Sufficient conditions for the convergence of the Magnus expansion

Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation $Y' = A(t) Y$. The first one provides a bound on the convergence domain based on the norm of the operator $A(t)$. The second condition links the convergence of the expansion with the structure of the spectrum of $Y(t)$, thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.Comment: 20 page

On an asymptotic method for computing the modified energy for symplectic methods

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory

Transformation kinetics of alloys under non-isothermal conditions

The overall solid-to-solid phase transformation kinetics under non-isothermal conditions has been modeled by means of a differential equation method. The method requires provisions for expressions of the fraction of the transformed phase in equilibrium condition and the relaxation time for transition as functions of temperature. The thermal history is an input to the model. We have used the method to calculate the time/temperature variation of the volume fraction of the favored phase in the alpha-to-beta transition in a zirconium alloy under heating and cooling, in agreement with experimental results. We also present a formulation that accounts for both additive and non-additive phase transformation processes. Moreover, a method based on the concept of path integral, which considers all the possible paths in thermal histories to reach the final state, is suggested.Comment: 16 pages, 7 figures. To appear in Modelling Simul. Mater. Sci. En

Genomic analysis reveals neutral and adaptive patterns that challenge the current management regime for East Atlantic cod Gadus morhua L

Challenging long‐held perceptions of fish management units can help to protect vulnerable stocks. When a fishery consisting of multiple genetic stocks is managed as a single unit, overexploitation and depletion of minor genetic units can occur. Atlantic cod (Gadus morhua) is an economically and ecologically important marine species across the North Atlantic. The application of new genomic resources, including SNP arrays, allows us to detect and explore novel structure within specific cod management units. In Norwegian waters, coastal cod (i.e. those not undertaking extensive migrations) are divided into two arbitrary management units defined by ICES: one between 62° and 70°N (Norwegian coastal cod; NCC) and one between 58° and 62°N (Norwegian coastal south; NCS). Together, these capture a fishery area of >25,000 km2 containing many spawning grounds. To assess whether these geographic units correctly represent genetic stocks, we analysed spawning cod of NCC and NCS for more than 8,000 SNPs along with samples of Russian White Sea cod, north‐east Arctic cod (NEAC: the largest Atlantic stock), and outgroup samples representing the Irish and Faroe Sea's. Our analyses revealed large differences in spatial patterns of genetic differentiation across the genome and revealed a complex biological structure within NCC and NCS. Haplotype maps from four chromosome sets show regional specific SNP indicating a complex genetic structure. The current management plan dividing the coastal cod into only two management units does not accurately reflect the genetic units and needs to be revised. Coastal cod in Norway, while highly heterogenous, is also genetically distinct from neighbouring stocks in the north (NEAC), west (Faroe Island) and the south. The White Sea cod are highly divergent from other cod, possibly yielding support to the earlier notion of subspecies rank.publishedVersio