New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

24A study on pollen morphology of the family Oleaceae Hoffmanns. & Link in Vietnam

This paper uses scanning electron microscopy (SEM) analyses and light microscopy (LM) to study pollen grains of species of the family Oleaceae Hoffmanns. & Link distributed in Vietnam, including 23 species belonging to 6 genera of the family (Oleaceae). Pollen grains of all species (Oleaceae) in Vietnam are isopolar, radial, 3- prolate pollen grains small size, with two types: subprolate type (Jasminum and Olea) and prolate spheroidal type (Jasminum, Fraxinus, Ligustrum, Osmanthus, and Chengiodendron). The surface of the pollen grains in most species is reticulate. These are important morphological features to classify the Nhai family through pollen morphology

Taxonomic studies in the genus Haplanthodes (Acanthaceae)

The endemic Indian genus Haplanthodes (Acanthaceae) is revised. Four species, viz. H. neilgherryensis, H. plumosa, H. tentaculata and H. verticillata and a new variety, H. neilgherryensis var. toranganensis are recognized. Lectotype is designated for Haplanthodes, Haplanthus plumosus and H. verticillaris. The nomenclature of the Linnaean name Ruellia tentaculata is also discussed. Micromorphology of seed using light microscope (LM) and scanning electron microscope (SEM) is observed for the first time, and two types, reticulate and micro papillate, have been recognized based on surface sculpturing pattern. The genus differs from the related genera Andrographis and Haplanthus by distinctly two grooved seeds with hygroscopic hairs. Pollen grains of the genus are oblate or prolate spheroidal, distinctly triangular, trizonocolporate with reticulate exine ornamentation.