The propagating source method has been extended to solve the Boltzmann equation with a quasi-linear diffusion scattering operator. A half-range polynomial expansion method is used to reduce the integral-diffusion form of the ‘collisional’ Boltzmann equation to an infinite set of linear hyperbolic partial differential equations in the harmonics of the polynomial expansion. The lowest-order truncation of the coupled set of equations yields an inhomogeneous form of the well-known telegrapher equation, which, unlike the homogeneous telegrapher equation, does not introduce physically unrealistic pulse solutions. Anisotropic quasi-linear scattering models for which the index $q$ of the power spectrum of magnetic fluctuations satisfies $1\,{<}\,q\,{<}\,2$ admit slow scattering through $90^{\circ}$ and no scattering through $90^{\circ}$ for $q \,{\ge}\,2$. Accordingly, four models that either allow or enhance scattering through $90^{\circ}$ are used to augment the standard quasi-linear model for pitch-angle scattering. These are mirroring, dynamical turbulence and two distinct wave-based models. In the case that mirroring is responsible for scattering particles through $90^{\circ}$, together with the standard QLT (quasi-linear theory) pitch-angle diffusion model for scattering within the forward and backward hemispheres, it is found that the QLT isotropic and anisotropic models are well approximated by relaxation time scattering models. As an application of the general study, the implications of the four models introduced to redress the difficulties faced by QLT in describing scattering through $90^{\circ}$ are briefly considered. An initial beam was found to relax more rapidly for either the dynamical turbulence or wave models with resonant scattering through $90^{\circ}$ than for mirroring models.

Our knowledge of the life of Justin “the Martyr” depends almost entirely on what he himself tells us, especially in the introduction to the Dialogue with Trypho, where he recounts his journey to the true philosophy, Christianity. It appears that he was born in the late first or early second century at Flavia Neapolis in Samaria into a Greek-speaking family. Although he refers to himself in one passage as a Samaritan by race (Dial. 120.6), this background seems unlikely, since there is no evidence that he was familiar with any Samaritan religious traditions. Rather, it appears that his ancestors were originally Greek or Roman colonists who settled in Flavia Neapolis after its establishment by Vespasian in A.D. 72. In any case, he certainly was not Jewish, for he did not encounter the Jewish scriptures until later in life.