Search results for: Ellipsoids

Authors: Leehter Yao, Kuei-Song Weng, Cherng-Dir Huang

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.

Keywords: Ellipsoids, genetic algorithm, classification, fuzzyc-means (FCM)

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Authors: Pavel Y. Tabakov, Kevin Duffy

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.

Keywords: Classification, clustering, data minig, genetic algorithms.

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Authors: Pi-Chung Hsu

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.

Keywords: Implicit surfaces, Soft objects, Super-quadrics.

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Authors: Sahar Noori, Seyed Amir Hossein, Mohammad Ebrahimi

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.

Keywords: Aerodynamic Heating, Blunt Body, Hypersonic Flow, Laminar, Turbulent.

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Authors: Lina Wu, Jia Liu, Ye Li

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.

Keywords: Bochner Formula, Stokes’ Theorem, Cauchy-Schwarz Inequality, first and second variation formulas, Hardy-Sobolev type inequalities, Liouville-type problem, p-harmonic map.

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