Integral Equations for Heat Kernel in Compound Media

By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator $-a^2\Delta$ in the case of compound media. In each of the media the parameter $a^2$ acquires a certain constant value. At the interface of the media the conditions are imposed which demand the continuity of the `temperature' and the `heat flows'. The integration in the equations is spread out only over the interface of the media. As a result the dimension of the initial problem is reduced by 1. The perturbation series for the integral equations derived are nothing else as the multiple scattering expansions for the relevant heat kernels. Thus a rigorous derivation of these expansions is given. In the one dimensional case the integral equations at hand are solved explicitly (Abel equations) and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions. Derivation of the asymptotic expansion of the integrated heat kernel for a compound media is considered by making use of the perturbation series for the integral equations obtained. The method proposed is also applicable to the configurations when the same medium is divided, by a smooth compact surface, into internal and external regions, or when only the region inside (or outside) this surface is considered with appropriate boundary conditions.Comment: 26 pages, no figures, no tables, REVTeX4; two items are added into the Reference List; a new section is added, a version that will be published in J. Math. Phy

Applying the ALARA concept to the evaluation of vesicoureteric reflux

The voiding cystourethrogram (VCUG) is a widely used study to define lower urinary tract anatomy and to diagnose vesicoureteric reflux (VUR) in children. We examine the technical advances in the VCUG and other examinations for reflux that have reduced radiation exposure of children, and we give recommendations for the use of imaging studies in four groups of children: (1) children with urinary tract infection, (2) siblings of patients with VUR, (3) infants with antenatal hydronephrosis (ANH), and (4) children with a solitary functioning kidney. By performing examinations with little to no radiation, carefully selecting only the children who need imaging studies and judiciously timing follow-up examinations, we can reduce the radiation exposure of children being studied for reflux