A relationship between two natural dynamical systems on groups is established thereby giving a new characterization of Engel elements. Using this connection, various closure properties for the sets of left and right Engel elements are established. Some other dynamical systems and their relationship with Engel elements and nilpotency for finite groups are also considered.

Bees (Anthophila) are one of the major groups of angiosperm-pollinating insects and accordingly are widely studied in both basic and applied research, for which it is essential to have a clear understanding of their phylogeny, and evolutionary history. Direct evidence of bee evolutionary history has been hindered by a dearth of available fossils needed to determine the timing and tempo of their...

This is the fourth installment intended to address outstanding taxonomic matters in bee genus Scaptotrigona Moure (Meliponini). Four new species of postica group (group B) are described and figured as: (Scaptotrigona) anaulax Engel, species, from Ecuador; S. (S.) vitorum semiflava Bolivia Peru; grueteri Bolivia.

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

We investigated the correlation of interictal epileptiform discharges (IED) in routine EEG 6 and 24 months after epilepsy surgery with regard to long-term seizure outcome. In 148 patients (74% temporal lobe epilepsy (TLE), 26% extratemporal epilepsy) EEG results (IED present or absent) were correlated with the postoperative outcome using the Engel classification 6 and 24 months after resection ...

An Engel curve describes how a consumer’s purchases of a good like food varies as the consumer’s total resources such as income or total expenditures vary. Engel curves may also depend on demographic variables and other consumer characteristics. A good’s Engel curve determines its income elasticity, and hence whether the good is an inferior, normal, or luxury good. Empirical Engel curves are cl...

For any group G, let C(G) denote the set of centralizers of G.We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n.In this note, we prove that every finite Cn-group with n ≤ 21 is soluble andthis estimate is sharp. Moreover, we prove that every finite Cn-group with|G| < 30n+1519 is non-nilpotent soluble. This result gives a partial answer to aconjecture raised by A. Ashrafi in ...

Here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. Also we find integers $n$ for which, these groups are $n$-central.