New mass spectrometric approaches to structural analysis in mixtures

Structural analysis of minor components in mixtures is a vital requirement in the development of any pharmaceutical compound. Mass spectrometry is uniquely able to give this kind of information on the trace amounts of material present as minor impurities in a drug substance. In this study we show that a combination of mass spectrometric analysers with different characteristics is an even more powerful approach with a higher chance of establishing a potential structure. In particular the advent of analysers capable of accurate mass measurement on small amounts of material has enabled structures to be proposed in situations where previously no real conclusions could be made. Copyright 1999 John Wiley & Sons, Ltd.     

Rational matrix approximation with a priori error bounds for non-symmetric matrix Riccati equations with analytic coefficients

In this paper the exact solution of the non-symmetric matrixRiccati equation with analytic coefficients is approximatedby a rational matrix function with a prefixed accuracy. Thisrational matrix function is locally defined as the exact solutionof a Riccati problem with matrix polynomial coefficients obtainedby truncation of the Taylor expansions of the matrix coefficientsof the original problem.     

Combined compact difference method for solving the incompressible Navier–Stokes equations

This paper presents a numerical method for solving the two‐dimensional unsteady incompressible Navier–Stokes equations in a vorticity–velocity formulation. The method is applicable for simulating the nonlinear wave interaction in a two‐dimensional boundary layer flow. It is based on combined compact difference schemes of up to 12th order for discretization of the spatial derivatives on equidistant grids and a fourth‐order five‐ to six‐alternating‐stage Runge–Kutta method for temporal integration. The spatial and temporal schemes are optimized together for the first derivative in a downstream direction to achieve a better spectral resolution. In this method, the dispersion and dissipation errors have been minimized to simulate physical waves accurately. At the same time, the schemes can efficiently suppress numerical grid‐mesh oscillations. The results of test calculations on coarse grids are in good agreement with the linear stability theory and comparable with other works. The accuracy and the efficiency of the current code indicate its potential to be extended to three‐dimensional cases in which full boundary layer transition happens. Copyright © 2011 John Wiley & Sons, Ltd.     

The point decomposition problem over hyperelliptic curves

Computing discrete logarithms is generically a difficult problem. For divisor class groups of curves defined over extension fields, a variant of the Index-Calculus called decomposition attack is used, and it can be faster than generic approaches. In this situation, collecting the relations is done by solving multiple instances of the point m-decomposition problem (PDP\(_m\)). An instance of this problem can be modelled as a zero-dimensional polynomial system. Solving is done with Gröbner bases algorithms, where the number of solutions of the system is a good indicator for the time complexity of the solving process. For systems arising from a PDP\(_m\) context, this number grows exponentially fast with the extension degree. To achieve an efficient harvesting, this number must be reduced as much as possible. Extending the elliptic case, we introduce a notion of summation ideals to describe PDP\(_m\) instances over higher genus curves, and compare to Nagao’s general approach to PDP\(_m\) solving. In even characteristic we obtain reductions of the number of solutions for both approaches, depending on the curve’s equation. In the best cases, for a hyperelliptic curve of genus g, we can divide the number of solutions by \(2^{(n-1)(g+1)}\). For instance, for a type II genus 2 curve defined over \(\mathbb {F}_{2^{93}}\) whose divisor class group has cardinality a near-prime 184 bits integer, the number of solutions is reduced from 4096 to 64. This is enough to build the matrix of relations in around 7 days with 8000 cores using a dedicated implementation.     

Noncommutative induced gauge theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field. We find that the gauge-invariant effective action involves, beyond the expected noncommutative version of the pure Yang–Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ⁴-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.