EACM is a comprehensive reference work covering the vast field of applied and computational mathematics. Applied mathematics itself accounts for at least 60 per cent of mathematics, and the emphasis on computation reflects the current and constantly growing importance of computational methods in all areas of applications. EACM emphasizes the strong links of applied mathematics with major areas of science, such as physics, chemistry, biology, and computer science, as well as specific fields like atmospheric ocean science. In addition, the mathematical input to modern engineering and technology form another core component of EACM.

such that the ray field is regular in the tube. For a point x 2 B T , define its and x 2 Œ0; T / by ray coordinates (rc) x 2 distg .x; / D distg .x; x / D x . Let x. ; / 2 B T be the point with the given rc ; ; in local coordinates 1 ; : : : ; n 1 on , one writes x. 1 ; : : : ; n 1 ; /. The map i W B T 3 x 7! f x ; x g 2 T WD Œ0; T /, which realizes the passage from the Cartesian to ray coordinates, induces the metric g WD i g on T (Fig. 1b; shaded), its length element in local coordinates taking

near-to-identity maps ˚h W Rd ! Rd . It should be mentioned (as noted in [11]) that the Butcher group is actually a group scheme, and thus equivalent to a commutative Hopf algebra. In the past few years, such a Hopf algebra turned out to have farreaching applications in several areas of mathematics and physics. The interested reader should consult [1] for a nice exposition of the different contexts where such algebraic structure appears. B-series and its generalizations play a central role in the

random input data. SIAM J. Numer. Anal. 46, 2411–2442 (2008) 27. Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008) 28. Poette, G., Despr´es, B., Lucor, D.: Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228, 2443–2467 (2009) 29. Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific, Singapore (1993)

kind permission from Springer Science+Business Media: Journal of Mathematical Biology, Multiscale modelling and nonlinear simulation of vascular tumour growth, Vol 58, Page 787, 2008) (9) Thus far, angiogenesis models have tended to focus on the mesoscale, with limited description of the biophysical details of cell-cell interactions, biochemical signaling and mechanical forces, and mechanotransduction-driven signaling processes. An important future direction for angiogenesis modeling involves

Fig. 4. We have recently adjusted the rate constants minimizing the errors between calculated and observed periods for a large class of experimental conditions. The new set of rates is the following: k1 D 2:0ŒM 3 s 1 , k2 D 1:8 106 ŒM 2 s 1 , k3 D 48ŒM 2 s 1 , k3r ŒH2 O D 2:8 108 ŒM 1 s 1 , k4 D 1:1 106 ŒM 2 s 1 , k5 D 3; 000ŒM 1s 1 , k6 D 6:6 109 ŒM 2 s 1 , ŒH2 O D 9:4Œs 1 , k7 D 40ŒM 1 s 1 , k6r k8 D 7:1ŒM 1 s 1 , k9 D 0:25ŒM 1 s 1 , and k10 D 0:053ŒM 1s 1 . The periods calculated