Radial basis functions (RBFs) are probably best known for their applications to scattered data problems. Until the 1990s, RBF theory only involved functions that were scalar-valued. Matrix-valued RBFs were subsequently introduced by Narcowich and Ward in 1994, when they constructed divergence-free vector-valued functions that interpolate data at scattered points. In 2002, Lowitzsch gave the first error estimates for divergence-free interpolants. However, these estimates are only valid when the target function resides in the native space of the RBF. In this paper we develop Sobolev-type error estimates for cases where the target function is less smooth than functions in the native space. In the process of doing this, we give an alternate characterization of the native space, derive improved stability estimates for the interpolation matrix, and give divergence-free interpolation and approximation results for band-limited functions. Furthermore, we introduce a new class of matrix-valued RBFs that can be used to produce curl-free interpolants.

Thesis (PhD) -- AUT University, 2007. / Includes bibliographical references. Also held in print (ix, 102 leaves ; 30 cm.) in City Campus Theses Collection (T 658.1511 KOE)

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [131]-136).

Thesis (Ph. D.)--University of Hong Kong, 1999. / Includes bibliographical references (leaves 102-105).

Thesis (Ph. D.)--Florida State University, 2005. / Advisor: David Peterson, Florida State University, College of Business, Dept. of Finance. Title and description from dissertation home page (viewed Jan. 24, 2006). Document formatted into pages; contains viii, 131 pages. Includes bibliographical references.

Author's thesis (doctoral)--Universität Frankfurt am Main, 2009. / Includes bibliographic references (p. 175-217).

Commutative algebra finds its roots in invariant theory and the connection is drawn from a modern standpoint. The Hilbert Basis Theorem and the Nullstellenstatz were considered lemmas for classical invariant theory. The Groebner basis is a modern tool used and is implemented with the computer algebra system Mathematica. Number 14 of Hilbert\'s 23 problems is discussed along with the notion of invariance under a group action of GLn(C). Computational difficulties are also discussed in reference to Groebner bases and Invariant theory.The straitening law is presented from a Groebner basis point of view and is motivated as being a key piece of machinery in proving First Fundamental Theorem of Invariant Theory. / Master of Science

From the beginning of the study of spaces in functional analysis, bases have been an indispensable tool for operating with vectors and functions over a concrete space. Bases can be organized by types, depending on their properties. This thesis is intended to give an overview of some bases and their relations. We study Hamel basis, Schauder basis and Orthonormal basis; we give some properties and compare them in different spaces, explaining the results. For example, an infinite dimensional Hilbert space will never have a basis which is a Schauder basis and a Hamel basis at the same time, but if this space is separable it has an orthonormal basis, which is also a Schauder basis. The project deals mainly with Banach spaces, but we also talk about the case when the space is a pre Hilbert space.

With significant advances in wired and wireless technologies and also increased shrinking in the size of VLSI circuits, many devices have become very large because they need to contain several large units. This large number of gates and in turn large number of transistors causes the devices to be more prone to faults. These faults specially in sensitive and critical applications may cause serious failures and hence should be avoided. On the other hand, some critical applications such as cryptosystems may also be prone to deliberately injected faults by malicious attackers. Some of these faults can produce erroneous results that can reveal some important secret information of the cryptosystems. Furthermore, yield factor improvement is always an important issue in VLSI design and fabrication processes. Digital systems such as cryptosystems and digital signal processors usually contain finite field operations. Therefore, error detection and correction of such operations have become an important issue recently. In most of the work reported so far, error detection and correction are applied using redundancies in space (hardware), time, and/or information (coding theory). In this work, schemes based on these redundancies are presented to detect errors in important finite field arithmetic operations resulting from hardware faults. Finite fields are used in a number of practical cryptosystems and channel encoders/decoders. The schemes presented here can detect errors in arithmetic operations of finite fields represented in different bases, including polynomial, dual and/or normal basis, and implemented in various architectures, including bit-serial, bit-parallel and/or systolic arrays.

With significant advances in wired and wireless technologies and also increased shrinking in the size of VLSI circuits, many devices have become very large because they need to contain several large units. This large number of gates and in turn large number of transistors causes the devices to be more prone to faults. These faults specially in sensitive and critical applications may cause serious failures and hence should be avoided. On the other hand, some critical applications such as cryptosystems may also be prone to deliberately injected faults by malicious attackers. Some of these faults can produce erroneous results that can reveal some important secret information of the cryptosystems. Furthermore, yield factor improvement is always an important issue in VLSI design and fabrication processes. Digital systems such as cryptosystems and digital signal processors usually contain finite field operations. Therefore, error detection and correction of such operations have become an important issue recently. In most of the work reported so far, error detection and correction are applied using redundancies in space (hardware), time, and/or information (coding theory). In this work, schemes based on these redundancies are presented to detect errors in important finite field arithmetic operations resulting from hardware faults. Finite fields are used in a number of practical cryptosystems and channel encoders/decoders. The schemes presented here can detect errors in arithmetic operations of finite fields represented in different bases, including polynomial, dual and/or normal basis, and implemented in various architectures, including bit-serial, bit-parallel and/or systolic arrays.