Differential Equations for Dyson Processes

We call "Dyson process" any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n by n Hermitian matrices, and the eigenvalues describe n curves. Given sets X_1,...,X_m the probability that for each k no curve passes through X_k at time \tau_k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process. Generalizing and strengthening earlier work, we assume that each X_k is a finite union of intervals and find for the Airy process a system of partial differential equations, with the end-points of the intervals of the X_k as independent variables, whose solution determines the probability that for each k no curve passes through X_k at time \tau_k. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.Comment: 36 pages, LaTeX. Version 3 corrects an error in the earlier version

Levelling the playing field for female academics to access scholarships and research funds

Although the balance of male to female academics in environmental health in Africa has improved over the years, there is still inequity between the academic genders. Recent evidence suggests that female academics are often found on a lower grade and salary on the University hierarchical scale compared with their male and slightly younger counterparts. To achieve equity in research and knowledge management of female environmental health issues in Africa, it is imperative that gender balance is achieved within academic practice. Lifestyle and work patterns of male and female academics require to be taken cognisance of by donor agencies when determining the eligibility criteria to apply for scholarships to study for higher degrees. Female academics often have to withdraw from the academic field for childbirth, child rearing in the first few years. In addition to fulfilling their role as mothers they are also the care givers within the family they are still expected to undertake the usual domestic chores of a wife and care for elderly relatives, unlike their male counterparts in comparison. All of which places female academics at a disadvantage in status upon return to work. After extensive consultation with both male and female academics working in the field of environmental health in Africa, the Africa Academy for Environmental Health (AAEH) advocate three recommendations for funding bodies and higher education institutions to adopt in their policies: (1) Increase age restrictions in donor scholarship and funding applications for female academics to accommodate their maternal and domestic responsibilities; (2) Scholarship awarding bodies should actively promote funding of split site and home based postgraduate studies. This would provide an equal opportunity for female academics to attain quality qualifications irrespective of their maternal and domestic commitments and ensure research is gender balanced and specific to the African context; and (3) Universities should implement effective mentoring schemes in place to support the needs of junior academics and develop their research and teaching skills