Functions and Sets of Smooth Substructure: Relationships and Examples

The past decade has seen the introduction of a number of classes of nonsmooth functions possessing smooth substructure, e.g., “amenable functions”, “partly smooth functions”, and “g ° F decomposable functions”. Along with these classes a number of structural properties have been proposed, e.g., “identifiable surfaces”, “fast tracks”, and “primal-dual gradient structures”. In this paper we examine the relationships between these various classes of functions and their smooth substructures. In the convex case we show that the definitions of identifiable surfaces, fast tracks, and partly smooth functions are equivalent. In the nonconvex case we discuss when a primal-dual gradient structure or g ° F decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal.     

Computing the Principal Branch of log-Gamma

Thelog-Gammafunction is an important special function of mathematics, and its principal branch is required in many applications. We develop here the mathematics required to evaluate the principal branch to arbitrary precision, including a new bound for the error in Stirling's asymptotic series. We conclude with a discussion of the implementation of the principal branch of the log-Gamma function in the Maple symbolic algebra system, starting with version Maple V, Release 3.     

The energy of signed measures

We generalize the concept of energy to complex measures of finite variation. We show that although the energy dimension of a measure can exceed that of its total variation, it is always less than the Hausdorff dimension of the measure. As an application we prove a variant of the uncertainty principle.

     

Theoretical evaluation of the collection efficiency at ring-disc microelectrodes

Ring-disc microelectrodes operating as collector-generators are capable of probing multiple aspects of mass transport and are becoming popular electrochemical detectors. In this paper, the key parameters affecting the efficiency of these detectors are modelled and compared to experimental measurements by us and others. The first parameters considered pertain to electrode geometry; the disc radius and inner and outer ring radii. Our model results and experimental measurements show that varying these parameters significantly changes the sensor’s efficiency. The influence of recess and protrusion of the active surfaces from the insulating face of the electrode is also studied over a reasonable range of distances. We find that in general, protrusions will increase the efficiency of the sensor whilst recesses decrease it. The other parameters considered pertain to the flux distribution at the surface of the ring and disc electrodes, focusing on the generator electrode. The flux distribution at the generator is influenced by the voltage applied and the relative rates of mass transport or electron transfer of the species used. We find that the sensor’s efficiency increases with increasing overpotential. When the level of overpotential is sufficiently high, D and k₀ have no effect on the value of the collection efficiency. However, to establish the required level of overpotential needed to ensure a maximum collection efficiency the reaction kinetics and diffusion coefficient of a species need to be specified. Our investigations highlight the best operating modes and points out problems and features of the ring-disc microelectrode and its use as a collector-generator.     

Visualising mouse neuroanatomy and function by metal distribution using laser ablation-inductively coupled plasma-mass spectrometry imaging

Metals have a number of important roles within the brain. We used laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS) to map the three-dimensional concentrations and distributions of transition metals, in particular iron (Fe), copper (Cu) and zinc (Zn) within the murine brain. LA-ICP-MS is one of the leading analytical tools for measuring metals in tissue samples. Here, we present a complete data reduction protocol for measuring metals in biological samples, including the application of a pyramidal voxel registration technique to reproducibly align tissue sections. We used gold (Au) nanoparticle and ytterbium (Yb)-tagged tyrosine hydroxylase antibodies to assess the co-localisation of Fe and dopamine throughout the entire mouse brain. We also examined the natural clustering of metal concentrations within the murine brain to elucidate areas of similar composition. This clustering technique uses a mathematical approach to identify multiple ‘elemental clusters’, avoiding user bias and showing that metal composition follows a hierarchical organisation of neuroanatomical structures. This work provides new insight into the distinct compartmentalisation of metals in the brain, and presents new avenues of exploration with regard to region-specific, metal-associated neurodegeneration observed in several chronic neurodegenerative diseases.     

The monic integer transfinite diameter

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval . The monic integer transfinite diameter is defined as the infimum of all such supremums. We show that if has length , then .

We make three general conjectures relating to the value of for intervals of length less than . We also conjecture a value for where . We give some partial results, as well as computational evidence, to support these conjectures.

We define functions and , which measure properties of the lengths of intervals with on either side of . Upper and lower bounds are given for these functions.

We also consider the problem of determining when is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.

     

Alpha,2-, alpha,3-, and alpha,4-dehydrophenol radical anions: formation,reactivity, and energetics leading to the heats of formation of alpha,2-, alpha,3-, and alpha,4-oxocyclohexadienylidene

We have regiospecifically generated the alpha,2-, alpha,3-, and alpha,4-dehydrophenoxide anions by collisional activation of o-, m-, and p-nitrobenzoate. The alpha,2 and alpha,4 isomers also were synthesized by reacting o-benzyne radical anion with carbon dioxide and electron ionization of p-diazophenol. All three dehydrophenol radical anions were differentiated from each other and identified by probing their chemical reactivity with several reagents. Each isomer was converted to phenoxide and its corresponding quinone as well. Thermochemical measurements were carried out on all three radical anions and their hydrogen-atom affinities, proton affinities, and electron binding energies are reported. These measured quantities are combined in thermodynamic cycles to derive the heats of formation of each of the radical anions and their corresponding carbenes (i.e., alpha,2-, alpha,3-, and alpha,4-dehydrophenol). These results are compared to MCQDPT2, G3, G2+(MP2), and B3LYP calculations and experimental data for appropriate reference compounds.     

Synthesis, structures and reactions of some metallocene alcohols

E-1-Ferrocenyl-4,4-dimethylpent-2-ene-1-one has been synthesised from the Friedel-Crafts acylation of ferrocene with E-3-tert-butylacryloylchloride and converted to 1-ferrocenyl-3-chloro-4,4-dimethylpentan-1-one using ethereal hydrogen chloride. This new chloro ketone has been converted into three new ferrocene alcohols: 1-ferrocenyl-3,4-dimethyl-4-hydroxypentan-1-one, 1-ferrocenyl-3-chloro-4,4-dimethylpentan-1-ol, and 2,2,6,6-tetramethyl-3-ferrocenyl-5-chloroheptan-3-ol. A new dinuclear ferrocene derivative, E,E-2,2,9,9-tetramethyl-5,6-diferrocenyl-deca-3,7-diene, was isolated after treatment of 1-ferrocenyl-3-chloro-4,4-dimethylpentan-1-ol with acidic alumina; its structure was confirmed by X-ray crystallography, whilst electrochemistry revealed metal-metal interactions of similar magnitude to those seen for other 1,2-bis(ferrocenyl)ethane derivatives. Crystal structures have also been determined for 2,2,6,6-tetramethyl-3-ferrocenyl-5-chloroheptan-3-ol, rac-1-hydroxy[3]ferrocenophane, rac-1S,3S-1,3-diphenyl-1-hydroxy[3]ferrocenophane, and of rac-1,1^′-diphenyl-1,1^′-(1,1^′- ruthenocenediyl)dimethanol and show an intramolecular Cl?H-O hydrogen bond, a tetramer based on O?H-O hydrogen bonds, no hydrogen bonding, and a dimer with inter- and intramolecular O?H-O hydrogen bonds, respectively.