Multiplication Groups of Topological Loops

In this brief survey, we discuss the following question: Which Lie groups can occur as the multiplication groups Mult(L) of connected topological loops L? We describe correspondences between the structure of the group Mult(L) and the structure of the loop L.     

Affine Reductive Spaces of Small Dimension and Left A-Loops

In this paper we determine the at least 4-dimensional affine reductive homogeneous manifolds for an at most 9-dimensional simple Lie group or an at most 6-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable left A-loops. Using this we classify all global almost differentiable left A-loops L having either a 6-dimensional semi-simple Lie group or the group as the group topologically generated by their left translations. Moreover, we determine all at most 5-dimensional left A-loops L with as the group topologically generated by their left translations.     

Density fingering in spatially modulated Hele-Shaw cells

Density fingering of the chlorite-tetrathionate reaction has been studied experimentally in a periodically heterogeneous Hele-Shaw cell where the heterogeneity is introduced in the form of spatial modulation of gap width along the front. Depending on the spatial wavelength, gap width, and chemical composition, three types of cellular structures have been observed. The initial evolution is characterized by dispersion curves, while the long time behavior is described by the change in the autocorrelation function of the front profile and in the mixing length of the patterns.     

Copper(II) carboxylate dimers prepared from ligands designed to form a robust π···π stacking synthon: supramolecular structures and molecular properties

The reactions of bifunctional carboxylate ligands (1,8-naphthalimido)propanoate, (L(C2)(-)), (1,8-naphthalimido)ethanoate, (L(C1)(-)), and (1,8-naphthalimido)benzoate, (L(C4)(-)) with Cu(2)(O(2)CCH(3))(4)(H(2)O)(2) in methanol or ethanol at room temperature lead to the formation of novel dimeric [Cu(2)(L(C2))(4)(MeOH)(2)] (1), [Cu(2)(L(C1))(4)(MeOH)(2)]·2(CH(2)Cl(2)) (2), [Cu(2)(L(C4))(4)(EtOH)(2)]·2(CH(2)Cl(2)) (3) complexes. When the reaction of L(C1)(-) with Cu(2)(O(2)CCH(3))(4)(H(2)O)(2) was carried out at -20 °C in the presence of pyridine, [Cu(2)(L(C1))(4)(py)(4)]·2(CH(2)Cl(2)) (4) was produced. At the core of complexes 1-3 lies the square Cu(2)(O(2)CR)(4) "paddlewheel" secondary building unit, where the two copper centers have a nearly square pyramidal geometry with methanol or ethanol occupying the axial coordination sites. Complex 4 contains a different type of dimeric core generated by two κ(1)-bridging carboxylate ligands. Additionally, two terminal carboxylates and four trans situated pyridine molecules complete the coordination environment of the five-coordinate copper(II) centers. In all four compounds, robust π···π stacking interactions of the naphthalimide rings organize the dimeric units into two-dimensional sheets. These two-dimensional networks are organized into a three-dimensional architecture by two different noncovalent interactions: strong π···π stacking of the naphthalimide rings (also the pyridine rings for 4) in 1, 3, and 4, and intermolecular hydrogen bonding of the coordinated methanol or ethanol molecules in 1-3. Magnetic measurements show that the copper ions in the paddlewheel complexes 1-3 are strongly antiferromagnetically coupled with -J values ranging from 255 to 325 cm(-1), whereas the copper ions in 4 are only weakly antiferromagnetically coupled. Typical values of the zero-field splitting parameter D were found from EPR studies of 1-3and the related known complexes [Cu(2)(L(C2))(4)(py)(2)]·2(CH(2)Cl(2))·(CH(3)OH), [Cu(2)(L(C3))(4)(py)(2)]·2(CH(2)Cl(2)) and [Cu(2)(L(C3))(4)(bipy)]·(CH(3)OH)(2)·(CH(2)Cl(2))(3.37) (L(C3)(-) = (1,8-naphthalimido)butanoate)), while its abnormal magnitude in [Cu(2)(L(C2))(4)(bipy)] was qualitatively rationalized by structural analysis and DFT calculations.     

Three-Dimensional Topological Loops with Solvable Multiplication Groups

We prove that each 3-dimensional connected topological loop L having a solvable Lie group of dimension ≤5 as the multiplication group of L is centrally nilpotent of class 2. Moreover, we classify the solvable non-nilpotent Lie groups G which are multiplication groups for 3-dimensional simply connected topological loops L and dim G ≤ 5. These groups are direct products of proper connected Lie groups and have dimension 5. We find also the inner mapping groups of L.     

Diffusion-driven pattern formation in ionic chemical solutions

The driving force in diffusion-driven pattern formation is the difference in the diffusional flux of the key species, which in the case of ionic systems builds up a local electric field at the concentration gradients. The arising additional migrational flux not only decreases but also enhances the instability of the base state, depending on the charge distribution among the components. The opposite charges on the slower diffusing autocatalyst and its reacting counterpart favor pattern formation and shift the onset of instability to a smaller difference in the diffusion coefficients. The same charges, in addition to having the opposite effect, may even lead to the complete stabilization of planar reaction fronts unstable in the neutral system.     

Scaling law of stable single cells in density fingering of chemical fronts

Stable single fingers evolving in a thin layer of hydrodynamically unstable reactive solution are investigated in moderately wide Hele-Shaw cells. The mixing length associated with the final pattern is shown to be proportional to the width of the container. The scaling law, which is best illustrated by the overlay of temporal average front profiles, is demonstrated for two chemical compositions and boundaries of different thermal conductances. The observed single finger is found to be independent of the orientation of the reaction vessel for a wide range of tilt angles.     

Migration-driven instability in the chlorite-tetrathionate reaction

We have studied the lateral stability of planar reaction-diffusion fronts in an autocatalytic reaction between aqueous ions in an externally imposed electric field. In our experiments, migration drives the pattern formation leading to cellular structures where the sufficiently greater migrational flux of the reactant with respect to that of the autocatalyst is the driving force. The difference in electric field strength between the two sides of the thin reaction front results from the significant increase in conductivity during the reaction. The results of the theoretical analysis based on the empirical rate-law model of the reaction reproduce the behavior observed experimentally.