The main goal of Chapter 11 is to demonstrate how the theory developed in the previous chapters can be used in the study of various Markov models that give rise to Markov chains with asymptotically zero drift. Some of those models are popular in stochastic modelling: random walks conditioned to stay positive, state-dependent branching processes or branching processes with migration, stochastic difference equations. In contrast with the general approach discussed here, the methods available in the literature for investigation of these models are mostly model tailored. We also introduce some new models to which our approach is applicable. For example, we introduce a risk process with surplus-dependent premium rate, which converges to the critical threshold in the nett-profit condition. Furthermore, we introduce a new class of branching processes with migration and with state-dependent offspring distributions.

We consider the other classical tests of general relativity. The first is the gravitational redshift (the change in the frequency of light). Next, we consider the geodesic radial motion, and we use the equations to find the time delay of a radar signal (or any light signal) moving in a gravitational field (the classic test is for bouncing a signal from a satellite in orbit, or on the Moon). We use the same geodesic equations to find the precession of the perihelion (closest distance to the Sun) of the ecliptic (motion of the planets around the Sun), specifically for the perihelion of Mercury’s ecliptic. Finally, we analyze the possible motions in the Schwarzschild metric, by using a nonrelativistic analogy (effective potential), both in the massive and in the null cases.

Core outcome sets are an agreed, standardised set of outcomes based on what key stakeholders consider the important outcomes in the management or prevention of a condition. They are crucial to unifying research data from different trials, allowing outcomes to be combined and compared. Core outcome sets in miscarriage prevention and management and ectopic pregnancy have been produced. The final outcomes for miscarriage management are: efficacy of miscarriage treatment, heavy vaginal bleeding, pelvic infection, maternal death, procedural related complications and patient satisfaction. The final outcomes for miscarriage prevention are: pregnancy loss, live birth, congenital abnormalities, fetal growth restriction, gestation at birth, pre-term birth, neonatal or infant death, maternal complications, compliance with intervention, patient satisfaction, maternal hospitalisation and neonatal or infant hospitalisation. The final outcomes for ectopic pregnancy are: treatment success, resolution time of EP, number of additional interventions, adverse events, mortality and severe morbidity and treatment satisfaction.

We describe gravitational waves. We start with radiation in the TT gauge, then the gravitational field of a mass distribution: after using the electromagnetic multipole expansion as an analogy, we describe the gravitational multipole expansion. We calculate gravitational radiation emitted from a source, then describe the pseudotensor of the gravitational field for the quadratic approximation, and use it to calculate the power radiated from through gravitational waves (the Einstein formula). Finally, we describe the exact, non-perturbative solution for gravitational waves with cylindrical symmetry found by Einstein and Rosen.

Obesity is a worsening global epidemic that is detrimentally affecting women’s reproductive health. Being obese not only decreases fertility, but also increases miscarriage rates and other pregnancy complications. Perhaps even more significant is the increasing evidence demonstrating associations between maternal obesity and long-term risks of obesity and metabolic disturbances in the offspring.

The mechanisms mediating the effects of obesity on early pregnancy are likely multifactorial, involving oocyte quality, embryo competence, endometrial receptivity, and fetal development. Early pregnancy is a critical time that strongly influences the fate of the pregnancy as well as the health of the next generation.