Two basic tools of free probability are the R-transform and the S-transform. These transforms were introduced by Voiculescu in the 1980s, and are used to understand the addition and the multiplication of two free random variables respectively. The R-transform has a natural and very useful multi-variable extension describing the addition of two free k-tuples of random variables, but the problem ...

OBJECTIVE To investigate the longitudinal associations between active commuting (walking and cycling to work) and body mass index (BMI). METHOD We used self-reported data on height, weight and active commuting from the Commuting and Health in Cambridge study (2009 to 2012; n=809). We used linear regression to test the associations between: a) maintenance of active commuting over one year and ...

We provide new interpretations for a subset of Raney numbers, involving threshold sequences and Motzkin-like paths with long up down steps. Given three integers n,k,ℓ such that n≥1,k≥2 0≤ℓ≤k−2, (k,ℓ)-threshold sequence length n is any strictly increasing S=(s1s2…sn) ki≤si≤kn+ℓ. These are in bijection the (ℓ+1)-tuples k-ary trees total internal nodes. prove this result using counting arguments b...

We improve on and generalize a 1960 result of Maltsev. For field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is copy $F$ defined $H(F)$, using existential formulas an arbitrary non-commuting pair $(u,v)$ as parameters. show interpreted computable $\Sigma_1$ no give two proofs. The first existence proof, relying Harrison-Trainor, Melnikov, R. Mill...

Let D be a division ring, n 2 a natural number, and C ⊆ Mn(D). Two matrices A and B are called C−commuting if there is C ∈ C that AB−BA = C. In this paper the C−commuting graph of Mn(D) is defined and denoted by ΓC(Mn(D)). Conditions are given that guarantee that the C−commuting graph is connected.

Let D be a division ring, n 2 a natural number, and C ⊆ Mn(D). Two matrices A and B are called C−commuting if there is C ∈ C that AB−BA = C. In this paper the C−commuting graph of Mn(D) is defined and denoted by ΓC(Mn(D)). Conditions are given that guarantee that the C−commuting graph is connected.

We present GDBR, Generalize DataBase Relation, an optimal, on-line O(n) algorithm for database relation generalization using concept hierarchies. The algorithm is a variant of attribute-oriented induction which generalizes database relations in either O(n log n) or O(np) time, where n is the number of input tuples and p is the number of tuples in the output relation. We augment the concept hier...

We will investigate so-called commuting operators and their relationship to bisymmetry and domination. In the case of bisymmetric aggregation operators we will show a sufficient condition ensuring that two operators commute, while for bisymmetric aggregation operators with neutral element we will even give a full characterization of commuting n-ary operators by means of unary commuting operators.