Minimality conditions on circularly ordered structures

We explore analogues of o‐minimality and weak o‐minimality for circularly ordered sets. Much of the theory goes through almost unchanged, since over a parameter the circular order yields a definable linear order. Working over ?? there are differences. Our main result is a structure theory (with infinitely many doubly transitive examples related to Jordan permutation groups) for ?₀‐categorical weakly circularly minimal structures. There is a 5‐homogeneous (or ‘5‐indiscernible’) example which is not 6‐homogeneous, but any example which is k‐homogeneous for some k ≥ 6 is k‐homogeneous for all k. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Visualizing zeptomole (electro)catalysis at single nanoparticles within an ensemble

The relationship between the structural properties, such as the size and the shape, of a catalytic nanoparticle and its reactivity is a key concept in (electro)catalysis. Current understanding of this relationship is mainly derived from studies involving large ensembles of nanoparticles (NPs). However, the results necessarily reflect the average catalytic behavior of an ensemble, even though the properties of individual particles may vary widely. Here, we demonstrate a novel approach using scanning electrochemical cell microscopy (SECCM) to locate and map the reactivity of individual NPs within an electrocatalytic ensemble, consisting of platinum NPs supported on a single carbon nanotube. Significantly, our studies show that subtle variations in the morphology of NPs lead to dramatic changes in (potential-dependent) reactivity, which has important implications for the design and assessment of NP catalysts. The instrumental approach described is general and opens up new avenues of research in functional imaging, nanoscale electron transfer, and catalysis.     

Uncountable Homogeneous Partial Orders

A partially ordered set (P, ≤) is called k‐homogeneous if any isomorphism between k‐element subsets extends to an automorphism of (P, ≤). Assuming the set‐theoretic assumption ⋄(ϰ₁), it is shown that for each k, there exist partially ordered sets of size ϰ₁ which embed each countable partial order and are k‐homogeneous, but not (k + 1)‐homogeneous. This is impossible in the countable case for k ≥ 4.     

Dinuclear cyano-bridged CoIII-FeII complexes as precursors for molecular mixed-valence complexes of higher nuclearity

The preparation and characterization of a series of trinuclear mixed-valence cyano-bridged Co(III)-Fe(II)-Co(III) compounds derived from known dinuclear [[L(n)Co(III)(mu-NC)]Fe(II)(CN)(5)](-) complexes (L(n)() = N(5) or N(3)S(2) n-membered pendant amine macrocycle) are presented. All of the new trinuclear complexes were fully characterized spectroscopically (UV-vis, IR, and (13)C NMR). Complexes exhibiting a trans and cis arrangement of the Co-Fe-Co units around the [Fe(CN)(6)](4-) center are described (i.e., cis/trans-[{L(n)Co(III)(mu-NC)](2)Fe(II)(CN)(4)](2+)), and some of their structures are determined by X-ray crystallography. Electrochemical experiments revealed an expected anodic shift of the Fe(III/II) redox potential upon addition of a tripositively charged [Co(III)L(n)] moiety. The Co(III/II) redox potentials do not change greatly from the di- to the trinuclear complex, but rather behave in a fully independent and noncooperative way. In this respect, the energies and extinction coefficients of the MMCT bands agree with the formal existence of two mixed-valence Fe(II)-CN-Co(III) units per molecule. Solvatochromic experiments also indicated that the MMCT band of these compounds behaves as expected for a class II mixed-valence complex. Nevertheless, its extinction coefficient is dramatically increased upon increasing the solvent donor number.     

Homogeneity in infinite permutation groups

It is shown that ifG is a permutation group on an infinite setX, andG is (k?1)-transitive but notk-transitive (wherek ≥ 5), then the following hold:

1. G is not (k + 3)-homogeneous.
2. IfG is (k + 2)-homogeneous, then the group induced byG on ak-subset ofX is the alternating groupA _k .

     

In-situ atomic force microscopy (AFM) imaging: influence of AFM probe geometry on diffusion to microscopic surfaces

The effect of AFM probe geometry on diffusion to micrometer-scale reactive (electrode) interfaces is considered. A disk-shaped substrate electrode was held at a potential to reduce a species of interest (aqueous Ru(NH 3) 6 (3+)) at a diffusion-controlled rate and the current response during AFM imaging provided information on local mass transport to the interface. This approach reveals how the AFM probe influences diffusion to a reactive surface, which is of importance in more clearly delineating the conditions under which in-situ AFM can be treated as a noninvasive probe of surface processes involving mass transport (e.g., electrode reactions and crystal dissolution and growth). An assessment has been made of three types of probes: V-shaped silicon nitride contact mode probes; single beam silicon probes; and batch-fabricated scanning electrochemical-atomic force microscopy (SECM-AFM) probes. Two disk electrodes, (6.1 microm and 1.6 microm diameter) have been considered as substrates. The results indicate that conventional V-shaped contact mode probes are the most invasive and that the batch-fabricated SECM-AFM probes are the least invasive to diffusion at both of the substrates used herein. The experimental data are complemented by the development of simulations based on a simple 2D model of the AFM probe and active surface site. The importance of probe parameters such as the cantilever size, tip cone height, and cone angle is discussed, and the implications of the results for studies in other areas, such as growth and dissolution processes, are considered briefly.     

Model theory of finite and pseudofinite groups

This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.