Decoupling the coupled DGLAP evolution equations: an analytic solution to pQCD

Using Laplace transform techniques, along with newly-developed accurate numerical inverse Laplace transform algorithms, we decouple the solutions for the singlet structure function $F_s(x,Q^2)$ and $G(x,Q^2)$ of the two leading-order coupled singlet DGLAP equations, allowing us to write fully decoupled solutions: F_s(x,Q^2)={\cal F}_s(F_{s0}(x), G_0(x)), G(x,Q^2)={\cal G}(F_{s0}(x), G_0(x)). Here ${\cal F}_s$ and $\cal G$ are known functions---found using the DGLAP splitting functions---of the functions $F_{s0}(x) \equiv F_s(x,Q_0^2)$ and $G_{0}(x) \equiv G(x,Q_0^2)$, the chosen starting functions at the virtuality $Q_0^2$. As a proof of method, we compare our numerical results from the above equations with the published MSTW LO gluon and singlet $F_s$ distributions, starting from their initial values at $Q_0^2=1 GeV^2$. Our method completely decouples the two LO distributions, at the same time guaranteeing that both distributions satisfy the singlet coupled DGLAP equations. It furnishes us with a new tool for readily obtaining the effects of the starting functions (independently) on the gluon and singlet structure functions, as functions of both $Q^2$ and $Q_0^2$. In addition, it can also be used for non-singlet distributions, thus allowing one to solve analytically for individual quark and gluon distributions values at a given $x$ and $Q^2$, with typical numerical accuracies of about 1 part in $10^5$, rather than having to evolve numerically coupled integral-differential equations on a two-dimensional grid in $x, Q^2$, as is currently done.Comment: 6 pages, 2 figure

Microstructure and mechanical behaviour of a Mg94Zn2Y4 alloy processed by equal channel angular pressing

A Mg94Zn2Y4 (at%) alloy containing the long-period stacking ordered (LPSO) phase and the Mg24Y5 phase was processed by equal channel angular pressing (ECAP). The ECAP processing develops a bimodal microstructure consisting of large deformed grains (Mg and LPSO) and sub-micron sized dynamically recrystallised (DRXed) grains. The DRXed grain boundaries are decorated with a large numbers of nano-sized Mg24Y5 precipitates. The presence of LPSO lamellae refined the deformed grains by kinking and promoted dynamic recrystallisation during the ECAP process. The ECAP processed alloy was then subjected to the small punch test (SPT). SPT result shows that the ECAP processing increased significantly the strength of the alloy. Under the biaxial tensile stress induced by SPT, the sample started to crack along the ECAP shear direction shortly after the linear elastic region on the load-displacement curve, and the DRXed grains are potential crack sources. These phenomena may be explained by different deformation behaviours of the fibre textured coarse grains and the random oriented DRXed grains, and the distribution of the DRXed grains