In the three main sections of this thesis (chapters II, III, and IV; chapter I consists of definitions) we explore three methods of studying Alexander polynomials of links which are alternatives to Fox' free differential calculus. In chapter II we work directly with a presentation of the link group and show how to obtain a presentation for the Alexander invariant. From this we deduce that the order ideal of the Alexander invariant is principal for links of two or three components (the case of one component is well known) but nonprincipal in general for links of four or more components. In any event we show that only one determinant is needed to obtain the Alexander polynomial. In chapter III we use surgery techniques to characterize Alexander invariants of links of two components in terms of their presentation matrices. We then use this to show that the Torres conditions characterize link polynomials when the linking number of the two components is zero or both components are unknotted and the linking number is two. Chapter IV uses Seifert surfaces to prove a generalization of a theorem of Kidwell which relates the individual degrees of the Alexander polynomial to the linking complexity, to present an algorithm for calculating the Alexander polynomial of a two-bridge link from a two-bridge diagram and to prove a conjecture of Kidwell in the special case of two-bridge links. These results are then used to generate link polynomials from allowable pairs (a concept introduced in chapter III) and these results in turn are used to produce methods of generating allowable pairs. / Science, Faculty of / Mathematics, Department of / Graduate

This paper shall be mostly concerned with the development and the properties of three quadratic polynomials. The primary interest will by with n-ary quadratic polynomials, called forms.

In this paper we determine when there exists a matrix M in PGL2(F), and its form, such that L_k = D^M_k where D^M_k is a higher-order Dickson polynomial. We first examine the cases where M has projective orders 3, 4, and 6. For the order 3 case, we find that M has entries in, at worst, a quadratic extension of F. This is also true for the orders 4 and 6, but requires a restriction on the coefficients of h(x), the characteristic polynomial of L. In all cases, an explicit formula for M is given, and in the order 4 case the meaning of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5.

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A subset of speech recognition is the use of speech recognition techniques for voice authentication. Voice authentication is an alternative security application to the other biometric security measures such as the use of fingerprints or iris scans. Voice authentication has advantages over the other biometric measures in that it can be utilized remotely, via a device like a telephone. However, voice authentication has disadvantages in that the authentication system typically requires a large memory and processing time than do fingerprint or iris scanning systems. Also, voice authentication research has yet to provide an authentication system as reliable as the other biometric measures. Most voice recognition systems use Hidden Markov Models (HMMs) as their basic probabilistic framework. Also, most voice recognition systems use a frame based approach to analyze the voice features. An example of research which has been shown to provide more accurate results is the use of a segment based model. The HMMs impose a requirement that each frame has conditional independence from the next. However, at a fixed frame rate, typically 10 ms., the adjacent feature vectors might span the same phonetic segment and often exhibit smooth dynamics and are highly correlated. The relationship between features of different phonetic segments is much weaker. Therefore, the segment based approach makes fewer conditional independence assumptions which are also violated to a lesser degree than for the frame based approach. Thus, the HMMs using segmental based approaches are more accurate. The speech polynomials (feature vectors) used in the segmental model have been shown to be Chebychev polynomials. Use of the properties of these polynomials has made it possible to reduce the computation time for speech recognition systems. Also, representing the spoken word waveform as a Chebychev polynomial allows for the recognition system to easily extract useful and repeatable features from the waveform allowing for a more accurate identification of the speaker. This thesis describes the segmental approach to speech recognition and addresses in detail the use of Chebychev polynomials in the representation of spoken words, specifically in the area of speaker recognition. .

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A method for finding simple roots of arbitrary polynomials based on divided differences is discussed. Theoretical background is presented for the case of simple roots. Numerical results are presented which show the algorithm finds simple and (usually) multiple zeros to an accuracy limited by the accuracy of polynomial evaluation. The method is designed for an SIMD parallel computer. The algorithm is compared to two other frequently used polynomial root finders, the Jenkins-Traub algorithm and Laguerre’s method. / Master of Science

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Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics and endpoint limit asymptotics on the interval. The zero spacing behavior follows, as well as estimates for the smallest and largest zeros. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval and the asymptotic properties of the associated Christoffel functions.