Digital Sums and Divide-and-Conquer Recurrences: Fourier Expansions and Absolute Convergence

We propose means for computing the Fourier expansions of periodic functions appearing in higher moments of the sum-of-digits function and in the solutions of some divide-and-conquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently compute numerically the coefficients involved to high precision.     

The fundamental cycle of concept construction underlying various theoretical frameworks

In this paper, the development of mathematical concepts over time is considered. Particular reference is given to the shifting of attention from step-by-step procedures that are performed in time, to symbolism that can be manipulated as mental entities on paper and in the mind. The development is analysed using different theoretical perspectives, including the SOLO model and various theories of concept construction to reveal a fundamental cycle underlying the building of concepts that features widely in different ways of thinking that occurs throughout mathematical learning.     

Thinking and Turing's test

The questions raised by A. M. Turing in his paper on thought and machines are discussed. Human thought is considered in turn as a concept of normal language usage, as a basic concept of psychology, and as the basis of intellectual activity. It is concluded that neither of these notions of thought identifies something specific that a human being can or cannot do. The imitation game proposed by Turing for deciding whether a machine can think is found to result from an arbitrary empoverishment of the channel of communication between the interrogator and the item under investigation. Turing's notions of thinking are shown to lead to logical difficulties. An alternative view of consciousness, that would place it beyond the reach of any finite test, is finally discussed.     

The accurate dimensional measurements within small internal structures

This paper shows that the dimension of internal structure of a component can be measured remotely to a resolution better than 5 μm by use of the combination of high resolution optical microscopy and image processing.A specialised high resolution, long working distance and diffraction limited lens was used to image within the component. A series of digital images were made to map spatially the interior of the component. An edge detector was then used to localise the specific location of features in order that an accurate internal measurement could be made. A visualisation of the internal surface finish was also achieved.     

Fluorination of diamond — C4F9I and CF3I photochemistry on diamond (100)

The radiation-induced decomposition of C₄F₉I and CF₃I overlayers at 119 K on diamond (100) surfaces has been shown to be an efficient route to fluorination of the diamond surface. X-ray photoelectron spectroscopy has been used for photoactivation as well as for studying the photodecomposition of the fluoroalkyl iodide molecules, the attachment of the photofragments to the diamond surface, and the thermal decomposition of the fluoroalkyl ligands. Measured chemical shifts agree well with ab initio calculations of both C 1s and F 1s binding energies. It is found that chemisorbed CF₃ groups on diamond (100) decompose by 300 K whereas C₄F₉ groups decompose over the range 300 to 700 K and this reactivity difference is rationalized on steric grounds. Both of these thermal decomposition processes produce surface C---F bonds on the diamond. The surface C---F species thermally decompose over a wide temperature range extending up to 1500 K. Hydrogen passivation of the diamond surface is ineffective in preventing free radical attack from the photodissociated products of the fluoroalkyl iodides; I atoms produced photolytically abstract H from surface C---H bonds to yield hydrogen iodide at 119 K allowing diamond fluorination. The attachment of chemisorbed F species to the diamond (100) surface causes band bending as the surface states are occupied as a result of chemisorption. This results in a shift to higher binding energy of the diamond-related C 1s levels present in the surface and subsurface regions which are sampled by XPS on the diamond. The use of photoactivation of fluoroalkyl iodides for the fluorination of diamond surfaces provides a convenient route compared to other methods involving the action of atomic F, molecular F₂, XeF₂ and F-containing plasmas.     

The Eta Invariant of Pin Manifolds with Cyclic Fundamental Groups

Periodica Mathematica Hungarica - Let ?= 2ν &;gt; 1. Let M be an orientable manifold of odd dimension m with $\pi _1 (M) = \mathbb{Z}_\ell $ whose universal cover $\tilde M$ is...