An upper bound on the Chebotarev invariant of a finite group

A subset {g ₁,..., g _d } of a finite group G invariably generates \(\left\{ {g_1^{{x_1}}, \ldots ,g_d^{{x_d}}} \right\}\) generates G for every choice of x _i ∈ G. The Chebotarev invariant C(G) of G is the expected value of the random variable n that is minimal subject to the requirement that n randomly chosen elements of G invariably generate G. The first author recently showed that \(C\left( G \right) \leqslant \beta \sqrt {\left| G \right|} \) for some absolute constant β. In this paper we show that, when G is soluble, then β is at most 5/3. We also show that this is best possible. Furthermore, we show that, in general, for each ε > 0 there exists a constant c _ε such that \(C\left( G \right) \leqslant \left( {1 + \in } \right)\sqrt {\left| G \right|} + {c_ \in }\).     

Unsymmetrical 1λ3-1,2,4,6-thiatriazinyls with aryl and trifluoromethyl substituents: synthesis, crystal structures, EPR spectroscopy, and voltammetry

A general synthetic route to 3-trifluoromethyl-5-aryl-1λ(3)-1,2,4,6-thiatriazinyl radicals was developed. X-ray structures were obtained for all five neutral radicals and show that they exist in the solid state as cofacial dimers linked by S···S contacts. X-ray structures were also obtained for two of the precursor chlorothiatriazines along with several aryl N-imidoylamidines, p-methoxybenzamidine, and N-chlorosulfonyl-N,N'-benzamidine. Cyclic voltammetric studies were performed on the [R(2)C(2)N(3)S](?) radicals in CH(3)CN and CH(2)Cl(2) with [(n)Bu(4)N][PF(6)] as the supporting electrolyte under vacuum conditions in an all-glass electrochemical cell. The results provide quasi-reversible formal potentials for the [R(2)C(2)N(3)S](-/0) process in the range of -0.61 to -0.47 V, irreversible peak potentials for the [R(2)C(2)N(3)S](0/+) process from 0.59 to 0.91 V at lower concentrations, and the appearance of a second, reversible oxidation process from 0.69 to 0.94 V at higher concentrations (versus the Fc(0/+) couple; Fc = ferrocene). This behavior was indicative of monomer-dimer equilibrium in solution, as ascertained from digital models of the voltammograms. There is a small but measurable trend in both the oxidation and reduction potentials with varying remote aryl substituents. EPR spectra were obtained for all five neutral radicals in CH(2)Cl(2) solutions, which confirm the concentration of the unpaired electron density on the heterocyclic core. Trends were also seen in the hyperfine splitting constants a(N) with varying remote aryl substituents. Calculations were performed for all three oxidation states of the [R(2)C(2)N(3)S](-/?/+) monomeric rings; the resulting theoretical redox energies correlate well with solution phase voltammetric data.     

Integration of a macro/micro architectured compartmentalised neuronal culture device using a rapid prototyping moulding process

The rapid prototyping of a reversible and one step moulded compartmentalised neuron glass/PDMS device with a thin wall barrier directly adjacent to the reservoirs is presented. A simple moulding technique to produce these devices results in a barrier of 560 μm where the 3 μm deep by 8 μm wide channels can be reversibly fabricated in either the glass base or PDMS compartmentalised mould depending on the type of application required. Using glass substrates with commercially laser engraved microchannels, both the PDMS planar and PDMS channelled device can be easily fabricated in a standard laboratory. The compartmentalised device has several advantages including good experimental accessibility and versatility with a variety of end user applications.     

Studies of Specific Adsorption Sites for Sodium Ion and Water on the Interface of Mixed Micelles of Novel Type II Lyomesophases

Novel aqueous type II lyotropic mesophases that align in magnetic fields have been synthesized for binary and ternary mixtures of amphiphiles. A system in which major components of the micelle bilayer are decyltrimethylammonium and dodecanoate ions is an aligning mesophase from 100% to 35% dodecanoate, but replacement of the cationic amphiphile with hexadecyltrimethylammonium facilitates the extension to aligning mesophases up to 100% replacement of the dodecanoate ions. A study of sodium and deuterium quadrupole splittings observed in their nuclear magnetic resonance spectra, confirms and extends a three site adsorption theory for sodium ion in the mixed head group interface. The first site is associated with more strongly hydrated ions and head groups and involves adjacent dodecanoate head groups, the second is not so strongly hydrated and involves one dodecanoate group while the third is a weakly bound water at - N(CH₃)⁺ groups and a sodium remote from the interface layer in close to average isotropic motion.

Mixed detergent mesophases of type II, which spontaneously align in the field, have also been synthesized from variable mixtures of hexadecyltrimethylammonium bromide and decylammonium chloride and from mixtures of potassium dodecanoate and sodium decylsulphate. In the mixed cationic system the sodium quadrupole splittings are always low but water is tightly bound to the -NH₃ ⁺ head group. In the anionic detergent mixture sodium appears about as strongly bound to dodecanoate and decylsulphate head groups. In most of the mixed mesophases some decanol is required (maximum 25% of the bilayer) in order to render the desired type II mesophase stable over a range of mixtures of the charged detergents. Specific site adsorption for sodium and water, peculiar to different head groups renders the effect of the added decanol as primarily a convenient diluent in the bilayer to preserve mesophase integrity.     

Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

Aluminium

Ohne Zusammenfassung     

14N nuclear quadrupole resonance in nicotinic acid derivatives

Nitrogen-14 nuclear quadrupole resonance (NQR) spectra of nicotinic acid derivatives, known as antitumor agents, are reported and analyzed in the framework of the Townes and Dailey theory. A strong correlation exists between the charge difference σ_NC ? π at the pyridinic nitrogen and the inhibition of acetylcholinesterase by nicotinic acid derivatives, in which the inhibition potency (pI₅₀) increases as the charge difference σ_NC ? π increases. There are significant differences between ¹⁴N NQR spectra for the pyridinic nitrogen sites of nicotinic acid and nicotinamide; these can be explained in terms of the differences in hydrogen bond lengths.