Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.

Selection rules for rovibronic transitions in symmetric top molecules

A simple diagrammatic rule is presented for determining the rotational selection rules governing transitions between any pair of vibronic states in electric dipole spectra of symmetric top molecules. The rule is useful in cases where degenerate vibronic levels with first-order Coriolis splittings occur, because it gives immediately the selection rule for the (+l) and (-l) components in any degenerate state. The rule is also helpful in determining the symmetry species and the effective zeta constants in overtone and combination levels involving degenerate vibrations. Particular attention is devoted to the conventions concerning the signs of zeta constants.

Raman selection rules for vibration-rotation transitions in symmetric top molecules

Symmetry restrictions on Raman selection rules can be obtained, quite generally, by considering a Raman allowed transition as the result of two successive dipole allowed transitions, and imposing the usual symmetry restrictions on the dipole transitions. This leads to the same results as the more familiar polarizability theory, but the vibration-rotation selection rules are easier to obtain by this argument. The selection rules for symmetric top molecules involving the (+l) and (-l) components of a degenerate vibrational level with first-order Coriolis splitting are derived in this paper. It is shown that these selection rules depend on the order of the highest-fold symmetry axis Cn, being different for molecules with n=3, n=4, or n ≧ 5; moreover the selection rules are different again for molecules belonging to the point groups Dnd with n even, and Sm with 1/2m even, for which the highest-fold symmetry axes Cn and Sm are related by m=2n. Finally it is shown that an apparent anomaly between the observed Raman and infra-red vibration-rotation spectra of the allene molecule is resolved when the correct selection rules are used, and a value for the A rotational constant of allene is derived without making use of the zeta sum rule.

Infrared and Raman selection rules for cyclopentane

Infra-red and Raman selection rules are obtained for the cyclopentane molecule, on the assumption that it has a free pseudo-rotation with a large potential hump at the D5h configuration. The selection rules obtained, which concern the vibrational, pseudo-rotational, and rotational quantum numbers, are summarized in tables 1, 2 and 3.

Assembly rules for protein networks derived from phylogenetic-statistical analysis of whole genomes

Background: We report an analysis of a protein network of functionally linked proteins, identified from a phylogenetic statistical analysis of complete eukaryotic genomes. Phylogenetic methods identify pairs of proteins that co-evolve on a phylogenetic tree, and have been shown to have a high probability of correctly identifying known functional links. Results: The eukaryotic correlated evolution network we derive displays the familiar power law scaling of connectivity. We introduce the use of explicit phylogenetic methods to reconstruct the ancestral presence or absence of proteins at the interior nodes of a phylogeny of eukaryote species. We find that the connectivity distribution of proteins at the point they arise on the tree and join the network follows a power law, as does the connectivity distribution of proteins at the time they are lost from the network. Proteins resident in the network acquire connections over time, but we find no evidence that 'preferential attachment' - the phenomenon of newly acquired connections in the network being more likely to be made to proteins with large numbers of connections - influences the network structure. We derive a 'variable rate of attachment' model in which proteins vary in their propensity to form network interactions independently of how many connections they have or of the total number of connections in the network, and show how this model can produce apparent power-law scaling without preferential attachment. Conclusion: A few simple rules can explain the topological structure and evolutionary changes to protein-interaction networks: most change is concentrated in satellite proteins of low connectivity and small phenotypic effect, and proteins differ in their propensity to form attachments. Given these rules of assembly, power law scaled networks naturally emerge from simple principles of selection, yielding protein interaction networks that retain a high-degree of robustness on short time scales and evolvability on longer evolutionary time scales.

Flexible rules cum constrained discretion: a new consensus in monetary policy

This paper demonstrates that recent influential contributions to monetary policy imply an emerging consensus whereby neither rigid rules nor complete discretion are found optimal. Instead, middle-ground monetary regimes based on rules (operative under 'normal' circumstances) to anchor inflation expectations over the long run, but designed with enough flexibility to mitigate the short-run effect of shocks (with communicated discretion in 'exceptional' circumstances temporarily overriding these rules), are gaining support in theoretical models and policy formulation and implementation. The opposition of 'rules versus discretion' has, thus, reappeared as the synthesis of 'rules cum discretion', in essence as inflation-forecast targeting. But such synthesis is not without major theoretical problems, as we argue in this contribution. Furthermore, the very recent real-world events have made it obvious that the inflation targeting strategy of monetary policy, which rests upon the new consensus paradigm in modern macroeconomics is at best a 'fair weather' model. In the turbulent economic climate of highly unstable inflation, deep financial crisis and world-wide, abrupt economic slowdown nowadays this approach needs serious rethinking to say the least, if not abandoning it altogether