Expansions for the Bollob��s-Riordan Polynomial of Separable Ribbon Graphs

We define 2-decompositions of ribbon graphs, which generalize 2-sums and tensor products of graphs. We give formulae for the Bollobás-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.     

Picard's successive approximation for non-linear two-point boundary-value problems

It is demonstrated that Picard's successive approximation provides a simple and efficient method for solving linear and non-linear two-point boundary-value problems. For problems, where intrinsic convergence is slow, the method can be readily modified to speed up convergence.     

A Series of Collaborations between Various Pharmaceutical Companies and Regulatory Authorities Concerning the Analysis of Biomolecules Using Capillary Electrophoresis: Additional Instruments/Buffer

An international project team (including members from US, Canada and UK) was formed from a number of interested biopharmaceutical companies and regulatory authorities to conduct a cross-organisation collaboration exercise. The results of the first comparison with eight different organisations that used instruments of the same equipment model, the same reagents, and the same methodology has been reported previously [1]. This report represents the addition of other instruments using a different run buffer. The relative migration times were different, as expected, prohibiting a direct comparison between companies. The within-organisation variability was low for both relative migration time (<0.34% RSD% for all companies save one) and the peak area (<5% RSD% for all companies save one) when measuring the purity of a representative IgG sample. The apparent molecular weight of bovine serum albumin was measured with good precision (less than 10% RSD% across all companies) to the theoretical value when all data is utilized (67.5 kDa compared to 66.4 kDa). For a representative IgG sample, the three main components, IgG Light Chain, IgG Non-glycosylated Heavy Chain, and IgG Heavy Chain, could not be separated, specifically the IgG Non-glycosylated Heavy Chain and IgG Heavy Chain. When the IgG Non-glycosylated Heavy Chain and IgG Heavy Chain were combined for all organisations, the fractional peak area for the IgG Light Chain and IgG Non-glycosylated Heavy Chain + IgG Heavy Chain peak also showed excellent agreement, with less than 7.5 and 3.5% RSD%, respectively. The value of this exercise is in demonstrating the reliability of CE for the determination of apparent size of biopharmaceutical proteins. This underpins the appropriate use of such CE data in support of regulatory submissions.     

Simultaneous diode laser based in situ quantification of oxygen, carbon monoxide, water vapor, and liquid water in a dense water mist environment

We designed a three-channel tunable diode laser absorption spectrometer to make simultaneous in situ measurements of oxygen, carbon monoxide and water vapor in fire environments to aid in the evaluation of water-based fire suppression systems. The instrument also provides simultaneous information on the liquid water content via optical density measurements by evaluation of the captured direct absorption waveforms. We evaluated the spectrometer in a 1350 L test enclosure containing a small propane flame and a piezoelectric generated water mist of sub-10 μm drops. Mist drop number densities were on the order of 10⁶ cm⁻³ resulting in transmission losses of greater than 99.99% for a sample pathlength of 21.6 cm. We were able to detect and quantify oxygen from transmission levels of 100% down to 0.01% with uncertainties of 0.1 and 0.4 vol%, respectively. The water vapor concentration uncertainty never exceeded 0.06 vol% even during times of heavy mist loading. Carbon monoxide levels produced in the test enclosure were below the 250 ppm detection limit determined from analysis of the noise levels in our detected signals. The liquid water content uncertainty was less than for values that typically reached .     

A Penrose polynomial for embedded graphs

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which cannot be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion–contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable chequerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G$G$ can be expressed as a sum of chromatic polynomials of twisted duals of G$G$. This allows us to obtain a new reformulation of the Four Colour Theorem.     

Partial duality and Bollobás and Riordan’s ribbon graph polynomial

Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan’s ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation which used the homfly polynomial of a knot.     

Correlation Analysis of the Alternating Step Generator

The alternating step generator is a well-known keystream generator consisting of two stop/go clocked LFSRs, LFSR₁ and LFSR₂, whose clocks are controlled by another LFSR, LFSR₃, which is clocked regularly. A probabilistic analysis of this generator is conducted which shows that the posterior probabilites of individual bits of the first derivatives of the regularly clocked LFSR₁ and LFSR₂ sequences, when conditioned on a given segment of the first derivative of the keystream sequence, can be computed efficiently in a number of probabilistic models of interest. The expected values of these probabilities, for a random keystream sequence, are derived by an approximate theoretical analysis and are also verified by systematic computer experiments. It is pointed out that these posterior probabilities can be enhanced in a resynchronization scenario and thus used for a low-complexity fast correlation attack on the two LFSRs. More generally, it is argued that even without resynchronization these probabilities may be significantly different from one half for fast correlation attacks based on iterative decoding algorithms to be successful, although with incresead complexity. A related method for computing the posterior probabilities of individual bits of the LFSR₃ sequence, when conditioned on both the keystream sequence and the LFSR₁ and LFSR₂ sequences, is also developed. As these posterior probabilities are much more different from one half, they can be used for a low-complexity fast correlation attack on LFSR₃, provided that the initial states of LFSR₁ and LFSR₂ are previously reconstructed.     

The development of information storage materials — How microscopy can help?

The response of giant magnetoresistance (GMR) devices depends critically on the film microstructure, with parameters such as layer thickness and interfacial abruptness being crucial. This paper presents results obtained using high resolution electron microscopy (HREM), chemical mapping and atom probe microanalysis. Local variations in the magnetic properties are induced by the microstructure and also when the films are patterned to form small elements. These lead to changes in the magnetization reversal mechanism. Some results of the studies of the magnetization reversal carried out using in situ in Lorentz transmission electron microscopy (LTEM) magnetizing experiments are also included.