Alternating sums of binomial coefficients with unit fraction arithmetic sequence coefficients

Expressions for finite sums involving the binomial coefficients with unit fraction coefficients whose denominators form an arithmetic sequence are determined.

Feature tracking on the unit-sphere

Techniques used in a previous study of the objective identification and tracking of meteorological features in model data are extended to the unit sphere. An alternative feature detection scheme is described based on cubic interpolation for the sphere and local maximization. The extension of the tracking technique, used in the previous study, to the unit sphere is described. An example of the application of these techniques to a global relative vorticity field from a model integration are presented and discussed.

Temperature trends derived from Stratospheric Sounding Unit radiances: the effect of increasing CO2 on the weighting function

Stratospheric Sounding Units (SSU) on the NOAA operational satellites have been the main source of near global temperature trend data above the lower stratosphere. They have been used extensively for comparison with model-derived trends. The SSU senses in the 15 micron band of CO2 and hence the weighting function is sensitive to changes in CO2 concentrations. The impact of this change in weighting function has been ignored in all recent trend analyses. We show that the apparent trends in global mean brightness temperature due to the change in weighting function vary from about -0.4 K/decade to 0.4 K/decade depending on the altitude sensed by the different SSU channels. For some channels, this apparent trend is of a similar size to the trend deduced from SSU data but ignoring the change in weighting function. In the mid-stratosphere, the revised trends are now significantly more negative and in better agreement with model-calculated trends.

Unity as a unit

The arguments for regarding the number 1 as a unit of the SI are reviewed. Examples of dimensionless quantities are presented, and problems associated with representing the values of dimensionless quantities of very small magnitude are discussed. The logarithmic ratio units bel, decibel and neper are discussed.

Chill unit models for blackcurrant (Ribes nigrum L.) cultivars 'Ben Gairn', 'Ben Hope' and 'Ben Tirran'

Temperate-zone crops require a period of winter chilling to terminate dormancy and ensure adequate bud break the following spring. The exact chilling requirement of blackcurrant (Ribes nigrum), a commercially important crop in northern Europe, is relatively unknown. Chill unit models have been successfully utilized to determine the optimum chilling temperature of a range of crops, with one chill unit equating to I h exposure to the optimum temperature for chill satisfaction. Two-year-old R. nigrum plants of the cultivars 'Ben Gairn', 'Ben Hope' and 'Ben Tirran' were exposed to temperatures of -10.1 degrees C. -3.4 degrees C. 0.1 degrees C, 1.5 degrees C, 2.1 degrees C, 3.4 degrees C or 8.9 degrees C (+/- 0.7 degrees C) for durations of 0, 2, 4, 6, 8 or 10 weeks and multiple regression analyses used to determine the optimum temperature for chill satisfaction. (C) 2009 Elsevier B.V. All rights reserved.

Hydrothermal synthesis and structural characterization of novel alpha-Keggin unit-supported Cu(II)- and Mn(II)-bipyridine complexes from a tri-lacunary precursor

Blue [{Cu(2,2'-bipy)(2)}(2){alpha-SiW12O40}] (bipy = bipyridyl) (1) and pale yellow [Mn(2,2'-bipy)(3)](2)[alpha-SiW12O40] (2) have been synthesized hydrothermally and characterized by IR spectroscopy and single crystal X-ray structure analysis. In 1, the [alpha-SiW12O40](4-) ion acts as a bridge between the two [{Cu(2,2'-bipy)(2)](2+) moieties via coordination through the terminal oxygen atoms, while in 2, the [Mn(2,2'-bipy)(3)](2+) ion balances the charge on the polyoxo anion without forming any covalent bond. To the best of our knowledge, this is the first example of transition metal-mediated transformation of [alpha-SiW9O34](10-) to [alpha-SiW12O40](4-).

Dioxadiaza- and trioxadiaza-macrocycles containing one dibenzofuran unit selective for cadmium

The binding properties of dioxadiaza-([17](DBF) N2O2) and trioxadiaza- ([22](DBF)N2O3), macrocyclic ligands containing a rigid dibenzofuran group ( DBF), to metal cations and structural studies of their metal complexes have been carried out. The protonation constants of these two ligands and the stability constants of their complexes with Ca2+, Ba2+, and Mn2+, Co2+, Ni2+, Cu2+, Zn2+ and Cd2+, were determined at 298.2 K in methanol-water ( 1 : 1, v/v), and at ionic strength 0.10 mol dm(-3) in KNO3. The values of the protonation constants of both ligands are similar, indicating that no cavity size effect is observed. Only mononuclear complexes of these ligands with the divalent metal ions studied were found, and their stability constants are lower than expected, especially for the complexes of the macrocycle with smaller cavity size. However, the Cd2+ complex with [ 17]( DBF) N2O2 exhibits the highest value of stability constant for the whole series of metal ions studied, indicating that this ligand reveals a remarkable selectivity for cadmium(II) in the presence of all the metal ions studied, except copper( II), indicating that this ligand reveals a remarkable selectivity for cadmium( II) in the presence of the mentioned metal ions. The crystal structures of H-2[17](DBF)N2O32+ (diprotonated form of the ligand) and of its cadmium complex were determined by X-ray diffraction. The Cd2+ ion fits exactly inside the macrocyclic cavity exhibiting coordination number eight by coordination to all the donor atoms of the ligand, and additionally to two oxygen atoms from one nitrate anion and one oxygen atom from a water molecule. The nickel( II) and copper( II) complexes with the two ligands were further studied by UV-vis-NIR and the copper( II) complexes also by EPR spectroscopic techniques in solution indicating square-pyramidal structures and suggesting that only one nitrogen and oxygen donors of the ligands are bound to the metal. However an additional weak interaction of the second nitrogen cannot be ruled out.

New dioxadiaza- and trioxadiaza-macrocycles containing one dibenzofuran unit with two amino pendant arms: synthesis, protonation and complexation studies

New dioxadiaza- and trioxadiaza-macrocycles containing one rigid dibenzofuran unit (DBF) and N-(2-aminoethyl) pendant arms were synthesized, N,N'-bis(2-aminoethyl)-[17]( DBF) N2O2 (L-1) and N,N'-bis(2-aminoethyl)-[22](DBF)N2O3 (L-2), respectively. The binding properties of both macrocycles to metal ions and structural studies of their metal complexes were carried out. The protonation constants of both compounds and the stability constants of their complexes with Co2+, Ni2+, Cu2+, Zn2+, Cd2+, and Pb2+ were determined at 298.2 K, in aqueous solutions, and at ionic strength 0.10 mol dm(-3) in KNO3. Mononuclear complexes with both ligands were formed, and dinuclear complexes were only found for L-2. The thermodynamic binding affinities of the metal complexes of L-2 are lower than those of L-1 as expected, but the Pb2+ complexes of both macrocycles exhibit close stability constant values. On the other hand, the binding affinities of Cd2+ and Pb2+ for L-1 are very high, when compared to those of Co2+, Ni2+ and Zn2+. These interesting properties were explained by the presence of the rigid DBF moiety in the backbone of the macrocycle and to the special match between the macrocyclic cavity size and the studied larger metal ions. To elucidate the adopted structures of complexes in solution, the nickel(II) and copper( II) complexes with both ligands were further studied by UV-vis-MR spectroscopy in DMSO-H2O 1 : 1 (v/v) solution. The copper(II) complexes were also studied by EPR spectroscopy in the same mixture of solvents. The crystal structure of the copper complex of L-1 was also determined. The copper(II) displays an octahedral geometry, the four nitrogen atoms forming the equatorial plane and two oxygen atoms, one from the DBF unit and the other one from the ether oxygen, in axial positions. One of the ether oxygens of the macrocycle is out of the coordination sphere. Our results led us to suggest that this geometry is also adopted by the Co2+ to Zn2+ complexes, and only the larger Cd2+ and Pb2+ manage to form complexes with the involvement of all the oxygen atoms of the macrocyclic backbone.

Determining feature relevance for the grouping of motor unit action potentials through generative topographic mapping

The study of motor unit action potential (MUAP) activity from electrornyographic signals is an important stage on neurological investigations that aim to understand the state of the neuromuscular system. In this context, the identification and clustering of MUAPs that exhibit common characteristics, and the assessment of which data features are most relevant for the definition of such cluster structure are central issues. In this paper, we propose the application of an unsupervised Feature Relevance Determination (FRD) method to the analysis of experimental MUAPs obtained from healthy human subjects. In contrast to approaches that require the knowledge of a priori information from the data, this FRD method is embedded on a constrained mixture model, known as Generative Topographic Mapping, which simultaneously performs clustering and visualization of MUAPs. The experimental results of the analysis of a data set consisting of MUAPs measured from the surface of the First Dorsal Interosseous, a hand muscle, indicate that the MUAP features corresponding to the hyperpolarization period in the physisiological process of generation of muscle fibre action potentials are consistently estimated as the most relevant and, therefore, as those that should be paid preferential attention for the interpretation of the MUAP groupings.

Perspex machine VII: The universal perspex machine

The perspex machine arose from the unification of projective geometry with the Turing machine. It uses a total arithmetic, called transreal arithmetic, that contains real arithmetic and allows division by zero. Transreal arithmetic is redefined here. The new arithmetic has both a positive and a negative infinity which lie at the extremes of the number line, and a number nullity that lies off the number line. We prove that nullity, 0/0, is a number. Hence a number may have one of four signs: negative, zero, positive, or nullity. It is, therefore, impossible to encode the sign of a number in one bit, as floating-, point arithmetic attempts to do, resulting in the difficulty of having both positive and negative zeros and NaNs. Transrational arithmetic is consistent with Cantor arithmetic. In an extension to real arithmetic, the product of zero, an infinity, or nullity with its reciprocal is nullity, not unity. This avoids the usual contradictions that follow from allowing division by zero. Transreal arithmetic has a fixed algebraic structure and does not admit options as IEEE, floating-point arithmetic does. Most significantly, nullity has a simple semantics that is related to zero. Zero means "no value" and nullity means "no information." We argue that nullity is as useful to a manufactured computer as zero is to a human computer. The perspex machine is intended to offer one solution to the mind-body problem by showing how the computable aspects of mind and. perhaps, the whole of mind relates to the geometrical aspects of body and, perhaps, the whole of body. We review some of Turing's writings and show that he held the view that his machine has spatial properties. In particular, that it has the property of being a 7D lattice of compact spaces. Thus, we read Turing as believing that his machine relates computation to geometrical bodies. We simplify the perspex machine by substituting an augmented Euclidean geometry for projective geometry. This leads to a general-linear perspex-machine which is very much easier to pro-ram than the original perspex-machine. We then show how to map the whole of perspex space into a unit cube. This allows us to construct a fractal of perspex machines with the cardinality of a real-numbered line or space. This fractal is the universal perspex machine. It can solve, in unit time, the halting problem for itself and for all perspex machines instantiated in real-numbered space, including all Turing machines. We cite an experiment that has been proposed to test the physical reality of the perspex machine's model of time, but we make no claim that the physical universe works this way or that it has the cardinality of the perspex machine. We leave it that the perspex machine provides an upper bound on the computational properties of physical things, including manufactured computers and biological organisms, that have a cardinality no greater than the real-number line.

Perspex machine VIII: axioms of transreal arithmetic

Transreal arithmetic is a total arithmetic that contains real arithmetic, but which has no arithmetical exceptions. It allows the specification of the Universal Perspex Machine which unifies geometry with the Turing Machine. Here we axiomatise the algebraic structure of transreal arithmetic so that it provides a total arithmetic on any appropriate set of numbers. This opens up the possibility of specifying a version of floating-point arithmetic that does not have any arithmetical exceptions and in which every number is a first-class citizen. We find that literal numbers in the axioms are distinct. In other words, the axiomatisation does not require special axioms to force non-triviality. It follows that transreal arithmetic must be defined on a set of numbers that contains{-8,-1,0,1,8,&pphi;} as a proper subset. We note that the axioms have been shown to be consistent by machine proof.

Generative topographic mapping applied to clustering and visualization of motor unit action potentials

The identification and visualization of clusters formed by motor unit action potentials (MUAPs) is an essential step in investigations seeking to explain the control of the neuromuscular system. This work introduces the generative topographic mapping (GTM), a novel machine learning tool, for clustering of MUAPs, and also it extends the GTM technique to provide a way of visualizing MUAPs. The performance of GTM was compared to that of three other clustering methods: the self-organizing map (SOM), a Gaussian mixture model (GMM), and the neural-gas network (NGN). The results, based on the study of experimental MUAPs, showed that the rate of success of both GTM and SOM outperformed that of GMM and NGN, and also that GTM may in practice be used as a principled alternative to the SOM in the study of MUAPs. A visualization tool, which we called GTM grid, was devised for visualization of MUAPs lying in a high-dimensional space. The visualization provided by the GTM grid was compared to that obtained from principal component analysis (PCA). (c) 2005 Elsevier Ireland Ltd. All rights reserved.