A new family of macrocycles produced by sequential Claisen-Schmidt condensations

[reaction: see text] Members of a new family of macrocycles have been synthesized in one step, from simple building blocks, by sequential Claisen-Schmidt condensations.     

Coupled perturbed modes and internal solitary waves

Coupled perturbed mode theory combines conventional coupled modes and perturbation theory. The theory is used to directly calculate mode coupling in a range-dependent shallow water problem involving propagation through continental shelf internal solitary waves. The solitary waves considered are thermocline depressions, separating well-mixed upper and lower layers. The method is fast and accurate. Results highlight mode coupling associated with internal solitary waves, and mode capture or loss to and from the discrete mode spectrum.     

Observation of quantum coherence in mesoscopic molecular magnets

By applying a transverse magnetic field B( perpendicular) of sufficient strength to the uniaxial molecular magnets Fe8 and Mn12, the tunneling splitting Delta(t) of their S = +/-10 magnetic ground states can be made large compared to perturbations such as hyperfine and dipolar interactions. We present evidence for such a Delta(t) from magnetic specific heat data below 1 K that is consistent with coherent quantum mechanical tunneling in a "mesoscopic" system under such conditions.     

A-Stability and Stochastic Mean-Square Stability

This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns mean-square stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic A-stability property of the Theta Method does not carry through to the mean-square context in general, and gave a condition under which unconditional stability holds. The main purpose of this note is to emphasize that the approach of Saito and Mitsui makes it possible to quantify precisely the point where unconditional stability is lost in terms of the ratio of the drift (deterministic) and diffusion (stochastic) coefficients. This leads to a concept akin to deterministic A()-stability that may be useful in the stability analysis of more general methods. It is also shown that mean-square A-stability is recovered if the Theta Method parameter is increased beyond its normal range to the value 3/2.     

Spectral reordering of a range-dependent weighted random graph

^** Email: aas96106{at}maths.strath.ac.uk Grindrod (2002. Phys. Rev. E, 66, 0667021–0667027) posedthe problem of reordering a range-dependent random graph andshowed that it is relevant to the analysis of data sets frombioinformatics. Reordering under a random graph hypothesis canbe regarded as an extension of clustering and fits into thegeneral area of data mining. Here, we consider a generalizationof Grindrod's model and show how an existing spectral reorderingalgorithm that has arisen in a number of areas may be interpretedfrom a maximum likelihood range-dependent random graph viewpoint.Looked at this way, the spectral algorithm, which uses eigenvectorinformation from the graph Laplacian, is found to be automaticallytuned to an exponential edge density. The connection is precisefor optimal reorderings, but is weaker when approximate reorderingsare computed via relaxation. We illustrate the performance ofthe spectral algorithm in the weighted random graph contextand give experimental evidence that it can be successful forother edge densities. We conclude by applying the algorithmto a data set from the biological literature that describescortical connectivity in the cat brain.     

Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward Euler converge with strong order of one half. The analysis exploits a relation between the backward and explicit Euler methods.     

Computing the field of values and pseudospectra using the Lanczos method with continuation

The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type.This work was supported by Engineering and Physical Sciences Research Council grants GR/H/52139 and GR/H/94528.