ALTERNATING UNITS AS FREE FACTORS IN THE GROUP OF UNITS OF INTEGRAL GROUP RINGS

Let G be a group of odd order that contains a non-central element x whose order is either a prime p >= 5 or 3(l), with l >= 2. Then, in U(ZG), the group of units of ZG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that < u(m), v(m)> = < u(m)> * < v(m)> congruent to Z * Z.

Involutions and free pairs of bicyclic units in integral group rings

If * : G -> G is an involution on the finite group G, then * extends to an involution on the integral group ring Z[G] . In this paper, we consider whether bicyclic units u is an element of Z[G] exist with the property that the group < u, u*> generated by u and u* is free on the two generators. If this occurs, we say that (u, u*)is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution *, Z[G] contains a free bicyclic pair.

Nil graded self-similar algebras

In [19], [24] we introduced a family of self-similar nil Lie algebras L over fields of prime characteristic p > 0 whose properties resemble those of Grigorchuk and Gupta-Sidki groups. The Lie algebra L is generated by two derivations v(1) = partial derivative(1) + t(0)(p-1) (partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...))))), v(2) = partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...)))) of the truncated polynomial ring K[t(i), i is an element of N vertical bar t(j)(p) =0, i is an element of N] in countably many variables. The associative algebra A generated by v(1), v(2) is equipped with a natural Z circle plus Z-gradation. In this paper we show that for p, which is not representable as p = m(2) + m + 1, m is an element of Z, the algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] andYa. S. Krylyuk [15] proved that for p = m(2) + m + 1 the algebra A is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always Z(p)-graded, graded nil, and are sums of two locally nilpotent subalgebras.

FREE GROUP ALGEBRAS IN DIVISION RINGS

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let sigma : L -> L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; sigma]. We show that D contains the group algebra kF of the free group F of rank 2.

A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Chemistry of pyrrocorphins: synthesis of isobacteriochlorins and pyrrocorphins bearing a methyl group at the meso position between rings A and D

A sigmatropic methyl shift from the angular position C-1 in ring to the position C-20 between rings and constitutes the crucial step in syntheses leading to a 20-methyl-isobacteriochlorin and to 20-methyl-pyrrocorphins which served as substrates in the investigation presented in the accompanying communication.

Sylow P-Subgroups of Abelian Group Rings

2000 Mathematics Subject Classification: Primary 20C07, 20K10, 20K20, 20K21; Secondary 16U60, 16S34.

Welfare outcomes for 3- and 6-month-old beef calves in a tropical environment castrated surgically or by applying rubber rings

Castration of cattle using rubber rings is becoming increasingly popular due to the perceived ease of the procedure and greater operator safety when compared with surgical castration. Few comparative studies have investigated the effects of different castration methods and calf age on welfare outcomes, particularly in a tropical environment. Thirty Belmont Red (a tropically adapted breed), 3-month-old (liveweight 71–119 kg) and 30, 6-month-old (liveweight 141–189 kg) calves were assigned to a two age × three castration (surgical, ring and sham) treatment factorial study (Surg3, Surg6, Ring3, Ring6, Sham3 and Sham6, n = 10 for each treatment group). Welfare outcomes were assessed post-castration using: behaviour for 2 weeks; blood parameters (cortisol and haptoglobin concentrations) to 4 weeks; wound healing to 5 weeks; and liveweights to 6 weeks. More Surg calves struggled during castration compared with Sham and Ring (P < 0.05, 90 ± 7% vs. 20 ± 9% and 24 ± 10%) and performed more struggles (1.9 ± 0.2, 1.1 ± 0.3 and 1.1 ± 0.3 for Surg, Sham and Ring, respectively), suggesting that surgical castration caused most pain during performance of the procedure. A significant (P < 0.05) time × castration method × age interaction for plasma cortisol revealed that concentrations decreased most rapidly in Sham; the Ring6 calves failed to show reduced cortisol concentrations at 2 h post-castration, unlike other treatment groups. By 7 h post-castration, all treatment groups had similar concentrations. A significant (P < 0.01) interaction between time and castration method showed that haptoglobin concentrations increased slightly to 0.89 and 0.84 mg/mL for Surg and Ring, respectively over the first 3 days post-castration. Concentrations for Surg then decreased to levels similar to Sham by day 21 and, although concentrations for Ring decreased on day 7 to 0.76 mg/mL, they increased significantly on day 14 to 0.97 mg/mL before reducing to concentrations similar to the other groups (0.66 mg/mL) by day 21. Significantly (P < 0.05) more of the wounds of the 3-month compared with the 6-month calves scored as ‘healed’ at day 7 (74% vs. 39%), while more (P = 0.062) of the Surg than Ring scored as ‘healed’ at day 21 (60% vs. 29%). At day 14 there were significantly (P < 0.05) fewer healed wounds in Ring6 compared with other treatment groups (13% vs. 40–60%). Liveweight gain was significantly (P < 0.05) greater in 3-month (0.53 kg/day) than in 6-month calves (0.44 kg/day) and in Sham calves (P < 0.001, 0.54 kg/day), than in Ring (0.44 kg/day) and Surg (0.48 kg/day) calves. Overall, welfare outcomes were slightly better for Surg than Ring calves due to reduced inflammation and faster wound healing, with little difference between age groups.

Welfare outcomes for 3- and 6-month-old beef calves in a tropical environment castrated surgically or by applying rubber rings

Castration of cattle using rubber rings is becoming increasingly popular due to the perceived ease of the procedure and greater operator safety when compared with surgical castration. Few comparative studies have investigated the effects of different castration methods and calf age on welfare outcomes, particularly in a tropical environment. Thirty Belmont Red (a tropically adapted breed), 3-month-old (liveweight 71–119 kg) and 30, 6-month-old (liveweight 141–189 kg) calves were assigned to a two age × three castration (surgical, ring and sham) treatment factorial study (Surg3, Surg6, Ring3, Ring6, Sham3 and Sham6, n = 10 for each treatment group). Welfare outcomes were assessed post-castration using: behaviour for 2 weeks; blood parameters (cortisol and haptoglobin concentrations) to 4 weeks; wound healing to 5 weeks; and liveweights to 6 weeks. More Surg calves struggled during castration compared with Sham and Ring (P < 0.05, 90 ± 7% vs. 20 ± 9% and 24 ± 10%) and performed more struggles (1.9 ± 0.2, 1.1 ± 0.3 and 1.1 ± 0.3 for Surg, Sham and Ring, respectively), suggesting that surgical castration caused most pain during performance of the procedure. A significant (P < 0.05) time × castration method × age interaction for plasma cortisol revealed that concentrations decreased most rapidly in Sham; the Ring6 calves failed to show reduced cortisol concentrations at 2 h post-castration, unlike other treatment groups. By 7 h post-castration, all treatment groups had similar concentrations. A significant (P < 0.01) interaction between time and castration method showed that haptoglobin concentrations increased slightly to 0.89 and 0.84 mg/mL for Surg and Ring, respectively over the first 3 days post-castration. Concentrations for Surg then decreased to levels similar to Sham by day 21 and, although concentrations for Ring decreased on day 7 to 0.76 mg/mL, they increased significantly on day 14 to 0.97 mg/mL before reducing to concentrations similar to the other groups (0.66 mg/mL) by day 21. Significantly (P < 0.05) more of the wounds of the 3-month compared with the 6-month calves scored as ‘healed’ at day 7 (74% vs. 39%), while more (P = 0.062) of the Surg than Ring scored as ‘healed’ at day 21 (60% vs. 29%). At day 14 there were significantly (P < 0.05) fewer healed wounds in Ring6 compared with other treatment groups (13% vs. 40–60%). Liveweight gain was significantly (P < 0.05) greater in 3-month (0.53 kg/day) than in 6-month calves (0.44 kg/day) and in Sham calves (P < 0.001, 0.54 kg/day), than in Ring (0.44 kg/day) and Surg (0.48 kg/day) calves. Overall, welfare outcomes were slightly better for Surg than Ring calves due to reduced inflammation and faster wound healing, with little difference between age groups.

Steroechemistry of the Pyrrolidine rings in the collagen structure

In the collagen triple-helical structure, large side groups occuring at location 3 in the repeating triplet sequences (Gly-Rz-Rz)n are appreciably constrained if a proline residue occurs as Rz in a neighbouring chain. The severity of the steric hindrance depends on the geometry of the prolyl ring. In this paper we propose two different puckerir.gs for the proline ring, the first one being energetically favorable for most types of residue sequences commonly found in collegen while the second is preferable when an amino acid residue with a large side group occurs at location 3 in a neighbouring chain. The puckering of the pyrrolidine ring of hydroxyproline, as proposed earlier, is quite favorable from energy as well as stereochemical considerations.

A density matrix renormalization group study of low-lying excitations of polythiophene within a Pariser-Parr-Pople model

Symmetrized density-matrix-renormalization-group calculations have been carried out, within Pariser-Parr-Pople Hamiltonian, to explore the nature of the ground and low-lying excited states of long polythiophene oligomers. We have exploited C-2 symmetry and spin parity of the system to obtain excited states of experimental interest, and studied the lowest dipole allowed excited state and lowest dipole forbidden two photon state, for different oligomer sizes. In the long system limit, the dipole allowed excited state always lies below the lowest dipole forbidden two-photon state which implies, by Kasha rule, that polythiophene fluoresces strongly. The lowest triplet state lies below two-photon state as usual in conjugated polymers. We have doped the system with a hole and an electron and obtained the charge excitation gap and the binding energy of the 1(1)B(u)(-) exciton. We have calculated the charge density of the ground, one-photon and two-photon states for the longer system size of 10 thiophene rings to characterize these states. We have studied bond order in these states to get an idea about the equilibrium excited state geometry of the system. We have also studied the charge density distribution of the singly and doubly doped polarons for longer system size, and observe that polythiophenes do not support bipolarons.

Nematogens with more flexible chains than aromatic rings in the core

Mesogens containing four rings in the main core can accommodate one terminal and two nearby lateral chains on each outside aromatic ring. These compounds containing six chains present an enantiotropic nematic range which is influenced by the rigidity of the links. The conformational behaviour of the first methyleneoxy group within the chains was investigated by one and two dimensional C-13 NMR. The sign of the jump in chemical shifts when entering the nematic phase indicates the folding of each lateral branch. Dipolar oscillations during cross-polarization contact provide the values of the bond order parameter. The two First lateral fragments do not behave in the same way, demonstrating the influence of the fragment along which the chain is back: folded.

Successful data recovery from oscillation photographs containing strong polycrystalline diffraction rings from an unknown small-molecule contaminant: preliminary structure solution of Salmonella typhimurium pyridoxal kinase (PdxK)

Pyridoxal kinase (PdxK; EC 2.7.1.35) belongs to the phosphotransferase family of enzymes and catalyzes the conversion of the three active forms of vitamin B-6, pyridoxine, pyridoxal and pyridoxamine, to their phosphorylated forms and thereby plays a key role in pyridoxal 5 `-phosphate salvage. In the present study, pyridoxal kinase from Salmonella typhimurium was cloned and overexpressed in Escherichia coli, purified using Ni-NTA affinity chromatography and crystallized. X-ray diffraction data were collected to 2.6 angstrom resolution at 100 K. The crystal belonged to the primitive orthorhombic space group P2(1)2(1)2(1), with unitcell parameters a = 65.11, b = 72.89, c = 107.52 angstrom. The data quality obtained by routine processing was poor owing to the presence of strong diffraction rings caused by a polycrystalline material of an unknown small molecule in all oscillation images. Excluding the reflections close to powder/polycrystalline rings provided data of sufficient quality for structure determination. A preliminary structure solution has been obtained by molecular replacement with the Phaser program in the CCP4 suite using E. coli pyridoxal kinase (PDB entry 2ddm) as the phasing model. Further refinement and analysis of the structure are likely to provide valuable insights into catalysis by pyridoxal kinases.

Model studies of Ziegler-Natta olefin polymerization using group 3 and group 4 metallocenes

Evidence for the stereochemical isomerization of a variety of ansa metallocene compounds is presented. For the scandocene allyl derivatives described here, we have established that the process is promoted by a variety of salts in both ether and hydrocarbon solvents and is not accelerated by light. A plausible mechanism based on an earlier proposal by Marks, et al., is offered as an explanation of this process. It involves coordination of anions and/or donor solvents to the metal center with cation assistance to encourage metalcyclopentadienyl bond heterolysis, rotation about the Si-Cp bond of the detached cyclopentadienide and recoordination of the opposite face. Our observations in some cases of thermodynamic racemic:meso ratios under the reaction conditions commonly used for the synthesis of the metallocene chlorides suggests that the interchange is faster than metallation, such that the composition of the reaction mixture is determined by thermodynamic, not kinetic, control in these cases.

Two new ansa-scandocene alkenyl compounds react with olefins resulting in the formation of η³-allyl complexes. Kinetics and labeling experiments indicate a tuck-in intermediate on the reaction pathway; in this intermediate the metal is bound to the carbon adjacent to the silyllinker in the rear of the metallocene wedge. In contrast, reaction of permethylscandocene alkenyl compounds with olefins results, almost exclusively, in vinylic C-H bond activation. It is proposed that relieving transition state steric interactions between the cyclopentadienyl rings and the olefin by either linking the rings together or using a larger lanthanide metal may allow for olefin coordination, stabilizing the transition state for allylic σ-bond metathesis.

A selectively isotopically labeled propylene, CH₂CD(¹³CH₃), was synthesized and its polymerization was carried out at low concentration in toluene solution using isospecific metallocene catalysts. Analysis of the NMR spectra (¹³C, ¹H, and ²H) of the resultant polymers revealed that the production of stereoerrors through chain epimerization proceeds exclusively by the tertiaryalkyl mechanism. Additionally, enantiofacial inversion of the terminally unsaturated polymer chain occurs by a non-dissociative process. The implications of these results on the mechanism of olefin polymerization with these catalysts is discussed.

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator

A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2^n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.

In Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.

A reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.

For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.