This monograph is devoted to a study of certain Hardy-type inequalities for non-convex domains in R^n. For convex domains, 1/4 is the precise value of the constant in such inequalities. This book is concerned with finding estimates on that constant for non-convex multi-dimensional domains. Some estimates were obtained earlier by other authors for simply connected planar domains with the help of complex-analytic methods. Our aim is to obtain lower bounds for the optimal constant by real-analytic methods.
Large sized planar structures are increasingly being employed in satellite and radar applications. But due to large electrical size and complex cellular patterns of modern designs, full-wave analysis of these structures requires enormous memory and processing requirements. Therefore conventional techniques based on linear meshing either fail to simulate such structures or require resources not available to a common antenna designer. A novel technique called Scale-changing Technique (SCT) addresses this problem by partitioning the cellular array geometry in numerous nested domains defined at different scale-levels in the array plane. Multi-modal networks, called Scale-changing Networks (SCN), are then computed to model the electromagnetic interaction between any two successive partitions. The cascade of these networks allows the computation of the equivalent surface impedance matrix of the complete array which in turn can be utilized to compute far-field scattering patterns. Since the computation of scale-changing networks is mutually independent, execution times can be reduced significantly by employing multiple processing units.
Spline surfaces defined on planar domains have been studied for more than 40 years and universally recognized as highly effective tools in approximation theory, computer-aided geometric design, computer-aided design, computer graphics and solutions of differential equations. Many methods and theories of bivariate polynomial splines on planar triangulations carry over. However, spherical Bezier-Bernstein polynomial splines defined on sphere have several significant differences from them because sphere is a closed manifold much different from planar domains. This book is based on the dissertation completed in the University of Georgia. It includes following contents: an overview of spherical splines, the method to construct a unique spherical Hermite interpolation splines by using minimal energy method, the estimation of approximation order under L2 and L-infinity norms, methods of hole filling and scattered data fitting with global r-th order continuity. Many examples in this book have demonstrated our theories and applications. This book is especially useful for people who have interest in CAGD, CAD &, CG, multivariate splines, geoscience and spline finite element methods.