We consider the affine varieties which arise by considering invertible polynomial maps from C2 to itself of less than or equal to a given-degree. These varieties arise naturally in the investigation of the long-standing Jacobian Conjecture. We start with some calculations in the lower degree cases. These calculations provide a proof of the Jacobian conjecture in these cases and suggest how the investigation in the higher degree cases should proceed. We then show how invertible polynomial maps can be decomposed as products of what we call triangular maps and we are able to prove a uniqueness result which gives a stronger version of Jung's theorem [j] which is one of the most important results in this area. Our proof also gives a new derivation of Jung's theorem from Segre's lemma. We give a different decomposition of an invertible polynomial map as a composition of "irreducible maps" and we are able to write down standard forms for these irreducibles. We use these standard forms to give a description of the structure of the varieties of invertible maps. We consider some interesting group actions on our varieties and show how these fit in with the structure we describe. Finally, we look at the problem of identifying polynomial maps of finite order. Our description of the structure of the above varieties allows us to solve this problem completely and we are able to show that the only elements of finite order are those which arise from conjugating linear elements of finite order.

Orientadores: Dimitar Kolev Dimitrov, Roberto Andreani / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T04:39:55Z (GMT). No. of bitstreams: 1 Rafaeli_FernandoRodrigo_D.pdf: 1231425 bytes, checksum: 33a23775a69f9b2b36c516f7cfcb0d0f (MD5) Previous issue date: 2010 / Resumo: Neste trabalho são obtidos resultados sobre o comportamento de zeros de polinômios ortogonais. Sabe-se que todos eles são reais e distintos e fazem papel importante de nós das mais utilizadas fórmulas de integração numérica, que são as fórmulas de quadratura de Gauss. São obtidos resultados sobre a localização e a monotonicidade dos zeros, considerados como funções dos correspondentes parâmetros, dos polinômios ortogonais clássicos. Apresentaremos também vários resultados que tratam da localização, monotonicidade e da assintótica de zeros de certas classes de polinômios ortogonais relacionados com as medidas clássicas / Abstract: Results concerning the behaviour of zeros of orthogonal polynomials are obtained. It is known that they are real and distinct and play as important role as node of the most frequently used rules for numerical integration, the Gaussian quadrature formulae. Result about the location and monotonicity of the zeros, considered as functions of parameters involved in the measure, are provided. We present various results that treat questions about location, monotonicity and asymptotics of zeros of certain classes of orthogonal polynomials with respect to measure that are closely related to the classical ones / Doutorado / Analise Aplicada / Doutor em Matemática Aplicada

A dissertation submitted to the Faculty of Science, School of Mathematics University of the Witwatersrand Johannesburg South Africa / Linear second order ordinary di erential boundary value problems feature prominently in many scienti c eld, such as physics and engineering. Solving these problems is often riddled with complications though a myriad of techniques have been devised to alleviate these di culties. One such method is by transforming a problem into a more readily solvable form or a problem which behaves in a manner which is well understood. The Darboux-Crum transformation is a particularly interesting transformation characterised by some surprising properties, and an increase in the number of works produced in the last few years related to this transformation has prompted this investigation. The classical orthogonal polynomials, namely those of Jacobi, Legendre, Hermite and Laguerre, have been nominated as test candidates and this work will investigate how these orthogonal families are a ected when transformed via Darboux-Crum transformations.

Methods of constructing orthogonal Latin of squares and incomplete balanced block designs are developed. The analysis of these designs is then derived. Particular care is taken in the determination of the number of degrees of freedom involved, a point which is usually neglected in other sources. The principle source of material for this thesis has been H.B. Mann's book, Analysis and Design of Experiments. / Thesis / Master of Arts (MA)

A new multiplexing technique based on the orthogonality of the signal waveform is proposed. It can bring extra capacity to the existing fiber-optic communication system. / A new multiplexing technique is proposed in this work, which is a promising alternative technique for next-generation high-capacity fiber-optic communication system. The concept of this technique is based on the orthogonality of the signal waveform. / Thesis / Master of Science (MSc) / A number of signals with orthogonal waveforms are multiplexed into a single optical fiber. In this scheme, the user’s information is encoded into the amplitude and the phase of the signal waveform. Then multiplexed signals can be transmitted through the fiber-optic communication system.

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In this thesis, we propose a new way to generate orthogonal code distinct from complete complementary (CC) code and Super CC code but it still have ideal auto-correlation and cross-correlation property. We also introduce the concept of correlation and propose six rules to determine if the code generated by different ways are the same.After that we use the rules on orthogonal matrix and find a new way to generate orthogonal matrix different from Hadamard marix. Then we will use this marix in 2-D orthogonal variable spreading factor (OVSF) code and generate similar codes.

Approved for public release; distribution is unlimited. / The major driver for broadband wireless communications has been reliable, high-data rate services. In wireless communication, the multipath fading constitutes a bottleneck for increasing data rates and causes performance degradation. To combat fading, we can use diversity. Wireless systems with multiple antennas at the transmitter and receiver have much larger capacity in fading channels than standard wireless systems. The objective of this thesis is to investigate the transmission scheme provided by matrix A and B in the 802.16 standard and show how it can be implemented. The research focuses on using maximal-ratio combining (MRC) to demodulate the transmitted symbols. Modifications to the existing matrix by using more frequency bands were introduced; this reduces the number of transmitting antennas and uses fewer time slots to transmit the same number of symbols. The modulator and demodulator design is also discussed. The performance of orthogonal and non-orthogonal space time codes (STC) are evaluated. / Civilian

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