and mental wellbeing. METHOD In 2009, cross-sectional postal questionnaire data were collected from a sample of working adults (aged 16 and over) in the Commuting and Health in Cambridge study. Travel behaviour and physical activity were ascertained using the Recent Physical Activity Questionnaire (RPAQ) ...

let $g$ be a non-abelian group and let $z(g)$ be the center of $g$. associate with $g$ there is agraph $gamma_g$ as follows: take $gsetminus z(g)$ as vertices of$gamma_g$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $gamma_g$ is called the non-commuting graph of $g$. in recent years many interesting works have been done in non-commutative graph of groups. computing the clique...

We formulate a systematic construction of commuting quantum traces for reflection algebras. This is achieved by introducing two dual sets of generalized reflection equations with associated consistent fusion procedures. Products of their respective solutions yield commuting quantum traces.

We prove that the commuting probability of a finite ring is no larger than the commuting probabilities of its subrings and quotients, and characterize when equality occurs in such a comparison.

We formulate a systematic construction of commuting quantum traces for reflection algebras. This is achieved by introducing two dual sets of generalized reflection equations with associated consistent fusion procedures. Products of their respective solutions yield commuting quantum traces.