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.

Some classes of semisimple group (and loop) algebras over finite fields

We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components We apply our work to obtain similar information about the loop algebras of mdecomposable RA loops and to produce negative answers to the isomorphism problem over various fields (C) 2010 Elsevier Inc All rights reserved

Star-group identities and groups of units

Analogous to *-identities in rings with involution we define *-identities in groups. Suppose that G is a torsion group with involution * and that F is an infinite field with char F not equal 2. Extend * linearly to FG. We prove that the unit group U of FG satisfies a *-identity if and only if the symmetric elements U(+) satisfy a group identity.

Locally nilpotent groups of units in tiled rings

We consider locally nilpotent subgroups of units in basic tiled rings A, over local rings O which satisfy a weak commutativity condition. Tiled rings are generalizations of both tiled orders and incidence rings. If, in addition, O is Artinian then we give a complete description of the maximal locally nilpotent subgroups of the unit group of A up to conjugacy. All of them are both nilpotent and maximal Engel. This generalizes our description of such subgroups of upper-triangular matrices over O given in M. Dokuchaev, V. Kirichenko, and C. Polcino Milies (2005) [3]. (C) 2010 Elsevier Inc. All rights reserved.

Group algebras of torsion groups and Lie nilpotence

Let * be an involution of a group algebra FG induced by an involution of the group G. For char F not equal 2, we classify the torsion groups G with no elements of order 2 whose Lie algebra of *-skew elements is nilpotent.

Group identities on symmetric units

Let F be an infinite field of characteristic different from 2, G a group and * an involution of G extended by linearity to an involution of the group algebra FG. Here we completely characterize the torsion groups G for which the *-symmetric units of FG satisfy a group identity. When * is the classical involution induced from g -> g(-1), g is an element of G, this result was obtained in [ A. Giambruno, S. K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443-461]. (C) 2009 Elsevier Inc. All rights reserved.

Understanding ring strain and ring flexibility in six - and eight-membered cyclic organometallic group 14 oxides

A simple model was developed for the approximation of ring strain energies of homo- and heterometallic, six- and eight-membered cyclic organometallic group 14 oxides and the degree of puckering of their ring conformations. The conformational energy of a ring is modelled as the sum of its angular strain components. The bending potential energy functions for the various endocyclic M–O–M′ and O–M–O linkages (M, M′=Si, Ge, Sn) were calculated at the B3LYP/(v)TZ level of theory using H3MOM′H3 and H2M(OH)2 as model compounds. For the six-membered rings, the minimum total angular contribution to ring strain, ERSGmin was calculated to decrease in the order: cyclo-(H2SiO)3 (13.0 kJ mol−1)>cyclo-H2Sn(OSiH2)2O (7.0 kJ mol−1)>cyclo-H2Ge(OSiH2)2O (4.9 kJ mol−1)>cyclo-H2Si(OSnH2)2O (3.4 kJ mol−1)>cyclo-(H2SnO)3 (1.7 kJ mol−1)>cyclo-H2Si(OGeH2)2O (0.8 kJ mol−1)≈cyclo-H2Ge(OSnH2)2O (0.7 kJ mol−1)>cyclo-H2Sn(OGeH2)2O (0.1 kJ mol−1)≈cyclo-(H2GeO)3 (0 kJ mol−1). All of the six-membered rings were predicted to adopt (nearly) planar conformations (a=0.996<a<1). By contrast, all eight-membered rings were predicted to adopt strainless, but puckered conformations. The degree of puckering was predicted to increase in the order: cyclo-(H2SiO)4 (a=0.983)<cyclo-H2Sn(OSiH2O)2SiH2 (a=0.959)<cyclo-(H2SiO)2(H2SnO)2 (a=0.942)< cyclo-H2Si(OSnH2O)2SiH2 (a=0.935)<cyclo-(H2SnO)4 (a=0.916)<cyclo-(H2GeO)4 (a=0.885). The differences in ring strain and the degree of puckering were linked to the different electronegativities of Si, Ge and Sn. The results obtained are consistent with experimental ring strain energies; reactivities towards ring opening polymerizations or ring expansion reactions and observed ring conformations of cyclic organometallic group 14 oxides.

Extending abelian groups to rings

For any abelian group G and any function f : G → G we define a commutative binary operation or `multiplication' on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p-group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p-group of odd order p².

Listening to student opinions about group assessment.

This paper illustrates the way two teacher-researchers are listening to mathematics education students’ voices in a Masters course. Group assignments have their advantages but it is difficult to ensure strong collaboration, high-level analysis and discussion, a good spread of work between group members, and positive social interactions. This research set out to explore one way of attending to these problems in a mathematics education Masters unit. Students submitted (unmarked) individual essays before combining them to create (graded) group assignments. They completed surveys about group work before and after this activity, and some were interviewed. Expecting individual work before group work led to increased levels of engagement, very high quality work, use of skills in analysis and critique, and good levels of student satisfaction.

On invariant Rings of Sylow subgroups of finite classical groups

In this thesis we study the invariant rings for the Sylow p-subgroups of the nite classical groups. We have successfully constructed presentations for the invariant rings for the Sylow p-subgroups of the unitary groups GU(3; Fq2) and GU(4; Fq2 ), the symplectic group Sp(4; Fq) and the orthogonal group O+(4; Fq) with q odd. In all cases, we obtained a minimal generating set which is also a SAGBI basis. Moreover, we computed the relations among the generators and showed that the invariant ring for these groups are a complete intersection. This shows that, even though the invariant rings of the Sylow p-subgroups of the general linear group are polynomial, the same is not true for Sylow p-subgroups of general classical groups. We also constructed the generators for the invariant elds for the Sylow p-subgroups of GU(n; Fq2 ), Sp(2n; Fq), O+(2n; Fq), O-(2n + 2; Fq) and O(2n + 1; Fq), for every n and q. This is an important step in order to obtain the generators and relations for the invariant rings of all these groups.

The Yang-Lee edge singularity in spin models on connected and non-connected rings

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Reactivity of mesenteric and aortic rings from trained rats fed with high caloric diet

A strong association between the benefits of physical exercise on the cardiovascular disease with an improvement of the endothelium-derived relaxing factor production has been consistently shown. The purpose of this study was to evaluate the effect of exercise training associated with high caloric diet in the reactivity of rat mesenteric and aortic rings. Experimental protocol consisted of 4 weeks of high caloric diet consumption previous to 4 weeks of run training (1.2 km/h, 0% grade, in sessions of 60 min, 5 days/week). Concentrations of triglycerides, glucose, insulin and nitrite/nitrate levels were measured and atherogenic index was calculated. Concentration-response curves to acetylcholine (10 nM-100 mu M), sodium nitroprusside (100 pM-100 nM) and phenylephrine (1 nM-3 mu M) were obtained. Exercise training reduced body mass (6%) and triglyceride levels (about 54%), without changes in glucose and insulin concentrations. An improvement of endothelium-dependent relaxation responses to acetylcholine in mesenteric and aortic rings was observed in trained group. No changes were seen for sodium nitroprusside and phenylephrine. In conclusion, our study is the first to show clearly that run training promotes an improvement of the endothelium-dependent relaxing response in aorta and mesenteric rings from rats fed with high caloric diet and that is associated with increase of NO production. (c) 2006 Elsevier B.V. All rights reserved.

Alternant and BCH codes over certain rings

Alternant codes over arbitrary finite commutative local rings with identity are constructed in terms of parity-check matrices. The derivation is based on the factorization of x s - 1 over the unit group of an appropriate extension of the finite ring. An efficient decoding procedure which makes use of the modified Berlekamp-Massey algorithm to correct errors and erasures is presented. Furthermore, we address the construction of BCH codes over Zm under Lee metric.

Exercise training improves relaxation response and SOD-1 expression in aortic and mesenteric rings from high caloric diet-fed rats

Background. Obesity has been associated with a variety of disease such as type II diabetes mellitus, arterial hypertension and atherosclerosis. Evidences have shown that exercise training promotes beneficial effects on these disorders, but the underlying mechanisms are not fully understood. The aim of this study was to investigate whether physical preconditioning prevents the deleterious effect of high caloric diet in vascular reactivity of rat aortic and mesenteric rings. Methods. Male Wistar rats were divided into sedentary (SD); trained (TR); sedentary diet (SDD) and trained diet (TRD) groups. Run training (RT) was performed in sessions of 60 min, 5 days/week for 12 weeks (70-80% VO2max). Triglycerides, glucose, insulin and nitrite/nitrate concentrations (NOx -) were measured. Concentration- response curves to acetylcholine (ACh) and sodium nitroprusside (SNP) were obtained. Expression of Cu/Zn superoxide dismutase (SOD-1) was assessed by Western blotting. Results. High caloric diet increased triglycerides concentration (SDD: 216 ± 25 mg/dl) and exercise training restored to the baseline value (TRD: 89 ± 9 mg/dl). Physical preconditioning significantly reduced insulin levels in both groups (TR: 0.54 ± 0.1 and TRD: 1.24 ± 0.3 ng/ml) as compared to sedentary animals (SD: 0.87 ± 0.1 and SDD: 2.57 ± 0.3 ng/ml). On the other hand, glucose concentration was slightly increased by high caloric diet, and RT did not modify this parameter (SD: 126 ± 6; TR: 140 ± 8; SDD: 156 ± 8 and TRD 153 ± 9 mg/dl). Neither high caloric diet nor RT modified NO x - levels (SD: 27 ± 4; TR: 28 ± 6; SDD: 27 ± 3 and TRD: 30 ± 2 μM). Functional assays showed that high caloric diet impaired the relaxing response to ACh in mesenteric (about 13%), but not in aortic rings. RT improved the relaxing responses to ACh either in aortic (28%, for TR and 16%, to TRD groups) or mesenteric rings (10%, for TR and 17%, to TRD groups) that was accompanied by up-regulation of SOD-1 expression and reduction in triglycerides levels. Conclusion. The improvement in endothelial function by physical preconditioning in mesenteric and aortic arteries from high caloric fed-rats was directly related to an increase in NO bioavailability to the smooth muscle mostly due to SOD-1 up regulation. © 2008 de Moraes et al; licensee BioMed Central Ltd.

Quaternion orders over quadratic integer rings from arithmetic fuchsian groups

In this paper we show that the quaternion orders OZ[ √ 2] ≃ ( √ 2, −1)Z[ √ 2] and OZ[ √ 3] ≃ (3 + 2√ 3, −1)Z[ √ 3], appearing in problems related to the coding theory [4], [3], are not maximal orders in the quaternion algebras AQ( √ 2) ≃ ( √ 2, −1)Q( √ 2) and AQ( √ 3) ≃ (3 + 2√ 3, −1)Q( √ 3), respectively. Furthermore, we identify the maximal orders containing these orders.