Abstract: The goal of this project is to investigate constant
properties (called the Liouville-type Problem) for a p-stable map
as a local or global minimum of a p-energy functional where
the domain is a Euclidean space and the target space is a
closed half-ellipsoid. The First and Second Variation Formulas
for a p-energy functional has been applied in the Calculus
Variation Method as computation techniques. Stokes’ Theorem,
Cauchy-Schwarz Inequality, Hardy-Sobolev type Inequalities, and
the Bochner Formula as estimation techniques have been used to
estimate the lower bound and the upper bound of the derived
p-Harmonic Stability Inequality. One challenging point in this project
is to construct a family of variation maps such that the images
of variation maps must be guaranteed in a closed half-ellipsoid.
The other challenging point is to find a contradiction between the
lower bound and the upper bound in an analysis of p-Harmonic
Stability Inequality when a p-energy minimizing map is not constant.
Therefore, the possibility of a non-constant p-energy minimizing
map has been ruled out and the constant property for a p-energy
minimizing map has been obtained. Our research finding is to explore
the constant property for a p-stable map from a Euclidean space into
a closed half-ellipsoid in a certain range of p. The certain range of
p is determined by the dimension values of a Euclidean space (the
domain) and an ellipsoid (the target space). The certain range of p
is also bounded by the curvature values on an ellipsoid (that is, the
ratio of the longest axis to the shortest axis). Regarding Liouville-type
results for a p-stable map, our research finding on an ellipsoid is a
generalization of mathematicians’ results on a sphere. Our result is
also an extension of mathematicians’ Liouville-type results from a
special ellipsoid with only one parameter to any ellipsoid with (n+1)
parameters in the general setting.
Abstract: This paper is devoted to predict laminar and turbulent
heating rates around blunt re-entry spacecraft at hypersonic
conditions. Heating calculation of a hypersonic body is normally
performed during the critical part of its flight trajectory. The
procedure is of an inverse method, where a shock wave is assumed,
and the body shape that supports this shock, as well as the flowfield
between the shock and body, are calculated. For simplicity the
normal momentum equation is replaced with a second order pressure
relation; this simplification significantly reduces computation time.
The geometries specified in this research, are parabola and ellipsoids
which may have conical after bodies. An excellent agreement is
observed between the results obtained in this paper and those
calculated by others- research. Since this method is much faster than
Navier-Stokes solutions, it can be used in preliminary design,
parametric study of hypersonic vehicles.
Abstract: A novel method of learning complex fuzzy decision regions in the n-dimensional feature space is proposed. Through the fuzzy decision regions, a given pattern's class membership value of every class is determined instead of the conventional crisp class the pattern belongs to. The n-dimensional fuzzy decision region is approximated by union of hyperellipsoids. By explicitly parameterizing these hyperellipsoids, the decision regions are determined by estimating the parameters of each hyperellipsoid.Genetic Algorithm is applied to estimate the parameters of each region component. With the global optimization ability of GA, the learned decision region can be arbitrarily complex.
Abstract: Super-quadrics can represent a set of implicit surfaces,
which can be used furthermore as primitive surfaces to construct a
complex object via Boolean set operations in implicit surface
modeling. In fact, super-quadrics were developed to create a
parametric surface by performing spherical product on two parametric
curves and some of the resulting parametric surfaces were also
represented as implicit surfaces. However, because not every
parametric curve can be redefined implicitly, this causes only implicit
super-elliptic and super-hyperbolic curves are applied to perform
spherical product and so only implicit super-ellipsoids and
hyperboloids are developed in super-quadrics. To create implicit
surfaces with more diverse shapes than super-quadrics, this paper
proposes an implicit representation of spherical product, which
performs spherical product on two implicit curves like super-quadrics
do. By means of the implicit representation, many new implicit curves
such as polygonal, star-shaped and rose-shaped curves can be used to
develop new implicit surfaces with a greater variety of shapes than
super-quadrics, such as polyhedrons, hyper-ellipsoids, superhyperboloids
and hyper-toroids containing star-shaped and roseshaped
major and minor circles. Besides, the newly developed implicit
surfaces can also be used to define new primitive implicit surfaces for
constructing a more complex implicit surface in implicit surface
modeling.
Abstract: A fuzzy classifier using multiple ellipsoids approximating decision regions for classification is to be designed in this paper. An algorithm called Gustafson-Kessel algorithm (GKA) with an adaptive distance norm based on covariance matrices of prototype data points is adopted to learn the ellipsoids. GKA is able toadapt the distance norm to the underlying distribution of the prototypedata points except that the sizes of ellipsoids need to be determined a priori. To overcome GKA's inability to determine appropriate size ofellipsoid, the genetic algorithm (GA) is applied to learn the size ofellipsoid. With GA combined with GKA, it will be shown in this paper that the proposed method outperforms the benchmark algorithms as well as algorithms in the field.
Abstract: The present study presents a new approach to automatic
data clustering and classification problems in large and complex
databases and, at the same time, derives specific types of explicit rules
describing each cluster. The method works well in both sparse and
dense multidimensional data spaces. The members of the data space
can be of the same nature or represent different classes. A number
of N-dimensional ellipsoids are used for enclosing the data clouds.
Due to the geometry of an ellipsoid and its free rotation in space
the detection of clusters becomes very efficient. The method is based
on genetic algorithms that are used for the optimization of location,
orientation and geometric characteristics of the hyper-ellipsoids. The
proposed approach can serve as a basis for the development of
general knowledge systems for discovering hidden knowledge and
unexpected patterns and rules in various large databases.