This paper concerns the identification and estimation of a shape-invariant Engel curve system with endogenous total expenditure. The shape-invariant specification involves a common shift parameter for each demographic group in a pooled system of Engel curves. Our focus is on the identification and estimation of both the nonparametric shape of the Engel curve and the parametric specification of ...

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.

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...

New material of the wasp family Maimetshidae (Apocrita) is presented from four Cretaceous amber deposits - the Neocomian of Lebanon, the Early Albian of Spain, the latest Albian/earliest Cenomanian of France, and the Campanian of Canada. The new record from Canadian Cretaceous amber extends the temporal and paleogeographical range of the family. New material from France is assignable to Guyotem...

We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of X. We describe algorithms answering the word problem, and bound its complexity under some additional assumptions. We give a partial algorithm that decides in a group generated by an automaton, given x, y, whether an Engel identity (r ̈ ̈ ̈ rrx, ys, ys, . . . , ys “ 1 for ...

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that 〈a, a〉 is nilpotent of class at most 4, whenever a is any element and b are right 4-Engel elements or a are left 4-Engel elements and b is an arbitrary element of G. Furthermore we prove that for any prime p and any element a of finite p-power order in a group G such that a ∈ L4(G), a, if p = 2, and ...

‎let $g$ be a finite group and $d_2(g)$ denotes the probability ‎that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $g$‎. ‎we ‎will obtain lower and upper bounds for $d_2(g)$ in the case where ‎the sets $e_g(x)={yin g:[y,x,x]=1}$ are subgroups of $g$ for ‎all $xin g$‎. ‎also the given examples illustrate that all the ‎bounds are sharp.

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...

‎let $g$ be a finite group and $d_2(g)$ denotes the probability‎ ‎that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $g$‎. ‎we‎ ‎will obtain lower and upper bounds for $d_2(g)$ in the case where‎ ‎the sets $e_g(x)={yin g:[y,x,x]=1}$ are subgroups of $g$ for‎ ‎all $xin g$‎. ‎also the given examples illustrate that all the‎ ‎bounds are sharp‎.