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This is a post-peer-review, pre-copyedit version of an article published in Advances in Computational Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s10444-017-9543-y We consider the Bessel functions Jν (z) and Yν (z) for ν > −1/2 and z ≥ 0. We derive a convergent expansion of Jν (z) in terms of the derivatives of (sin z)/z, and a convergent expansion of Yν (z) in terms of derivatives of (1−cos z)/z, derivatives of (1 − e−z)/z and (2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments. This research was supported by the Spanish Ministry of "Economía y Competitividad", project MTM2014-52859-P. The Universidad Pública de Navarra is acknowledged by its financial support.

Many distributed services need to be scalable: internet search, electronic commerce, e-government... In order to achieve scalability, high availability and fault tolerance, such applications rely on replicated components. Because of the dynamics of growth and volatility of customer markets, applications need to be hosted by adaptive, highly scalable systems. In particular, the scalability of the reliable multicast mechanisms used for supporting the consistency of replicas is of crucial importance. Reliable multicast might propagate updates in a pre-determined order (e.g., FIFO, total or causal). Since total order needs more communication rounds than causal order, the latter appears to be the preferable candidate for achieving multicast scalability, although the consistency guarantees based on causal order are weaker than those of total order. This paper provides a historical survey of different scalability approaches for reliable causal multicast protocols., This work was supported by European Regional Development Fund (FEDER) and Ministerio de Economia y Competitividad (MINECO) under research Grant TIN2012-37719-C03-01.

Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be used for the non-stiff part of the problem. However, for systems with a large number of equations, memory storage requirement is also an important issue. When the high dimension of the problem compromises the computer memory capacity, it is important to incorporate low memory usage to some other properties of the scheme. In this paper we consider Additive Semi-Implicit Runge-Kutta (ASIRK) methods, a class of implicitexplicit Runge-Kutta methods for additive differential systems. We construct two second order 3-stage ASIRK schemes with low-storage requirements. Having in mind problems with stiffness parameters, besides accuracy and stability properties, we also impose stiff accuracy conditions. The numerical experiments done show the advantages of the new methods., 23 pages, 52 figures in Journal of Scientific Computing, 2015

An injective coloring of a graph is a vertex labeling such that two vertices sharing a common neighbor get different labels. In this work we introduce and study what we call additive colorings. An injective coloring $$c:V(G)\rightarrow \mathbb {Z}$$c:V(G)?Z of a graph $$G$$G is an additive coloring if for every $$uv, vw$$uv,vw in $$E(G)$$E(G), $$c(u)+c(w)\ne 2c(v)$$c(u)+c(w)?2c(v). The smallest integer $$k$$k such that an injective (resp. additive) coloring of a given graph $$G$$G exists with $$k$$k colors (resp. colors in $$\{1,\ldots ,k\}$${1,?,k}) is called the injective (resp. additive) chromatic number (resp. index). They are denoted by $$\chi _i(G)$$?i(G) and $$\chi '_a(G)$$?a?(G), respectively. In the first part of this work, we present several upper bounds for the additive chromatic index. On the one hand, we prove a super linear upper bound in terms of the injective chromatic number for arbitrary graphs, as well as a linear upper bound for bipartite graphs and trees. Complete graphs are extremal graphs for the super linear bound, while complete balanced bipartite graphs are extremal graphs for the linear bound. On the other hand, we prove a quadratic upper bound in terms of the maximum degree. In the second part, we study the computational complexity of computing $$\chi '_a(G)$$?a?(G). We prove that it can be computed in polynomial time for trees. We also prove that for bounded treewidth graphs, to decide whether $$\chi '_a(G)\le k$$?a?(G)≤k, for a fixed $$k$$k, can be done in polynomial time. On the other hand, we show that for cubic graphs it is NP-complete to decide whether $$\chi '_a(G)\le 4$$?a?(G)≤4. We also prove that for every $$\epsilon >0$$∈>0 there is a polynomial time approximation algorithm with approximation factor $$n^{1/3+\epsilon }$$n1/3+∈ for $$\chi '_a(G)$$?a?(G), when restricted to split graphs. However, unless $$\mathsf P =\mathsf NP $$P=NP, for every $$\epsilon >0$$∈>0 there is no polynomial time approximation algorithm with approximation factor $$n^{1/3-\epsilon }$$n1/3-∈ for $$\chi '_a(G)$$?a?(G), even when restricted to split graphs.

International audience; Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics.

Both the physiological and the pathological morphogenetic processes that we can meet in embryogenesis, neogenesis and degenerative dysgenesis present common features: they are ruled by three different kinds of mechanisms, one related to cell migration, the second to cell differentiation and the third to cell proliferation. We deal here with an application to the cambial growth which essentially involves the third type of mechanism. Woody plants produce secondary tissue (secondary xylem and phloem) from a meristematic tissue called vascular cambium, responsible for the radial growth of a tree. This paper focuses on the formation of secondary xylem, considered in two dimensions in a cross-section framework. A new discrete modelling approach is used, based on the cellular scale, in order to attain a more accurate understanding of how the elementary microscopic behaviour of each cell takes part in the macroscopic morphogenesis. The mathematical model essentially uses an occurrence method simulating the main features of radial growth with simple geometric rules, such as Thom's division rule (Thom,1972)to account for the cell proliferation. The study applies to concrete instances in which the changes made in the geometrical cellular patterns of the vascular cambium clearly affect the shape of the tree, as in Pinus radiata (D. Don.).