Inverse problem for an inhomogeneous Schr\"odinger equation

An inverse problem is considered for an inhomogeneous Schr\"odinger equation. Assuming that the potential vanishes outside a finite interval and satisfies some other technical assumptions, one proves the uniqueness of the recovery of this potential from the knowledge of the wave function at the ends of the above interval for all energies. An algorithm is given for the recovery of the potential from the above data

Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\"odinger equation in R1+4\R^{1+4}

We obtain global well-posedness, scattering, uniform regularity, and global $L^6_{t,x}$ spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schr\"odinger equation in $\R\times\R^4$. Our arguments closely follow those of Colliander-Keel-Staffilani-Takaoka-Tao, though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the $L^6_{t,x}$-norm

Interviewer: 'Are women and girls ever responsible for the domestic violence they encounter?' Student: 'No, well, unless they did something really, really bad …'

Research shows the ‘gendered nature’ of domestic violence, with Women’s Aid (a UK-based charity) estimating that 1 in 4 women are affected (2014). This paper reports on a project - funded by Comic Relief, completed by Nottinghamshire Domestic Violence Forum (now known as Equation) and evaluated by Nottingham Trent University. The project adopts a Whole School Approach in seeking to prevent domestic violence. Students at three secondary schools attended between one and five blocks of work, and special events. There is evidence of positive developments - with young people showing understanding of domestic violence as well as the margins between healthy and unhealthy relationships. However, not all students could reply ‘never’ to the question of ‘are women and girls to blame for the domestic violence they experience?’, remarking that if the woman had done something ‘really, really bad’ then violence might be justified. We argue that young people’s uncertainties need to be situated within the gender-unequal socio-contexts of contemporary society, and further call for a WSA to domestic violence prevention to be a compulsory part of the UK national curriculum

PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation

We discuss several PT-symmetric deformations of superderivatives. Based on these various possibilities, we propose new families of complex PT-symmetric deformations of the supersymmetric Korteweg-de Vries equation. Some of these new models are mere fermionic extensions of the former in the sense that they are formulated in terms of superspace valued superfields containing bosonic and fermionic fields, breaking however the supersymmetry invariance. Nonetheless, we also find extensions, which may be viewed as new supersymmetric Korteweg-de Vries equation. Moreover, we show that these deformations allow for a non-Hermitian Hamiltonian formulation and construct three charges associated to the corresponding flow.Comment: 10 page

Discrete--time ratchets, the Fokker--Planck equation and Parrondo's paradox

Parrondo's games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature. Here we review two recent approaches, based on the Fokker-Planck equation, that rigorously establish the connection between Parrondo's games and a physical model known as the flashing Brownian ratchet. This gives rise to a new set of Parrondo's games, of which the original games are a special case. For the first time, we perform a complete analysis of the new games via a discrete-time Markov chain (DTMC) analysis, producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs.Comment: 17 pages, 5 figure