Currently, there exist several methods for finding roots of polynomial functions. From elementary processes such as the quadratic formula and the Rational Root Theorem to calculus-based ideas, choosing an appropriate means of solving often depends on the conditions of the given polynomial. This report will explore several solving methods and discuss their advantages as well as their limitations. / text

Made available in DSpace on 2014-06-11T19:25:29Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-08Bitstream added on 2014-06-13T18:53:33Z : No. of bitstreams: 1 sa_fl_me_guara.pdf: 527655 bytes, checksum: ce3c2e41805e19e7e9763063d33b1f56 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Universidade Estadual Paulista (UNESP) / Neste trabaho realizamos um estudo detalhado do posicionamento dos zeros de Yang- Lee do modelo de Blume-Emery-Griffiths unidimensional atraves de metodos analiticos e numericos. Em particular, analisamos o efeito de uma rede dinamica (aneis conexos e desconexos) sobre tais zeros. Nossos resultados numericos e um calculo via ponto de sela indicam que estes ultimos tendem aos zeros do modelo definido sobre um anel conexo (condicões periodicas de contorno) no limite termodinâmico. Conjecturamos a existência de uma região no espaço de parâmetros do modelo para a qual os zeros correspondem a campos magneticos puramente imaginarios independentemente da temperatura. Nossos resultados mostram que, ao contrario do que sugere resultados anteriores para o modelo de Blume-Capel, nao ha uma relacao direta entre os mínimos de energia e a posicao dos zeros de Yang-Lee. Para o caso de um anel conexo deduzimos uma equação aproximada para a curva dos zeros de Yang-Lee a partir dos autovalores da matriz de transferencia. Resultados numericos e analíticos mostram que mesmo com alguns acoplamentos antiferromagneticos temos zeros para campos magneticos puramente imaginarios. Por fim, calculamos numericamente a densidade dos zeros proximos a ponta da curva a qual pertencem (singularidade da ponta de Yang-Lee) obtendo atraves de ajustes numericos e relações de escala de tamanho finito uma densidade que diverge na ponta com expoente crítico proximo de -1/2 mesmo quando o campo magnetico nao þe puramente imaginario e a rede þe dinamica. / In this work we carry out a detailed study of the position of the Yang-Lee zeros of the one-dimensional Blume-Emery-Griffiths model through analytic and numerical methods. In particular, we analyze the effect of a dynamical lattice (connected and non-connected rings) over such zeros. Our numerical results and a saddle point caulculation indicate that such zeros tend to overlap the zeros of the model defined on one-ring (periodic boundary conditions) in the thermodynamic limit. We conjecture the existence of a region in the parameter space of the model where the zeros correspond to purely imaginary magnetic fields independently of the temperature. Here we show that, contrary to the previous results for the Blume-Capel model, there is no straightforward relationship between the energy minima and zeros position. For the connected ring we deduce the approximate equation for the Yang-Lee zeros curve from the eigenvalues of the transfer matrix. Our numerical and analytic results show that even with some antiferromagnetic couplings we have zeros at purely imaginary magnetic field. Finally, we calculate numerically the density of the zeros close to the edge of the curves (Yang-Lee edge singularity) obtaining, through numerical fits and finite size scaling relations, a density which diverges at the edge with critical exponent approximately -1/2 even when the magnetic field is not purely imaginary and the lattice is dynamic.

We use the behavioural approach and commutative algebra to define and characterize poles and zeros of multidimensional (nD) linear systems. In the case of a system with a standard input output structure we provide new definitions and characterizations of system, controllable and uncontrollable zeros and demonstrate strong relationships between the controllable poles and zeros and properties of the system transfer matrix, and we show that the uncontrollable zeros are in fact uncontrollable poles. We also show that we can regard the zero as a form of pole with respect to an additional form of input output structure imposed on the zero output sub-behaviour. In the case when the behaviour has a latent variable description we make a further distinction of the zeros into several other classes including observable, unobservable and invariant zeros. In addition we also introduce their corresponding controllable and uncontrollable zeros, such as the observable controllable, unobservable controllable, invariant controllable, observable uncontrollable, unobservable uncontrollable and invariant uncontrollable etc. We again demonstrate strong relationships between these and other types of zeros and provide physical interpretations in terms of exponential and polynomial exponential trajectories. In the 1D case of a state-space model we show that the definitions and characterizations of the observable controllable and invariant zeros correspond to the transmission zeros and the invariant zeros in the classical 1D framework. This then completes the correspondences between the behavioural definitions of poles and zeros and those classical poles and zeros which have an interpretation in nD.

The thesis covers the following topics: Assuming a partonic structure of the Pomeron we give predictions for diffractive heavy flavour production at the Tevatron and the LHC and diffractive Higgs production at the LHC in leading-order approximation. For the latter the dominant background processes are also considered. We obtain results on single as well as double diffractive cross sections assuming a Donnachie-Landshoff-type Pomeron flux factor. Measurements of the single inclusive jet cross section at the Tevatron by the CDF Collaboration maybe suggest a modified picture of QCD in the large E(_T) range. A parameter fit of a neutral heavy vector boson Z(^1) to the CDF data, in leading order perturbation theory, is performed, and the question of how the corresponding single inclusive jet cross-sections and the dijet angular distributions at the LHC are affected by this additional Z' is discussed. We test a model, motivated by the former R(_b) and R(_c) "crisis", which contains extra vector fermions. We suggest an alternative indirect test of the possible existence of new heavy quark flavours at the LEP2 collider, which turns out to give the clearest signal. We calculate qq cross sections within this framework, including one loop corrections. Hadronic antenna patterns can provide a valuable diagnostic tool for probing the origin of the reported excess of high x, Q(^2) events at HERA. We present quantitative predictions for the distributions of soft particles and jets in standard deep inelastic scattering eq → eq events and in events corresponding to the production of a narrow colour-triplet scalar resonance. We also study the patterns corresponding to Higgs production and decay in high- energy hadron-hadron collisions. In particular, the signal gg → H → bb and background gg → bb processes are shown to have very different radiation patterns, and this may provide a useful additional method for distinguishing Higgs signal events from the QCD background. The process eq → eq + γ exhibits radiation zeros, i.e. configurations of the final-state particles for which the scattering amplitude vanishes. We study these zeros for both e(^+)u and e(^+)d scattering. The latter exhibits a new type of zero which to our knowledge has not previously been identified. The observability of radiation zeros at HERA is discussed. In the framework of this new type of zero we also study the process e(^+)e(^-) → qqγ. We calculate the positions of these zeros for u-quark and d-quark production and assess the feasibifity of identifying the zeros in experiments at high energies. The radiation zeros are shown to occur also for massive quarks, and we discuss how the bbγ final state may offer a particularly clean environment in which to observe them.

In this thesis, we put restrictions on the coefficients of polynomials and give bounds concerning the number of zeros in a specific region. Our results generalize a number of previously known theorems, as well as implying many new corollaries with hypotheses concerning monotonicity of the modulus, real, as well as real and imaginary parts of the coefficients separately. We worked with Enestr\"{o}m-Kakeya type hypotheses, yet we were only concerned with the number of zeros of the polynomial. We considered putting the same type of restrictions on the coefficients of three different types of polynomials: polynomials with a monotonicity``flip" at some index $k$, polynomials split into a monotonicity condition on the even and odd coefficients independently, and ${\cal P}_{n,\mu}$ polynomials that have a gap in between the leading coefficient and the proceeding coefficient, namely the $\mu^{\mbox{th}}$ coefficient.

In this thesis, we explore the effect of restricting the coefficients of polynomials on the bounds for the number of zeros in a given region. The results presented herein build on a body of work, culminating in the generalization of bounds among three classes of polynomials. The hypotheses of monotonicity on each class of polynomials were further subdivided into sections concerning r reversals among the moduli, real parts, and both real and imaginary parts of the coefficients.

An important part of stereo vision consists of finding and matching points in two images which correspond to the same physical element in the scene. We show that zeros of curvature of curves are perspective projection invariants and can therefore be used to find corresponding points. They can be used to help solve the registration problem (Longuet-Higgins, 1982) and to obtain the correct depth when a curve enters the forbidden zone (Krol and van de Grind, 1982). They are also relevant to theories for representing image curves. We consider the stability of these zeros of curvature.

In the study of mathematics, one of the most useful, relevant topics explored in secondary mathematics remains the zeros of polynomials. This paper will present various ways to explore this topic while preserving the fundamental concept as a whole. In addition, this paper will reveal some distinct relationships between roots and their behavior within the different branches of mathematics. The purpose of this paper is to show how this topic can be inserted at key points in the developmental curriculum to preserve the autonomy of this vital mathematical concept, allowing students to experience the behavior and value of this topic in a variety of contexts. / text

Made available in DSpace on 2014-06-11T19:30:51Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-03-16Bitstream added on 2014-06-13T19:19:32Z : No. of bitstreams: 1 andrea_cq_dr_ilha.pdf: 1305115 bytes, checksum: 49a5c189c623805f391f849cb2048b5f (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho são propostas metodologias para a utilização dos zeros em sistemas de controle. Neste contexto, propõe-se o uso dos zeros no projeto de sistemas de rastreamento de sinais de referência e apresenta-se um processo sistemático para alocação de zeros aplicado a sistemas de controle. Objetivando-se solucionar o problema de rastreamento de sinais de referência propõe-se uma metodologia de variação dos zeros, sendo considerado a existência de um sinal de entrada exógena de perturbação ou distúrbio na planta. Neste contexto, em um primeiro momento, projeta-se um controlador a fim de diminuir o efeito desta perturbação no desempenho do sistema. O processo de atenuação do efeito do sinal exógeno na saída do sistema é abordado de duas maneiras: com a realimentação dos estados reconstruídos da planta através do estimador, neste caso utiliza-se um controlador H2 e utilizando-se a realimentação dinâmica da saída do sistema através de um controlador K(s) com índice de desempenho H 2 ou H8. A seguir, através da modificação dos zeros, minimiza a norma H8 entre o sinal de referência e o erro entre o sinal de saída e a referência, constituindo portanto um rastreador de sinal. Os projetos são equacionados utilizando-se inequações matriciais lineares, que... / Methodologies that use zeros in control systems were proposed. In this context, were proposed the use of zero for tracking system and a systematic process for zeros location used in control system. A methodology were proposed to vary the zeros, that is useful to solve the tracking problem, when the existence of input perturbation or disturbance signal is considered in the plant. Initially, a controller is designed in order to attenuate the effect of disturbance signal. The attenuation of the effect of the signal disturbance is made in two ways. In the first moment, is used the feedback of states of the plant thought of the estimator, and in this case is designed a H2 controller. In the other hand, is used the dynamic output feedback, and so it is designed a H2 or H8 controller for obtain this attenuation. Following, it is minimized the H8-norm between the reference signal and error signal, where the error signal is the difference between the output signal and the reference signal, it becoming a system tracking. The designs was formulated in terms of linear matrix inequalities, that allow describe the problems of convex optimization. A methodology for to obtain the global optimum solution using zeros variation is presented. The methodology of zeros used to solve... (Complete abstract, click electronic access below)

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-01-25Bitstream added on 2014-06-13T19:55:24Z : No. of bitstreams: 1 coelho_ja_me_sjrp.pdf: 348011 bytes, checksum: 34e9942042312b5c3c0bd68f0673e590 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Nesta dissertação estudamos a conjectura de Smale junto com a maioria dos resultados sobre este assunto e sua interpretação eletrostática. Mostramos alguns testes numéricos sobre a citada conjectura, para podermos analisar sua interpretação eletrostática. / In this dissertation we study a conjecture of smale together with majority of results about this topic and its electrostatic interpretation. We show some numerical tests concerning the conjecture in order to analise its electrostatic interpretation.