Operations between sets in geometry

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space $\R^n$. For example, it is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and associative if and only if it is $L_p$ addition for some $1\le p\le\infty$. It is also demonstrated that if $n\ge 2$, an operation * between compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and has the identity property (i.e., $K*\{o\}=K=\{o\}*K$ for all compact convex sets $K$, where $o$ denotes the origin) if and only if it is Minkowski addition. Some analogous results for operations between star sets are obtained. An operation called $M$-addition is generalized and systematically studied for the first time. Geometric-analytic formulas that characterize continuous and GL(n)-covariant operations between compact convex sets in terms of $M$-addition are established. The term "polynomial volume" is introduced for the property of operations * between compact convex or star sets that the volume of $rK*sL$, $r,s\ge 0$, is a polynomial in the variables $r$ and $s$. It is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, associative, and has polynomial volume if and only if it is Minkowski addition

On the large-Q^2 behavior of the pion transition form factor

We study the transition of non-perturbative to perturbative QCD in situations with possible violations of scaling limits. To this end we consider the singly- and doubly-virtual pion transition form factor $\pi^0\to\gamma\gamma$ at all momentum scales of symmetric and asymmetric photon momenta within the Dyson-Schwinger/Bethe-Salpeter approach. For the doubly virtual form factor we find good agreement with perturbative asymptotic scaling laws. For the singly-virtual form factor our results agree with the Belle data. At very large off-shell photon momenta we identify a mechanism that introduces quantitative modifications to Efremov-Radyushkin-Brodsky-Lepage scaling.Comment: 5 pages, 7 figures, v3:contents revised, version published in PL

Subcellular mRNA localisation at a glance.

mRNA localisation coupled to translational regulation provides an important means of dictating when and where proteins function in a variety of model systems. This mechanism is particularly relevant in polarised or migrating cells. Although many of the models for how this is achieved were first proposed over 20 years ago, some of the molecular details are still poorly understood. Nevertheless, advanced imaging, biochemical and computational approaches have started to shed light on the cis-acting localisation signals and trans-acting factors that dictate the final destination of localised transcripts. In this Cell Science at a Glance article and accompanying poster, we provide an overview of mRNA localisation, from transcription to degradation, focusing on the microtubule-dependent active transport and anchoring mechanism, which we will use to explain the general paradigm. However, it is clear that there are diverse ways in which mRNAs become localised and target protein expression, and we highlight some of the similarities and differences between these mechanisms.This work was supported by a Wellcome Trust Senior Research Fellowship to I.D. supporting R.M.P. [grant number: 096144], a studentship from the Wellcome Trust to A.D. [grant number: 097304], the University of Cambridge, ISSF to T.T.W. [grant number 097814].This is the final version of the article. It first appeared from the Company of Biologists via http://dx.doi.org/10.1242/jcs.11427

Transplantation in children

Kidney transplantation in very young children, less than 2 years of age, has usually failed, mainly because of difficulties maintaining these patients on hemodialysis long enough to permit retransplantation after loss of the original graft. Liver replacement in the very young child has been associated with a higher frequency of vascular and biliary obstruction than in the older child, due to the small size of these structures. Such accidents have contributed to unsatisfactory results with biliary atresia. Transplantation of kidney or liver into older children has been more successful than transplantation of these organs into adults. Related or cadaveric kidney transplantation in the child has been followed by at least a 60 per cent patient survival for 6 to 13 years and a very acceptable quality of life. Liver replacement for diseases other than biliary atresia has been followed by a 56 per cent 1 year survival rate, and two children have survived for more than 5 years