Standard approaches to measuring the welfare gains from variety take variety as exogenous to consumers. Empirically we observe an Engel curve for variety, with richer consumers purchasing a larger set of varieties than poor ones. I present a tractable model for analyzing the welfare gains from variety in the presence of this non-homotheticity. The model generates variety Engel curves through di...

An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two Engel structures are locally diffeomorphic. We investigate the deformation space of Engel structures obtained by deforming certain canonical Engel structures on four-manifolds with boundary. When the manifold is RP 3 × I where I is a closed interval, we show that this deformation space conta...

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph EG as follows: Take G\L(G) as vertices of EG and join two distinct vertices x and y whenever [x,k y] 6= 1 and [y,k x] 6= 1 for all positive integers k. We call EG, the Engel graph of G. In this paper we study the graph theoretical properties of EG.

In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime p and any left 3-Engel element x of finite p-power order in a group G, x is in the Baer radical of G. Also it is proved that 〈x, y〉 is nilpotent of class 4 for every two left 3-Engel elements in a group G.

IN this paper we shall study the connection between local and global nilpotency for groups which satisfy certain chain conditions. In the first section, we shall prove that a group satisfying the chain condition on centralizers has a unique maximal normal nilpotent subgroup (the Fitting subgroup), i.e. the group generated by all normal nilpotent subgroups must again be nilpotent. This generaliz...

We consider decidability problems associated with Engel’s identity ([· · · [[x, y], y], . . . , y] = 1 for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given x, y, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk’s 2-group is not Engel. We consider next the problem of recognizing Eng...

The inter-rater reliability, expressed as kappa score, k, of the Engel and International League Against Epilepsy (ILAE) classifications of epilepsy surgery seizure outcome has not previously been evaluated. In a consecutive series of 76 patients (40 male; 25 children), 75 undergoing resective and 1 disconnective surgery at a mean age of 27.5 years (13 months-62 years), one observer classified 8...

A third species of the rare augochlorine bee genus Chlerogelloides Engel et al. (Halictinae, Augochlorini) is described and figured. Chlerogelloides nexosasp. n. is most similar to the type species, Chlerogelloides femoralis Engel et al., in that both have modified midlegs in the males. The former, however, can be distinguished on the basis of its terminalia, which in some respects more closely...

The Extended Engel Expansion is an algorithm that leads to unique series expansions of q-series. Various examples related to classical partition theorems, including the RogersRamanujan identities, have been given recently. The object of this paper is to show that the new and elegant Rogers-Ramanujan generalization found by Garrett, Ismail, and Stanton also ts into this framework. This not only ...