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 Rogers-Ramanujan 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...

By relating Engel curves and social learning, we explain the existence of differently shaped Engel curves—an interesting phenomenon in the theory of demand. A formal approach to cultural learning within a population of consumers accounts for some cognitive foundations of these demand patterns. We find that a changing influence of an individual’s role models due to her increasing income, which e...

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 R be a prime ring of characteristic different from 2, d a nonzero derivation of R, and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I. We prove that if [I, I]I ≠ 0, then d(I)I = 0. 1. Introduction. Let R be a prime ring and d a nonzero derivation of R. Define [x, y] 1 = [x, y] = xy − yx, then an Engel condition is a polynomial [x, y] k = [[x, y] k−1 ,y]

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 ...

1 Medical Genetics Service, Hospital de Cĺınicas de Porto Alegre, Rua Ramiro Barcelos, 2350, 90035-903 Porto Alegre, RS, Brazil 2 PPG Ciências Farmacêiticas, Universidade Federal do Piauı́, 64049-550 Terezina, PI, Brazil 3 Genetics Service, Hospital Universitário, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil 4 PPG BioSaúde, Universidade Luterana do Brasil, Avenida ...

4. Prentice RL, Manson JE, Langer RD, Anderson GL, Pettinger M, Jackson RD, et al. Benefits and risks of postmenopausal hormone therapy when it is initiated soon after menopause. Am J Epidemiol. 2009; 170 (1): 12-23. References Mailing address: Flávio Danni Fuchs • Rua Ramiro Barcelos 2350, Sala 2060 Rio Branco 90035-903 Porto Alegre, RS Brazil E-mail: [email protected] Manuscript received D...

*Departamento de Zoologia e Antropologia da Faculdade de Cie“ ncias da Universidade do Porto, Prac a GomesTeixeira, 4099-002 Porto, Portugal. CIIMAR, Rua dos Bragas 177, 4050-123 Porto, Portugal. Faculdade de Cie“ ncias da Sau¤ de da Universidade Fernando Pessoa, Rua Carlos da Maia 296, 4200-150 Porto, Portugal. ISPA, Rua Jardim doTabaco 34, 1149-041 Lisboa, Portugal. Corresponding author, e-ma...

We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on step 3 Carnot group dimension 5. prove conjectured cut times Yu. Sachkov for Cartan problem. Along proof, we obtain comparison with known in Engel group, and sufficient (generic) condition uniqueness length minimizer between two points. Hence reduce optimal synthesis to solving certain system equation...