Bicyclic units, Bass cyclic units and free groups

Let G be a finite group and ZG its integral group ring. We show that if alpha is a nontrivial bicyclic unit of ZG, then there are bicyclic units beta and gamma of different types, such that and are non-abelian free groups. In the case when G is non-abelian of order coprime to 6 we prove the existence of a bicyclic unit u and a Bass cyclic unit v in ZG, such that < u(m), v > is a free non-abelian group for all sufficiently large positive integers m.

INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS

Let ZG be the integral group ring of the finite nonabelian group G over the ring of integers Z, and let * be an involution of ZG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (u(k,m)(x*), u(k,m)(x*)) or (u(k,m)(x), u(k,m)(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ZG.

BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ZG of a solvable and finite group G, such that u has infinite order modulo the center of U(ZG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ZG which is either a Bass cyclic unit or a bicyclic unit.

The effects of stocking density in physiological parameters and growth of the endangered teleost species piabanha, Brycon insignis (Steindachner, 1877)

This study investigated the effects of stocking density on the growth and fatty acid (FA) of Brycon insignis metabolism. Fingerlings (360) were distributed into eight ponds at two stocking densities (105 and 210 g/m(3)). The analysis of growth showed that the condition factor (K) and the coefficient of variation (CV) for body mass were not affected by stocking density. However, final body mass and length, specific growth rate (SGR), and weight gain (WG) were higher in the low stocking density group, which also presented a higher feed efficiency (FE) and survival (S). By contrast, muscle protein levels were higher in the high stocking density group. The plasma and muscle lipid content were not affected by stocking density, but fish reared at lower stocking density presented higher lipid concentration in the liver, with no differences in hepatosomatic index values. Even with the differences observed in metabolic and growth parameters, plasma cortisol was not affected by stocking density. The FA profile in the muscle and liver neutral fraction were not affected by stocking density, but the FA in the polar fractions differed between the two stocking densities. In the liver, total polyunsaturated fatty acids (PUFA) and PUFA n - 3 increased in higher stocking density, mainly due to an increase in docosahexaenoic acid (DHA). In addition, PUFA n - 6 were also increased in the higher stocking density group, mainly due to an increase in arachidonic acid (AA) and docosadienoic acid (22:2n - 6). In the muscle polar fraction, the saturated fatty acids (SFA) and monounsaturated fatty acids (MUFA) decreased in the animals from the higher stocking density group, and this reduction was compensated by an increase in PUFA n - 3 and PUFA n - 6, mainly the FA with 20-22 carbons (20:4n - 6: 22:4:n - 6; 22:5n - 6, 22:5n - 3, and 22:6n - 3). A different profile was observed for the C18 PUFAs, mainly 18:2n - 6 and 18:4n - 6, which were higher in the lower density stocking group. The data suggest that when living in high stocking density, B. insignis differentially utilizes the hepatic lipids as energy source and remodels the membrane fatty acids, with higher amounts of DHA in the polar muscle fraction compensated for by a decrease in MUFA. The zootechnical and physiological indices reveal that the lower stocking density group achieve overall better performance. (C) 2010 Elsevier B.V. All rights reserved.

Fatty acids as trophic biomarkers in vitellogenic females in an impounded tropical river

Fatty acids have been used in marine biogeochemistry as food chain biomarkers, but in freshwater these studies are rare. In order to evaluate the fatty acid potential as biomarkers in freshwater, their profile was analyzed during vitellogenesis in two fish species, in both waterfall and reservoir environments of the Paraiba do Sul River Basin. Detrivorous Hypostomus affinis and omnivorous Geophagus brasiliensis seem to elongate and desaturate polyunsaturated fatty acids (PUFA) and transfer them to the ovaries` phospholipids. Waterfall Geophagus brasiliensis have more highly unsaturated fatty acids in the liver, but in the reservoir, accumulation mainly occurs in muscle and ovary triglycerides, suggesting trophic opportunism and a plasticity during vitellogenesis. In Hypostomus affinis, PUFA alteration occurs only in the reservoir, suggesting a high phytoplankton occurrence. Eutrophication and water speed is reflected in Hypostomus affinis ovaries by higher PUFAn3 and bacterial fatty acids. As in marine environments, analysis of mono- and polyunsaturated fatty acids during vitellogenesis can be used as a tool in food chain studies in freshwater.