Nonnormality and stochastic differential equations

A highly nonnormal Jacobian may give rise to large transients. This behaviour has been shown to have implications for (a) the relevance of linearising a nonlinear system and (b) the timestep restrictions required to keep a numerical method stable. Here, we show that nonnormality also manifests itself for stochastic differential equations. We give an example of a family of systems that is stable without noise, but can be made exponentially unstable in mean-square by a noise perturbation that shrinks to zero as the nonnormality increases. We then show via finite-time convergence theory that an Euler approximation shares the same property, giving a discrete analogue of the result. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65C30, 34F05     

Efficient algorithms for the matrix cosine and sine

Several improvements are made to an algorithm of Higham and Smith for computing the matrix cosine. The original algorithm scales the matrix by a power of 2 to bring the ∞-norm to 1 or less, evaluates the [8/8] Padé approximant, then uses the double-angle formula cos (2A)=2cos ²A−I to recover the cosine of the original matrix. The first improvement is to phrase truncation error bounds in terms of ‖A²‖^1/2 instead of the (no smaller and potentially much larger quantity) ‖A‖. The second is to choose the degree of the Padé approximant to minimize the computational cost subject to achieving a desired truncation error. A third improvement is to use an absolute, rather than relative, error criterion in the choice of Padé approximant; this allows the use of higher degree approximants without worsening an a priori error bound. Our theory and experiments show that each of these modifications brings a reduction in computational cost. Moreover, because the modifications tend to reduce the number of double-angle steps they usually result in a more accurate computed cosine in floating point arithmetic. We also derive an algorithm for computing both cos (A) and sin (A), by adapting the ideas developed for the cosine and intertwining the cosine and sine double angle recurrences. AMS subject classification 65F30 Numerical Analysis Report 461, Manchester Centre for Computational Mathematics, February 2005. Gareth I. Hargreaves: This work was supported by an Engineering and Physical Sciences Research Council Ph.D. Studentship. Nicholas J. Higham: This work was supported by Engineering and Physical Sciences Research Council grant GR/T08739 and by a Royal Society–Wolfson Research Merit Award.     

Communicability betweenness in complex networks

Betweenness measures provide quantitative tools to pick out fine details from the massive amount of interaction data that is available from large complex networks. They allow us to study the extent to which a node takes part when information is passed around the network. Nodes with high betweenness may be regarded as key players that have a highly active role. At one extreme, betweenness has been defined by considering information passing only through the shortest paths between pairs of nodes. At the other extreme, an alternative type of betweenness has been defined by considering all possible walks of any length. In this work, we propose a betweenness measure that lies between these two opposing viewpoints. We allow information to pass through all possible routes, but introduce a scaling so that longer walks carry less importance. This new definition shares a similar philosophy to that of communicability for pairs of nodes in a network, which was introduced by Estrada and Hatano [E. Estrada, N. Hatano, Phys. Rev. E 77 (2008) 036111]. Having defined this new communicability betweenness measure, we show that it can be characterized neatly in terms of the exponential of the adjacency matrix. We also show that this measure is closely related to a Fréchet derivative of the matrix exponential. This allows us to conclude that it also describes network sensitivity when the edges of a given node are subject to infinitesimally small perturbations. Using illustrative synthetic and real life networks, we show that the new betweenness measure behaves differently to existing versions, and in particular we show that it recovers meaningful biological information from a protein-protein interaction network.     

Air-stable, highly fluorescent primary phosphanes

Light without fright: a synthetic route to fluorescent primary phosphanes (RPH(2)) that are resistant to air oxidation both in the solid state and in chloroform solution is described. These versatile precursors undergo hydrophosphination to give tripodal ligands and subsequently fluorescent transition-metal complexes.     

The relevance of optical transducers to the process industries

The first part of the paper deals with the requirements of the process industries for measurement transducers and systems. It classifies the industries and discusses the measurands involved. Reference is made to the desirable characteristics and performances of transducers as discrete units and as system components.The second half of the paper examines these requirements in relation to the potential of optical transducers. It concludes that although these new transducers could have some useful characteristics involving simplicity, safety and freedom from interference, there are many obvious design and development problems and limitations which have to be overcome before optical transducers can replace their electrical counterparts particularly in control systems.     

Analysis of the dynamics of local error control via a piecewise continuous residual

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for thematlab ode23 algorithm [10] when applied to a variety of problems. Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in this way. Supported by the Engineering and Physical Sciences Research Council under grants GR/H94634 and GR/K80228. Supported by the Office of Naval Research under grant N00014-92-J-1876 and by the National Science Foundation under grant DMS-9201727.     

Phenol derivatized hexahydropyrimidines prepared from Mannich condensations

Six new phenol derivatized hexahydropyrimidines (1,3-bis-(2-hydroxy-3,5-di(X)benzyl) hexahydropyrimidine and 5,5′-dimethyl-1,3-bis-(2-hydroxy-3,5-di(X)benzyl) hexahydropyrimidine where X = Me, t-Bu, and Cl) were synthesized in one-pot in 33-79% yield using Mannich condensations.     

New99mTc-diiodine substituted IDA derivative (DIIODIDA) for hepatobiliary imaging

A new diiodine substituted IDA derivative, 2,4-diiodine-6-methyl IDA (DIIODIDA) was synthesized and labeled with^99mTc. It was established that^99mTc-DIIODIDA had high radiochemical purity. Biodistribution and influence of bilirubin on^99mTc-DIIODIDA biokinetics has been studied in rats and compared to the corresponding results for^99mTc-SOLCOIODIDA. Related to^99mTc-SOLCOIODIDA,^99mTc-DIIODIDA has much better biliary exretion (55.18 versus 43.63%). No change of^99mTc-DIIODIDA biokinetics, under influence of bilirubin was noticed. Biliary excretion of^99mTc-SOLCOIODIDA has been reduced for about 60%. The protein binding of^99mTc-DIIODIDA and^99mTc-SOLCOIODIDA were also determined, using in vitro method of precipitation. These results showed that^99mTc-DIIODIDA hepatobiliary imaging agent is superior to the presently used^99mTc-monoiodine IDA derivatives.