Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Dietary modification for women after breast cancer treatment : a narrative review

Diet is thought to account for about 25% of cancers in developed countries. It is well documented that the risks associated with both the breast cancer itself and its treatments are important for women previously treated for breast cancer. Women are at risk of recurrence of the primary disease and prone to develop treatment-induced co-morbidities, some of which are thought to be modified by diet. With a view to making dietary recommendations for the breast cancer patients we encounter in our clinical nursing research, we mined the literature to scope the most current robust evidence concerning the role of the diet in protecting women against the recurrence of breast cancer and its potential to ameliorate some of the longer-term morbidities associated with the disease. We found that the evidence about the role of the diet in breast cancer recurrence is largely inconclusive. However, drawing on international guidelines enabled us to make three definitive recommendations. Women at risk of breast cancer recurrence, or who experience co-morbidities as a result of treatment, should limit their exposure to alcohol, moderate their nutritional intake so it does not contribute to postmenopausal weight gain, and should adhere to a balanced diet. Nursing education planned for breast cancer patients about dietary issues should ideally be individually tailored, based on a good understanding of the international recommendations and the evidence underpinning them