## Dissertations

#### Title

Null spherical t-designs

2016

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### Subject Categories

Discrete Mathematics and Combinatorics | Mathematics

#### College

College of Science

#### Department/Unit

Mathematics and Statistics Department

Ederlina G. Nocon

#### Defense Panel Chair

Arlene A. Pascasio

#### Defense Panel Member

Severino V. Gervacio
Yvette F. Lim
Fidel R. Nemenzo
Jose Tristan F. Reyes

#### Abstract/Summary

The survey paper  of Eiichi Bannai and Etsuko Bannai provided an overview of the study of spherical designs and algebraic combinarotics. In the survey paper the authors focused on the study of "good" nite subsets of the unit sphere in n-dimension, n{u100000}1 and that part of their problem is to de ne what "good nite subsets" should mean. However, up to today, no de nite answer is known and it is unrealistic to expect a single good answer. A possible point of view that one could take is to de ne a good subset of the unit sphere to be the one that globally approximates the whole sphere using only a nite number of point. A reasonable de nition to what it means for a nite subset to approximate the sphere was given by Delsarte-Goethals-Seidel in 1966 as follows: a nite subset X on n{u100000}1 is called a spherical t-design on n{u100000}1, if for any polynomial f(x) = f(x1 x2 : : : xn) of degree at most t, the value of the integral of f(x) on n{u100000}1 (divided by the volume of n{u100000}1) is just the average value of f(x) on the nite set X that is, 1 j n{u100000}1j Z x2 n{u100000}1 f(x)d (x) = 1 jX j X x2X f(x) where is a Lesbegue measure on n{u100000}1: In

In one of the talks on Algebraic Combinatorics at Shanghai Jiao Tong University on May 2012, Eiichi Bannai de ned the notion of a null spherical t-design on the unit sphere in n-dimension. For any non-negative integers n t such that n > 1 and t 0 a pair (X !) is a null spherical t-design on n{u100000}1 if X is a nite subset of n{u100000}1 and ! is a non-zero weight function on X that satis es X x2X !(x)f(x) = 0 for any homogeneous harmonic polynomial f(x) in n variables of degree at most t: This de nition generalizes the notion of the usual spherical t-designs on the unit sphere by allowing non-zero weights.

In this study, properties of null spherical t-designs similar to properties of spherical t-designs are presented. Construction of null spherical designs is also provided using known spherical designs. Null spherical designs are also described using the Gegenbauer polynomials and characteristic matrices. Bounds on the number of points in a null spherical design are determined. In particular, we conjecture that the minimum number of points in a null spherical t-design on n{u100000}1 is 2(t + 1):

html

English

Electronic

CDTG006753

#### Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

#### Physical Description

1 computer optical disc; 4 3/4 in

#### Keywords

Combinatorial designs and configurations

COinS