THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell-Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n >= 1, the class of m-string braid groups B(m)(S(2)\{x(1), ... , x(n)}) of the n-punctured sphere includes the usual Artin braid groups B(m) (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (C) over tilde (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B(m) is completely determined for all m is an element of N (respectively, for all m not equal 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n >= 2, we prove that the lower central series (respectively, derived series) of B(m)(S(2)\{x(1), ... , x(n)}) is constant from the commutator subgroup onwards for all m >= 3 (respectively, m >= 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B(2)(S(2)\{x(1), x(2)}) admits various interpretations, as the Baumslag-Solitar group BS(2, 2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B(2)(S(2)\{x(1), x(2)}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product Z(2) * Z. Further, its lower central series quotients Gamma(i)/Gamma(i+1) are direct sums of copies of Z(2), the number of summands being determined explicitly. In the case m >= 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Gamma(i)/Gamma(i+1) are 2-elementary finitely-generated groups.

Topical application of the lectin Artin M accelerates wound healing in rat oral mucosa by enhancing TGF-β and VEGF production

The lectin Artin M has been shown to accelerate the wound-healing process. The aims of this study were to evaluate the effects of Artin M on wound healing in the palatal mucosa of rats and to investigate the effects of Artin M on transforming growth factor beta (TGF-β) and vascular endothelial growth factor (VEGF) secretion by rat gingival fibroblasts. A surgical wound was created on the palatal mucosa of 72 rats divided into three groups according to treatment: C - Control (nontreated), A - Artin M gel, and V - Vehicle. Eight animals per group were sacrificed at 3, 5, and 7 days postsurgery for histology, immunohistochemistry and determination of the levels of cytokines, and growth factors. Gingival fibroblasts were incubated with 2.5 μg/mL of Artin M for 24, 48, and 72 hours. The expression of VEGF and TGF-β was determined by enzyme-linked immunosorbent assay. Histologically, at day 7, the Artin M group showed earlier reepithelialization, milder inflammatory infiltration, and increased collagen fiber formation, resulting in faster maturation of granular tissue than in the other groups (p < 0.05). Artin M-induced cell proliferation in vivo and promoted a greater expression of TGF-β and VEGF in both experiments (p < 0.05). Artin M was effective in healing oral mucosa wounds in rats and was associated with increased TGF-β and VEGF release, cell proliferation, reepithelialization, and collagen deposition and arrangement of fibers. © 2013 by the Wound Healing Society.

Estudo da equivalência entre a lectina artin M obtida a partir da semente da jaca e a sua forma recombinante na afinidade por glicanas

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Avaliação do efeito da Artin M no processo de repação em mucosa mastigatória

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.

On some Finite-Dimensional Representations of Artin Braid Group

Валентин В. Илиев - Авторът изучава някои хомоморфни образи G на групата на Артин на плитките върху n нишки в крайни симетрични групи. Получените пермутационни групи G са разширения на симетричната група върху n букви чрез подходяща абелева група. Разширенията G зависят от един целочислен параметър q ≥ 1 и се разцепват тогава и само тогава, когато 4 не дели q. В случая на нечетно q са намерени всички крайномерни неприводими представяния на G, а те от своя страна генерират безкрайна редица от неприводими представяния на групата на плитките.

Working in groups

This chapter is a condensation of a guide written by the chapter authors for both university teachers and students (Fowler et al., 2006). All page references given in this chapter are to this guide, unless otherwise stated. University students often work in groups. It may be a formal group (i.e. one that has been formed for a group presentation, writing a report, or completing a project) or an informal group (i.e. some students have decided to form a study group or undertake research together). Whether formal or informal, this chapter aims to make working in groups easier for you. Health care professionals also often work in groups. Yes, working in groups will extend well beyond your time at university. In fact, the skills and abilities to work effectively in groups are some of the most sought-after attributes in health care professionals. It seems obvious, then, that taking the opportunity to develop and enhance your skills and abilities for working in groups is a wise choice.