We develop a construction of Engel structures on 4– manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4–manifolds admit an Engel structure. We also show that, given two Engel manifolds M1, M2 satisfying a certain condition on the characteristic foliations, there is an Engel structure on M1#M2#(S ×S) which is closely related to the ...

We obtain an explicit bound for the nilpotency class of n-Engel Lie algebras of characteristic zero. 1991 Mathematics Subject Classification (Amer. Math. Soc.): 17B30.

We have proved that 4-Engel groups are locally nilpotent. The proof is based upon detailed computations by both hand and machine. Here we elaborate on explicit computer calculations which provided some of the motivation behind the proof. In particular we give details on the hardest coset enumerations now required to show in a direct proof that 4-Engel p-groups are locally finite for 5 ≤ p ≤ 31....

OBJECTIVE To investigate the influence of patient's age and seizure onset on surgical outcome of temporal lobe epilepsy (TLE). METHOD A retrospective observational investigation performed from a cohort of patients from 2000 to 2012. RESULTS A total of 229 patients were included. One-hundred and eleven of 179 patients (62%) were classified as Engel I in the group with < 50 years old, whereas...

In the paper we consider some commutator-type and power-type matrix equations in the group UT(∞,K) of infinite dimensional unitriangular matrices over a fieldK. We introduce a notion of a power outer commutator ω1k k (x1, ..., xk) and a power Engel commutator e1k k (x, y) as outer (respectively Engel) commutators modified by allowing powers of letters instead of letters alone. For a given infin...

Objective To present a surgical series of patients with low grade temporal gliomas causing intractable epilepsy, focusing on long-term seizure outcome.Method A retrospective study was conducted with patients with temporal low-grade gliomas (LGG).Results Sixty five patients with were operated in our institution. Males were more affected than females and the mean age at surgery was 32.3 ± 8.4 (9-...

They imparted a value for individuals, the importance of humility, an enthusiasm for sharing knowledge, and an appreciation of history. Photo: Kellie K. Magill. fi eld, and poorer countries could also develop their own biotechnology approaches. This more democratic approach to agro-biotech could conceivably gain back some of the trust that giant companies like Monsanto and DuPont lost in the GM...