Michael Engel

Results on Engel Fuzzy Subgroups

‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of ...

4-Engel Groups are Locally Nilpotent

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.

Moving up the Residential Hierarchy: A New Remedy for an Old Injury Arising from Housing Discrimination

Group rings satisfying generalized Engel conditions

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1)  y]=[[x ,_( n)  y]  , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y)   ,_( n(x,y))  y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group an...

On the additive maps satisfying Skew-Engel conditions

Let $R$ be a prime ring, $I$ be any nonzero ideal of $R$ and $f:Irightarrow R$ be an additivemap. Then skew-Engel condition $langle... langle langle$$f(x),x^{n_1} rangle,x^{n_2} rangle ,...,x^{n_k} rangle=0$ implies that $f (x)=0$ $forall,xin I$ provided $2neq$ char $(R)>n_1+n_2+...+n_k, $ where $n_1,n_2,...,n_k$ are natural numbers. This extends some existing results. In the end, we also gener...

Complexity in Surfaces of Densest Packings for Families of Polyhedra

Elizabeth R. Chen, Daphne Klotsa, Michael Engel, Pablo F. Damasceno, and Sharon C. Glotzer School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Applied Physics Program, University of Michigan, Ann Arbor, Michigan 48109, USA Department of Materials Science and...