Zero divisors and units with small supports in group algebras of torsion-free groups

Kaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field 𝔽, the group ring 𝔽[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in 𝔽[G] whose supports have size 3. For any field 𝔽 and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥10. If 𝔽 = 𝔽₂ is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields.     

A Straightforward Method for the Synthesis of Functionalized Trisubstituted Alkenes through Na2S/Al2O3 Catalyzed Knoevenagel Condensation

Na₂S/Al₂O₃ catalyzed the Knoevenagel condensation between an aldehyde and an active methylene compound to yield various trisubstituted alkenes.     

Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

Let be a bounded starshaped domain. In this note we consider critical points of the functional

where of class satisfies the natural growth

for some and 0$">, is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).

     

Synthesis and characterization of new 13 and 14-membered macrocycles and their transition metal complexes

Two new macrocyclic ligands 1,4,7,9,12-pentaaza-10,11-dioxo-8,9,12,13-bis-(1-oxo-3-thio-2-hydropyrimidine)-trideca-7,13-diene, (L¹) and 1,4,7,9,12-pentaaza-10,12-dioxo-8,9,13,14-bis-(1-oxo-3-thio-2-hydropyrimidine)-tetradeca-7,14-diene, (L²) and their complexes with Cr^III, Mn^II, Fe^III, Co^II, Ni^II, Cu^II and Zn^II have been synthesized, and characterized by elemental analysis, i.r., ¹H-n.m.r., e.p.r., u.v.–vis. spectroscopy, magnetic susceptibility and conductance measurements. The conductivity measurements suggest that the complexes of divalent metal ions are 1:1 electrolytes whereas the trivalent metal ions are non-electrolytes. On the basis of electronic spectra and magnetic moment measurements the Cr^III and Fe^III complexes are octahedral, while the divalent metal complexes are tetrahedral except for the Ni^II and Cu^II complexes which are proposed to have square planar geometry. All the ligands and their complexes have been screened against gram-positive bacteria Staphylococcus aureus and gram-negative bacteria E. coli. The results show that they inhibit the growth of bacteria.     

Thermodynamics of the solvent extraction of thorium and europium nitrates by neutral phosphorylated ligands

Solvent extraction of tetravalent thorium and trivalent europium ions from nitrate media into dichloromethane solution of triphenylphosphine oxide (TPPO) has been studied. The extractant was shown to be more efficient for europium than for thorium. A conventional log-log analysis reveals that the extraction of both metal nitrates takes place via the formation of the species with 1 : 2 metal to ligand ratio. Thermodynamic parameters i.e., ΔG°, ΔH° and ΔS° of the extraction process using 1,2-dichloroethane as diluent have been calculated based on the influence of the temperature on extraction equilibria in the range 293-313 K. While the extraction of europium is controlled by enthalpy changes, the extraction of thorium is an endothermic process and is driven by entropy changes. A comparison of these data with those obtained for the extraction of europium and thorium nitrates by two other related phosphorylated ligands, tri-n-octylphosphine oxide (TOPO) and diphenyl-N,N-dimethylcarbamoylmethylphosphine oxide (DФDMCMPO), indicates that DФDMCMPO coordinates presumably as a chelating ligand. This revised version was published online in July 2006 with corrections to the Cover Date.     

An enzymatic flow analysis method for the determination of phosphatidylcholine in sediment pore waters and extracts

A sensitive and selective flow injection method for the determination of phosphatidylcholine (PC) in sediment pore waters and extracts is described. It involves the use of phospholipase C, alkaline phosphatase and choline oxidase co-immobilized on controlled pore glass in a packed column reactor. The final product of the enzymatic reaction of phosphatidylcholine is hydrogen peroxide, and this is detected by measuring the chemiluminescence emission resulting from cobalt(II) catalysed reaction with luminol. The flow injection method is rapid (30 injections/h), reproducible (1.4% R.S.D. at 3 μM PC, n = 10) with a detection limit of 0.14 μM (estimated from 3σ_n−1 of the measured blank). A linear calibration response was obtained over a concentration range of 0.5-9 μM (r = 0.999). The method has been applied to the determination of phosphatidylcholine in sediment extracts and sediment pore waters.     

Synthesis of new 14- and 16-membered macrocycles and their transition metal complexes

Summary Benzoylacetic acid (1 mol) interacts with ethylenediamine or with propanediamine (2 mol) to yield new N₄ macrocycles 1,5,8,12-tetraazacyclotetradeca-2,4,9,11-tetraphenyl-3, 10-dicarboxylic-4,11-diacetic acid- 1,8-diene (L¹) and 1,5,9,13-tetraazacyclohexadeca-2,4,10,12-tetraphenyl-3, 11-dicarboxylic-4,12-diacetic acid-1,9-diene (L²), respectively. These macrocycles have been successfully complexed with Cr^III, Fe^III, Mn^II, Co^II, Ni^II, Cu^II and Zn^II. The complexes of the divalent metal ions are non-electrolytes, while those of Fe^III and Cr^III are 1:1 electrolytes in DMSO. On the basis of ligand field spectra and magnetic moments an octahedral geometry has been proposed for all the complexes.     

Influences of reinforcing particle and interface bonding strength on material properties of Mg/nano-particle composites

In the present study, an effective model is proposed to predict the effective elastic behavior of the three-phase composite containing spherical inclusions, each of which is surrounded by an interphase layer. The constitutive equations are derived for the stress and strain of each phase of the composite subjected to a far-field tension. Based on these constitutive laws, the effective bulk, shear and Young’s modulus are obtained. A statistical debonding criterion is adopted to characterize the varying probability of the evolution of interphase debonding. Influences of debonding damage, particle volume fraction, interphase properties and bonding strength on overall mechanical behavior of composites are also discussed. Numerical analyses are carried out on particle-reinforced composites and the predictions have a good agreement with the experimental results.